KILLED proof of input_XC0Y1uhC7W.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 442 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 41 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 304 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 117 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 184 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 116 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 222 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 94 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 138 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 43 ms] (50) CpxRNTS (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 427 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 157 ms] (56) CpxRNTS (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 5468 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 2615 ms] (62) CpxRNTS (63) CompletionProof [UPPER BOUND(ID), 0 ms] (64) CpxTypedWeightedCompleteTrs (65) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (66) CpxRNTS (67) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (68) CdtProblem (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 6 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 5 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 171 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 1 ms] (106) CdtProblem (107) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 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) CdtRewritingProof [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), 0 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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRewritingProof [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)) p(s(x)) -> x f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *(s(x), s(y))) 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)) p(s(x)) -> x f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *'(s(x), s(y))) 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)) p(s(x)) -> x f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *(s(x), s(y))) 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] p(s(x)) -> x [1] f(s(x), s(y)) -> f(-(min(s(x), s(y)), max(s(x), s(y))), *(s(x), s(y))) [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] p(s(x)) -> x [1] f(s(x), s(y)) -> f(minus(min(s(x), s(y)), max(s(x), s(y))), times(s(x), s(y))) [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] p(s(x)) -> x [1] f(s(x), s(y)) -> f(minus(min(s(x), s(y)), max(s(x), s(y))), times(s(x), s(y))) [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 p :: 0:s -> 0:s f :: 0:s -> 0:s -> f Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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: p_1 f_2 (c) The following functions are completely defined: times_2 minus_2 min_2 max_2 plus_2 Due to the following rules being added: minus(v0, v1) -> 0 [0] And the following fresh constants: const ---------------------------------------- (12) 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] p(s(x)) -> x [1] f(s(x), s(y)) -> f(minus(min(s(x), s(y)), max(s(x), s(y))), times(s(x), s(y))) [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 p :: 0:s -> 0:s f :: 0:s -> 0:s -> f const :: f Rewrite Strategy: INNERMOST ---------------------------------------- (13) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (14) 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] p(s(x)) -> x [1] f(s(x), s(y)) -> f(minus(s(min(x, y)), s(max(x, y))), plus(s(x), times(s(x), y))) [4] 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 p :: 0:s -> 0:s f :: 0:s -> 0:s -> f const :: f Rewrite Strategy: INNERMOST ---------------------------------------- (15) 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 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(1 + min(x, y), 1 + max(x, y)), plus(1 + x, times(1 + x, y))) :|: z' = 1 + y, x >= 0, y >= 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 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x 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 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(1 + min(z - 1, z' - 1), 1 + max(z - 1, z' - 1)), plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: z - 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { max } { minus } { min } { plus } { p } { times } { f } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(1 + min(z - 1, z' - 1), 1 + max(z - 1, z' - 1)), plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: z - 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {times}, {f} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(1 + min(z - 1, z' - 1), 1 + max(z - 1, z' - 1)), plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: z - 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {times}, {f} ---------------------------------------- (23) 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' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(1 + min(z - 1, z' - 1), 1 + max(z - 1, z' - 1)), plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: z - 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {times}, {f} Previous analysis results are: max: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (25) 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' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 4 }-> f(minus(1 + min(z - 1, z' - 1), 1 + max(z - 1, z' - 1)), plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: z - 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {times}, {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 + z' }-> f(minus(1 + min(z - 1, z' - 1), 1 + s'), plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= z' - 1 + (z - 1), z - 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {times}, {f} Previous analysis results are: max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] ---------------------------------------- (29) 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 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 + z' }-> f(minus(1 + min(z - 1, z' - 1), 1 + s'), plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= z' - 1 + (z - 1), z - 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {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] ---------------------------------------- (31) 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' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 + z' }-> f(minus(1 + min(z - 1, z' - 1), 1 + s'), plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= z' - 1 + (z - 1), z - 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {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] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 + z' }-> f(minus(1 + min(z - 1, z' - 1), 1 + s'), plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= z' - 1 + (z - 1), z - 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {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] ---------------------------------------- (35) 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 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 + z' }-> f(minus(1 + min(z - 