KILLED proof of input_e5GOlqMvww.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) CompletionProof [UPPER BOUND(ID), 0 ms] (16) CpxTypedWeightedCompleteTrs (17) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 491 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 92 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 376 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 126 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 234 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 57 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 283 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 93 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 424 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 87 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 6262 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 3809 ms] (60) CpxRNTS (61) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 312 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 207 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 298 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 17 ms] (92) CdtProblem (93) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 373 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 12 ms] (98) CdtProblem (99) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 397 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 459 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 482 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 10 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 530 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 537 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 2 ms] (134) CdtProblem (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 6 ms] (150) CdtProblem (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 3 ms] (160) CdtProblem (161) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: min(0, y) -> 0 min(x, 0) -> 0 min(s(x), s(y)) -> s(min(x, y)) max(0, y) -> y max(x, 0) -> x max(s(x), s(y)) -> s(max(x, y)) +(0, y) -> y +(s(x), y) -> s(+(x, y)) -(x, 0) -> x -(s(x), s(y)) -> -(x, y) *(x, 0) -> 0 *(x, s(y)) -> +(x, *(x, y)) f(s(x)) -> f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: min(0', y) -> 0' min(x, 0') -> 0' min(s(x), s(y)) -> s(min(x, y)) max(0', y) -> y max(x, 0') -> x max(s(x), s(y)) -> s(max(x, y)) +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) -(x, 0') -> x -(s(x), s(y)) -> -(x, y) *'(x, 0') -> 0' *'(x, s(y)) -> +'(x, *'(x, y)) f(s(x)) -> f(-(max(*'(s(x), s(x)), +'(s(x), s(s(s(0'))))), max(s(*'(s(x), s(x))), +'(s(x), s(s(s(s(0')))))))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: min(0, y) -> 0 min(x, 0) -> 0 min(s(x), s(y)) -> s(min(x, y)) max(0, y) -> y max(x, 0) -> x max(s(x), s(y)) -> s(max(x, y)) +(0, y) -> y +(s(x), y) -> s(+(x, y)) -(x, 0) -> x -(s(x), s(y)) -> -(x, y) *(x, 0) -> 0 *(x, s(y)) -> +(x, *(x, y)) f(s(x)) -> f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: min(0, y) -> 0 [1] min(x, 0) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(0, y) -> y [1] max(x, 0) -> x [1] max(s(x), s(y)) -> s(max(x, y)) [1] +(0, y) -> y [1] +(s(x), y) -> s(+(x, y)) [1] -(x, 0) -> x [1] -(s(x), s(y)) -> -(x, y) [1] *(x, 0) -> 0 [1] *(x, s(y)) -> +(x, *(x, y)) [1] f(s(x)) -> f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: + => plus - => minus * => times ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: min(0, y) -> 0 [1] min(x, 0) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(0, y) -> y [1] max(x, 0) -> x [1] max(s(x), s(y)) -> s(max(x, y)) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(x, times(x, y)) [1] f(s(x)) -> f(minus(max(times(s(x), s(x)), plus(s(x), s(s(s(0))))), max(s(times(s(x), s(x))), plus(s(x), s(s(s(s(0)))))))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: min(0, y) -> 0 [1] min(x, 0) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(0, y) -> y [1] max(x, 0) -> x [1] max(s(x), s(y)) -> s(max(x, y)) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(x, times(x, y)) [1] f(s(x)) -> f(minus(max(times(s(x), s(x)), plus(s(x), s(s(s(0))))), max(s(times(s(x), s(x))), plus(s(x), s(s(s(s(0)))))))) [1] The TRS has the following type information: min :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s max :: 0:s -> 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s minus :: 0:s -> 0:s -> 0:s times :: 0:s -> 0:s -> 0:s f :: 0:s -> f Rewrite Strategy: INNERMOST ---------------------------------------- (11) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: minus(v0, v1) -> null_minus [0] f(v0) -> null_f [0] min(v0, v1) -> null_min [0] max(v0, v1) -> null_max [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] And the following fresh constants: null_minus, null_f, null_min, null_max, null_plus, null_times ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: min(0, y) -> 0 [1] min(x, 0) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(0, y) -> y [1] max(x, 0) -> x [1] max(s(x), s(y)) -> s(max(x, y)) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(x, times(x, y)) [1] f(s(x)) -> f(minus(max(times(s(x), s(x)), plus(s(x), s(s(s(0))))), max(s(times(s(x), s(x))), plus(s(x), s(s(s(s(0)))))))) [1] minus(v0, v1) -> null_minus [0] f(v0) -> null_f [0] min(v0, v1) -> null_min [0] max(v0, v1) -> null_max [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] The TRS has the following type information: min :: 0:s:null_minus:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_min:null_max:null_plus:null_times 0 :: 0:s:null_minus:null_min:null_max:null_plus:null_times s :: 0:s:null_minus:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_min:null_max:null_plus:null_times max :: 0:s:null_minus:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_min:null_max:null_plus:null_times plus :: 0:s:null_minus:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_min:null_max:null_plus:null_times minus :: 0:s:null_minus:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_min:null_max:null_plus:null_times times :: 0:s:null_minus:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_min:null_max:null_plus:null_times f :: 0:s:null_minus:null_min:null_max:null_plus:null_times -> null_f null_minus :: 0:s:null_minus:null_min:null_max:null_plus:null_times null_f :: null_f null_min :: 0:s:null_minus:null_min:null_max:null_plus:null_times null_max :: 0:s:null_minus:null_min:null_max:null_plus:null_times null_plus :: 0:s:null_minus:null_min:null_max:null_plus:null_times null_times :: 0:s:null_minus:null_min:null_max:null_plus:null_times Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 null_minus => 0 null_f => 0 null_min => 0 null_max => 0 null_plus => 0 null_times => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> f(minus(max(times(1 + x, 1 + x), plus(1 + x, 1 + (1 + (1 + 0)))), max(1 + times(1 + x, 1 + x), plus(1 + x, 1 + (1 + (1 + (1 + 0))))))) :|: x >= 0, z = 1 + x f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 max(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 max(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y max(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 max(z, z') -{ 1 }-> 1 + max(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 min(z, z') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 1 }-> plus(x, times(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = x times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (15) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: min_2 f_1 (c) The following functions are completely defined: times_2 minus_2 max_2 plus_2 Due to the following rules being added: minus(v0, v1) -> 0 [0] And the following fresh constants: const ---------------------------------------- (16) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: min(0, y) -> 0 [1] min(x, 0) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(0, y) -> y [1] max(x, 0) -> x [1] max(s(x), s(y)) -> s(max(x, y)) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(x, times(x, y)) [1] f(s(x)) -> f(minus(max(times(s(x), s(x)), plus(s(x), s(s(s(0))))), max(s(times(s(x), s(x))), plus(s(x), s(s(s(s(0)))))))) [1] minus(v0, v1) -> 0 [0] The TRS has the following type information: min :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s max :: 0:s -> 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s minus :: 0:s -> 0:s -> 0:s times :: 0:s -> 0:s -> 0:s f :: 0:s -> f const :: f Rewrite Strategy: INNERMOST ---------------------------------------- (17) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (18) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: min(0, y) -> 0 [1] min(x, 0) -> 0 [1] min(s(x), s(y)) -> s(min(x, y)) [1] max(0, y) -> y [1] max(x, 0) -> x [1] max(s(x), s(y)) -> s(max(x, y)) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] times(x, 0) -> 0 [1] times(x, s(0)) -> plus(x, 0) [2] times(x, s(s(y'))) -> plus(x, plus(x, times(x, y'))) [2] f(s(x)) -> f(minus(max(plus(s(x), times(s(x), x)), s(plus(x, s(s(s(0)))))), max(s(plus(s(x), times(s(x), x))), s(plus(x, s(s(s(s(0))))))))) [5] minus(v0, v1) -> 0 [0] The TRS has the following type information: min :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s max :: 0:s -> 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s minus :: 0:s -> 0:s -> 0:s times :: 0:s -> 0:s -> 0:s f :: 0:s -> f const :: f Rewrite Strategy: INNERMOST ---------------------------------------- (19) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 const => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> f(minus(max(plus(1 + x, times(1 + x, x)), 1 + plus(x, 1 + (1 + (1 + 0)))), max(1 + plus(1 + x, times(1 + x, x)), 1 + plus(x, 1 + (1 + (1 + (1 + 0))))))) :|: x >= 0, z = 1 + x max(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 max(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y max(z, z') -{ 1 }-> 1 + max(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 min(z, z') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 2 }-> plus(x, plus(x, times(x, y'))) :|: z' = 1 + (1 + y'), x >= 0, y' >= 0, z = x times(z, z') -{ 2 }-> plus(x, 0) :|: x >= 0, z' = 1 + 0, z = x times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 ---------------------------------------- (21) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + 0)))), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + (1 + 0))))))) :|: z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 ---------------------------------------- (23) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { max } { minus } { min } { plus } { times } { f } ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + 0)))), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + (1 + 0))))))) :|: z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {max}, {minus}, {min}, {plus}, {times}, {f} ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + 0)))), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + (1 + 0))))))) :|: z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {max}, {minus}, {min}, {plus}, {times}, {f} ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: max after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + 0)))), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + (1 + 0))))))) :|: z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {max}, {minus}, {min}, {plus}, {times}, {f} Previous analysis results are: max: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: max after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + 0)))), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + (1 + 0))))))) :|: z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minus}, {min}, {plus}, {times}, {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + 0)))), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + (1 + 0))))))) :|: z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minus}, {min}, {plus}, {times}, {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + 0)))), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + (1 + 0))))))) :|: z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minus}, {min}, {plus}, {times}, {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + 0)))), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + (1 + 0))))))) :|: z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {min}, {plus}, {times}, {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + 0)))), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + (1 + 0))))))) :|: z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {min}, {plus}, {times}, {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: min after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + 0)))), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + (1 + 0))))))) :|: z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {min}, {plus}, {times}, {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: ?, size: O(n^1) [z] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: min after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + 0)))), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + (1 + 0))))))) :|: z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {times}, {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + 0)))), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + (1 + 0))))))) :|: z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {times}, {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + 0)))), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + (1 + 0))))))) :|: z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {times}, {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + 0)))), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + plus(z - 1, 1 + (1 + (1 + (1 + 0))))))) :|: z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + 2*z }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + s3), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + s4))) :|: s3 >= 0, s3 <= z - 1 + (1 + (1 + (1 + 0))), s4 >= 0, s4 <= z - 1 + (1 + (1 + (1 + (1 + 0)))), z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s2 :|: s2 >= 0, s2 <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: z + 2*z*z' ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + 2*z }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + s3), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + s4))) :|: s3 >= 0, s3 <= z - 1 + (1 + (1 + (1 + 0))), s4 >= 0, s4 <= z - 1 + (1 + (1 + (1 + (1 + 0)))), z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s2 :|: s2 >= 0, s2 <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: ?, size: O(n^2) [z + 2*z*z'] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + z + 2*z*z' + 4*z' ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + 2*z }-> f(minus(max(plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + s3), max(1 + plus(1 + (z - 1), times(1 + (z - 1), z - 1)), 1 + s4))) :|: s3 >= 0, s3 <= z - 1 + (1 + (1 + (1 + 0))), s4 >= 0, s4 <= z - 1 + (1 + (1 + (1 + (1 + 0)))), z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s2 :|: s2 >= 0, s2 <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (55) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 14 + s13 + s3 + s4 + 10*z + 4*z^2 }-> f(s14) :|: s8 >= 0, s8 <= 1 + (z - 1) + 2 * ((z - 1) * (1 + (z - 1))), s9 >= 0, s9 <= 1 + (z - 1) + s8, s10 >= 0, s10 <= 1 + s3 + s9, s11 >= 0, s11 <= 1 + (z - 1) + 2 * ((z - 1) * (1 + (z - 1))), s12 >= 0, s12 <= 1 + (z - 1) + s11, s13 >= 0, s13 <= 1 + s4 + (1 + s12), s14 >= 0, s14 <= s10, s3 >= 0, s3 <= z - 1 + (1 + (1 + (1 + 0))), s4 >= 0, s4 <= z - 1 + (1 + (1 + (1 + (1 + 0)))), z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s2 :|: s2 >= 0, s2 <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s7 :|: s5 >= 0, s5 <= z + 2 * ((z' - 2) * z), s6 >= 0, s6 <= z + s5, s7 >= 0, s7 <= z + s6, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 14 + s13 + s3 + s4 + 10*z + 4*z^2 }-> f(s14) :|: s8 >= 0, s8 <= 1 + (z - 1) + 2 * ((z - 1) * (1 + (z - 1))), s9 >= 0, s9 <= 1 + (z - 1) + s8, s10 >= 0, s10 <= 1 + s3 + s9, s11 >= 0, s11 <= 1 + (z - 1) + 2 * ((z - 1) * (1 + (z - 1))), s12 >= 0, s12 <= 1 + (z - 1) + s11, s13 >= 0, s13 <= 1 + s4 + (1 + s12), s14 >= 0, s14 <= s10, s3 >= 0, s3 <= z - 1 + (1 + (1 + (1 + 0))), s4 >= 0, s4 <= z - 1 + (1 + (1 + (1 + (1 + 0)))), z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s2 :|: s2 >= 0, s2 <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s7 :|: s5 >= 0, s5 <= z + 2 * ((z' - 2) * z), s6 >= 0, s6 <= z + s5, s7 >= 0, s7 <= z + s6, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] f: runtime: ?, size: O(1) [0] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 14 + s13 + s3 + s4 + 10*z + 4*z^2 }-> f(s14) :|: s8 >= 0, s8 <= 1 + (z - 1) + 2 * ((z - 1) * (1 + (z - 1))), s9 >= 0, s9 <= 1 + (z - 1) + s8, s10 >= 0, s10 <= 1 + s3 + s9, s11 >= 0, s11 <= 1 + (z - 1) + 2 * ((z - 1) * (1 + (z - 1))), s12 >= 0, s12 <= 1 + (z - 1) + s11, s13 >= 0, s13 <= 1 + s4 + (1 + s12), s14 >= 0, s14 <= s10, s3 >= 0, s3 <= z - 1 + (1 + (1 + (1 + 0))), s4 >= 0, s4 <= z - 1 + (1 + (1 + (1 + (1 + 0)))), z - 1 >= 0 max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 max(z, z') -{ 2 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0 min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 min(z, z') -{ 2 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s2 :|: s2 >= 0, s2 <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s7 :|: s5 >= 0, s5 <= z + 2 * ((z' - 2) * z), s6 >= 0, s6 <= z + s5, s7 >= 0, s7 <= z + s6, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] min: runtime: O(n^1) [2 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] f: runtime: INF, size: O(1) [0] ---------------------------------------- (61) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: min(0, z0) -> 0 min(z0, 0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) f(s(z0)) -> f(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))) Tuples: MIN(0, z0) -> c MIN(z0, 0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(0, z0) -> c3 MAX(z0, 0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) -'(z0, 0) -> c8 -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, 0) -> c10 *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) S tuples: MIN(0, z0) -> c MIN(z0, 0) -> c1 MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(0, z0) -> c3 MAX(z0, 0) -> c4 MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) -'(z0, 0) -> c8 -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, 0) -> c10 *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) K tuples:none Defined Rule Symbols: min_2, max_2, +_2, -_2, *_2, f_1 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c, c1, c2_1, c3, c4, c5_1, c6, c7_1, c8, c9_1, c10, c11_2, c12_4, c13_4, c14_4, c15_4 ---------------------------------------- (63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: -'(z0, 0) -> c8 +'(0, z0) -> c6 MIN(z0, 0) -> c1 MAX(0, z0) -> c3 MIN(0, z0) -> c MAX(z0, 0) -> c4 *'(z0, 0) -> c10 ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: min(0, z0) -> 0 min(z0, 0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) f(s(z0)) -> f(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) K tuples:none Defined Rule Symbols: min_2, max_2, +_2, -_2, *_2, f_1 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c12_4, c13_4, c14_4, c15_4 ---------------------------------------- (65) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: min(0, z0) -> 0 min(z0, 0) -> 0 min(s(z0), s(z1)) -> s(min(z0, z1)) f(s(z0)) -> f(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) S tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c12_4, c13_4, c14_4, c15_4 ---------------------------------------- (67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1)) = 0 POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = x_1 + x_2 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c13(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c14(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c15(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c12_4, c13_4, c14_4, c15_4 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) by F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_4, c14_4, c15_4, c12_4 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) by F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c14_4, c15_4, c12_4, c13_4 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) by F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c15_4, c12_4, c13_4, c14_4 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) by F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c12_4, c13_4, c14_4, c15_4 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) by F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c12_4, c13_4, c14_4, c15_4, c12_1 ---------------------------------------- (79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = x_1 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1)) = [1] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c13(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c14(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c15(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = [1] POL(s(x_1)) = 0 ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c12_4, c13_4, c14_4, c15_4, c12_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) by F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c12_4, c13_4, c14_4, c15_4, c12_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) by F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), s(+(z1, s(s(s(0)))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c12_4, c13_4, c14_4, c15_4, c12_1, c12_2 ---------------------------------------- (85) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), 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max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), 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s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), 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*'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c12_4, c13_4, c14_4, c15_4, c12_1, c_1 ---------------------------------------- (87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = x_1 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1)) = [1] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c13(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c14(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c15(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = [1] POL(s(x_1)) = 0 ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c12_4, c13_4, c14_4, c15_4, c12_1, c_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) by F(s(z0)) -> c12(F(-(max(+(s(z0), *(s(z0), z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(+(s(z0), *(s(z0), z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_4, c14_4, c15_4, c12_4, c12_1, c_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) by F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), 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*'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_4, c14_4, c15_4, c12_4, c12_1, c_1, c13_1 ---------------------------------------- (93) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = x_1 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1)) = [1] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c14(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c15(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = [1] POL(s(x_1)) = 0 ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), 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s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_4, c14_4, c15_4, c12_4, c12_1, c_1, c13_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) by F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, 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s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_4, c14_4, c15_4, c12_4, c12_1, c_1, c13_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) by F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), s(+(z1, s(s(s(0)))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), 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MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_4, c14_4, c15_4, c12_4, c12_1, c_1, c13_1, c13_3 ---------------------------------------- (99) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), 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s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_4, c14_4, c15_4, c12_4, c12_1, c_1, c13_1, c1_1 ---------------------------------------- (101) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = x_1 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1)) = [1] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c14(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c15(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = [1] POL(s(x_1)) = 0 ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, 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s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_4, c14_4, c15_4, c12_4, c12_1, c_1, c13_1, c1_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) by F(s(z0)) -> c13(F(-(max(+(s(z0), *(s(z0), z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(+(s(z0), *(s(z0), z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), 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c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), 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MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, 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s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, 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s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c14_4, c15_4, c12_4, c12_1, c_1, c13_4, c13_1, c1_1 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) by F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), 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c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), 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c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, 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s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c14_4, c15_4, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_2 ---------------------------------------- (107) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), 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c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, 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+(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c14_4, c15_4, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c3_1 ---------------------------------------- (109) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = x_1 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1)) = [1] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c14(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c15(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = [1] POL(s(x_1)) = 0 ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), 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s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c14_4, c15_4, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c3_1 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) by F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), 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c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), 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s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c14_4, c15_4, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c3_1 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) by F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), s(+(z1, s(s(s(0)))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), 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s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), 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z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c14_4, c15_4, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c3_1, c14_1 ---------------------------------------- (115) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = x_1 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1)) = [1] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c14(x_1)) = x_1 POL(c14(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c15(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = [1] POL(s(x_1)) = 0 ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), 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c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), 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max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), 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s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c14_4, c15_4, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c3_1, c14_1 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) by F(s(z0)) -> c14(F(-(max(+(s(z0), *(s(z0), z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(+(s(z0), *(s(z0), z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), 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c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c15_4, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) by F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), 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max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), 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MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), 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+'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c15_4, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_3 ---------------------------------------- (121) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), 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F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), 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+(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), 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s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c15_4, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c4_1 ---------------------------------------- (123) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = x_1 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1)) = [1] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c14(x_1)) = x_1 POL(c14(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c15(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = [1] POL(s(x_1)) = 0 ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), 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s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c15_4, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c4_1 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) by F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), 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c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, 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MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), 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s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c15_4, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c4_1 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) by F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), s(+(z1, s(s(s(0)))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> 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MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), 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c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), 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s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c15_4, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c4_1, c15_1 ---------------------------------------- (129) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) We considered the (Usable) Rules:none And the Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> 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+(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = x_1 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1)) = [1] POL(MAX(x_1, x_2)) = 0 POL(MIN(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c14(x_1)) = x_1 POL(c14(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(max(x_1, x_2)) = [1] POL(s(x_1)) = 0 ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), 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MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), 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s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), 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s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), 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-> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), 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s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c15_4, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c4_1, c15_1 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) by F(s(z0)) -> c15(F(-(max(+(s(z0), *(s(z0), z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(+(s(z0), *(s(z0), z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), 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-> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), 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c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), 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s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c12(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), *'(s(x0), s(x0))) by F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), 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s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), 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F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), 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MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) by F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), 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s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), 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s(s(s(s(0)))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), 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F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), 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MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), 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s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) by F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), 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s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) S tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), 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F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), 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MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), 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s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1 ---------------------------------------- (139) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) by F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), 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c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, 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MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), 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s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) K tuples: MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_1 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1 ---------------------------------------- (141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) by MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), 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c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), 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s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(s(+(z0, 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-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) K tuples: F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MAX_2, +'_2, -'_2, *'_2, F_1, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_2, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1, c2_1 ---------------------------------------- (143) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) by F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), 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c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, 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s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), 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-> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), 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-'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) K tuples: F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MAX_2, +'_2, -'_2, *'_2, F_1, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_2, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1, c2_1 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) by F(s(z0)) -> c12(F(-(max(+(s(z0), *(s(z0), z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), 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c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), 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c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), 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s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(+(s(z0), *(s(z0), z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) K tuples: F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MAX_2, +'_2, -'_2, *'_2, F_1, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_2, c12_4, c12_1, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1, c2_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) by F(s(z0)) -> c12(-'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c12(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), 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MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), 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s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(+(s(z0), *(s(z0), z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) K tuples: F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MAX_2, +'_2, -'_2, *'_2, F_1, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_2, c12_4, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1, c2_1, c12_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) by F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), 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s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), 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+(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(+(s(z0), *(s(z0), z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) K tuples: F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MAX_2, +'_2, -'_2, *'_2, F_1, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_2, c12_4, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1, c2_1, c12_1 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) by F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), 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c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), 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-'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(+(s(z0), *(s(z0), z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) K tuples: F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MAX_2, +'_2, -'_2, *'_2, F_1, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_2, c12_4, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1, c2_1, c12_1 ---------------------------------------- (153) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z1))) -> c12(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) by F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), 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s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), 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s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), 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*'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(+(s(z0), *(s(z0), z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(-'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(+(s(s(z0)), +(s(s(z0)), 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+(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(+(s(z0), *(s(z0), z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) K tuples: F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MAX_2, +'_2, -'_2, *'_2, F_1, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_2, c12_4, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1, c2_1, c12_1 ---------------------------------------- (155) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) by F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), 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s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(+(s(z0), *(s(z0), z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) K tuples: F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MAX_2, +'_2, -'_2, *'_2, F_1, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_2, c12_4, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1, c2_1, c12_1 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) by F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), 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c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), 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max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(+(s(z0), *(s(z0), z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) K tuples: F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: MAX_2, +'_2, -'_2, *'_2, F_1, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_2, c12_4, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1, c2_1, c12_1 ---------------------------------------- (159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) by MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: +'(s(z0), z1) -> c7(+'(z0, z1)) -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), 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MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(+(s(z0), *(s(z0), z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) K tuples: F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: +'_2, -'_2, *'_2, F_1, MIN_2, MAX_2 Compound Symbols: c7_1, c9_1, c11_2, c12_4, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1, c2_1, c12_1, c5_1 ---------------------------------------- (161) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(s(z0), z1) -> c7(+'(z0, z1)) by +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), +'(s(0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), 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c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, 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-> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), 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s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c15(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), +'(s(x0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), +'(s(0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(+(s(z0), *(s(z0), z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) K tuples: F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: -'_2, *'_2, F_1, MIN_2, MAX_2, +'_2 Compound Symbols: c9_1, c11_2, c12_4, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1, c2_1, c12_1, c5_1, c7_1 ---------------------------------------- (163) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 8 trailing tuple parts ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) max(0, z0) -> z0 max(z0, 0) -> z0 max(s(z0), s(z1)) -> s(max(z0, z1)) Tuples: -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c12(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), *'(s(z1), s(z1))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0))))), *'(s(0), s(0))) F(s(s(z1))) -> c12(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), *'(s(s(z1)), s(s(z1)))) F(s(x0)) -> c13(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), +'(s(x0), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(s(z0))) -> c13(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), +'(s(s(z0)), s(s(s(0))))) F(s(z1)) -> c13(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), +'(s(z1), s(s(s(0))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(z0)) -> c13(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), +'(s(z0), s(s(s(0))))) F(s(s(z1))) -> c13(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), +'(s(s(z1)), s(s(s(0))))) F(s(x0)) -> c14(F(-(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), s(max(*(s(x0), s(x0)), +(x0, s(s(s(s(0))))))))), -'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))), MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))), *'(s(x0), s(x0))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c14(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z1)) -> c14(F(-(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(z1)) -> c14(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z1))) -> c14(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), *'(s(s(z1)), s(s(z1)))) F(s(0)) -> c14(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c14(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), 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c14(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), *'(s(z1), s(z1))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c14(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c14(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z1))) -> c14(F(-(max(+(s(s(z1)), +(s(s(z1)), 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max(s(+(s(z1), *(s(z1), z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), s(z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), s(+(z1, s(s(s(s(0))))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(s(+(z0, *(s(z0), z0)))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c15(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), +(s(z1), s(s(s(0))))), max(s(+(s(z1), *(s(z1), z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(s(z0))) -> c15(F(-(max(*(s(s(z0)), s(s(z0))), s(s(+(z0, s(s(s(0))))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0)))))), +'(s(s(z0)), s(s(s(s(0)))))) F(s(z1)) -> c15(F(-(max(+(s(z1), *(s(z1), z1)), s(+(z1, s(s(s(0)))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))))), -'(max(*(s(z1), s(z1)), +(s(z1), s(s(s(0))))), max(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0))))))), MAX(s(*(s(z1), s(z1))), +(s(z1), s(s(s(s(0)))))), +'(s(z1), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(z0)) -> c15(F(-(max(s(+(z0, *(s(z0), z0))), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0)))))), +'(s(z0), s(s(s(s(0)))))) F(s(s(z1))) -> c15(F(-(max(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))))), -'(max(*(s(s(z1)), s(s(z1))), +(s(s(z1)), s(s(s(0))))), max(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0))))))), MAX(s(*(s(s(z1)), s(s(z1)))), +(s(s(z1)), s(s(s(s(0)))))), +'(s(s(z1)), s(s(s(s(0)))))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), s(max(*(s(z0), s(z0)), +(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), s(s(+(z0, s(s(s(s(0)))))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(+(s(z0), *(s(z0), z0))), s(+(z0, s(s(s(s(0))))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(*(s(z0), 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s(s(s(0)))))), *'(s(z0), s(z0))) F(s(s(z0))) -> c12(F(-(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0)))), +(s(s(z0)), s(s(s(s(0)))))))), -'(max(*(s(s(z0)), s(s(z0))), +(s(s(z0)), s(s(s(0))))), max(s(*(s(s(z0)), s(s(z0)))), +(s(s(z0)), s(s(s(s(0))))))), MAX(*(s(s(z0)), s(s(z0))), s(+(s(z0), s(s(s(0)))))), *'(s(s(z0)), s(s(z0)))) F(s(0)) -> c12(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), s(+(0, s(s(s(0)))))), *'(s(0), s(0))) F(s(z0)) -> c12(F(-(max(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), max(s(+(s(z0), *(s(z0), z0))), +(s(z0), s(s(s(s(0)))))))), -'(max(*(s(z0), s(z0)), +(s(z0), s(s(s(0))))), max(s(*(s(z0), s(z0))), +(s(z0), s(s(s(s(0))))))), MAX(*(s(z0), s(z0)), s(+(z0, s(s(s(0)))))), *'(s(z0), s(z0))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0)))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0)))))) F(s(0)) -> c13(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0)))))) F(s(0)) -> c13(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(*(s(0), s(0)), +(s(0), s(s(s(0)))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), s(s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(+(s(0), 0)), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))) F(s(0)) -> c15(F(-(max(*(s(0), s(0)), s(s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))) F(s(0)) -> c15(F(-(max(+(s(0), 0), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0)))))))), -'(max(*(s(0), s(0)), +(s(0), s(s(s(0))))), max(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))), MAX(s(*(s(0), s(0))), +(s(0), s(s(s(s(0))))))) K tuples: F(s(x0)) -> c12(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c13(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c1(MAX(*(s(x0), s(x0)), +(s(x0), s(s(s(0)))))) F(s(x0)) -> c1(+'(s(x0), s(s(s(0))))) F(s(x0)) -> c3(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c3(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c14(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) F(s(x0)) -> c4(MAX(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0))))))) F(s(x0)) -> c4(+'(s(x0), s(s(s(s(0)))))) F(s(x0)) -> c15(-'(max(*(s(x0), s(x0)), +(s(x0), s(s(s(0))))), max(s(*(s(x0), s(x0))), +(s(x0), s(s(s(s(0)))))))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) Defined Rule Symbols: *_2, +_2, -_2, max_2 Defined Pair Symbols: -'_2, *'_2, F_1, MIN_2, MAX_2, +'_2 Compound Symbols: c9_1, c11_2, c12_4, c_1, c13_4, c13_1, c1_1, c14_4, c3_1, c14_1, c15_4, c4_1, c15_1, c2_1, c12_1, c5_1, c7_1, c13_3, c15_3