1, z' - 1), 1 + s'), plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= z' - 1 + (z - 1), z - 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {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] ---------------------------------------- (37) 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' ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 5 + z' }-> f(minus(1 + min(z - 1, z' - 1), 1 + s'), plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= z' - 1 + (z - 1), z - 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {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] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 8 + s' + 2*z' }-> f(s3, plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: s2 >= 0, s2 <= z - 1, s3 >= 0, s3 <= 1 + s2, s' >= 0, s' <= z' - 1 + (z - 1), z - 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 + s1 :|: s1 >= 0, s1 <= 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {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] ---------------------------------------- (41) 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' ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 8 + s' + 2*z' }-> f(s3, plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: s2 >= 0, s2 <= z - 1, s3 >= 0, s3 <= 1 + s2, s' >= 0, s' <= z' - 1 + (z - 1), z - 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 + s1 :|: s1 >= 0, s1 <= 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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}, {p}, {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'] ---------------------------------------- (43) 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 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 8 + s' + 2*z' }-> f(s3, plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: s2 >= 0, s2 <= z - 1, s3 >= 0, s3 <= 1 + s2, s' >= 0, s' <= z' - 1 + (z - 1), z - 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 + s1 :|: s1 >= 0, s1 <= 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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: {p}, {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'] ---------------------------------------- (45) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 8 + s' + 2*z' }-> f(s3, plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: s2 >= 0, s2 <= z - 1, s3 >= 0, s3 <= 1 + s2, s' >= 0, s' <= z' - 1 + (z - 1), z - 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 + s1 :|: s1 >= 0, s1 <= 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s5 :|: s5 >= 0, s5 <= 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: {p}, {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'] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 8 + s' + 2*z' }-> f(s3, plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: s2 >= 0, s2 <= z - 1, s3 >= 0, s3 <= 1 + s2, s' >= 0, s' <= z' - 1 + (z - 1), z - 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 + s1 :|: s1 >= 0, s1 <= 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s5 :|: s5 >= 0, s5 <= 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: {p}, {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'] p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: p after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 8 + s' + 2*z' }-> f(s3, plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: s2 >= 0, s2 <= z - 1, s3 >= 0, s3 <= 1 + s2, s' >= 0, s' <= z' - 1 + (z - 1), z - 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 + s1 :|: s1 >= 0, s1 <= 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s5 :|: s5 >= 0, s5 <= 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'] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (51) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 8 + s' + 2*z' }-> f(s3, plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: s2 >= 0, s2 <= z - 1, s3 >= 0, s3 <= 1 + s2, s' >= 0, s' <= z' - 1 + (z - 1), z - 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 + s1 :|: s1 >= 0, s1 <= 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s5 :|: s5 >= 0, s5 <= 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'] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (53) 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' ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 8 + s' + 2*z' }-> f(s3, plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: s2 >= 0, s2 <= z - 1, s3 >= 0, s3 <= 1 + s2, s' >= 0, s' <= z' - 1 + (z - 1), z - 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 + s1 :|: s1 >= 0, s1 <= 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s5 :|: s5 >= 0, s5 <= 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'] p: runtime: O(1) [1], size: O(n^1) [z] times: runtime: ?, size: O(n^2) [z + 2*z*z'] ---------------------------------------- (55) 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' ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 8 + s' + 2*z' }-> f(s3, plus(1 + (z - 1), times(1 + (z - 1), z' - 1))) :|: s2 >= 0, s2 <= z - 1, s3 >= 0, s3 <= 1 + s2, s' >= 0, s' <= z' - 1 + (z - 1), z - 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 + s1 :|: s1 >= 0, s1 <= 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s5 :|: s5 >= 0, s5 <= 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'] p: runtime: O(1) [1], size: O(n^1) [z] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (57) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 9 + s' + 2*z*z' + 6*z' }-> f(s3, s10) :|: s9 >= 0, s9 <= 1 + (z - 1) + 2 * ((z' - 1) * (1 + (z - 1))), s10 >= 0, s10 <= 1 + (z - 1) + s9, s2 >= 0, s2 <= z - 1, s3 >= 0, s3 <= 1 + s2, s' >= 0, s' <= z' - 1 + (z - 1), z - 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 + s1 :|: s1 >= 0, s1 <= 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s5 :|: s5 >= 0, s5 <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s8 :|: s6 >= 0, s6 <= z + 2 * ((z' - 2) * z), s7 >= 0, s7 <= z + s6, s8 >= 0, s8 <= z + s7, 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'] p: runtime: O(1) [1], size: O(n^1) [z] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (59) 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 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 9 + s' + 2*z*z' + 6*z' }-> f(s3, s10) :|: s9 >= 0, s9 <= 1 + (z - 1) + 2 * ((z' - 1) * (1 + (z - 1))), s10 >= 0, s10 <= 1 + (z - 1) + s9, s2 >= 0, s2 <= z - 1, s3 >= 0, s3 <= 1 + s2, s' >= 0, s' <= z' - 1 + (z - 1), z - 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 + s1 :|: s1 >= 0, s1 <= 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s5 :|: s5 >= 0, s5 <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s8 :|: s6 >= 0, s6 <= z + 2 * ((z' - 2) * z), s7 >= 0, s7 <= z + s6, s8 >= 0, s8 <= z + s7, 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'] p: runtime: O(1) [1], size: O(n^1) [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] ---------------------------------------- (61) 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: ? ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 9 + s' + 2*z*z' + 6*z' }-> f(s3, s10) :|: s9 >= 0, s9 <= 1 + (z - 1) + 2 * ((z' - 1) * (1 + (z - 1))), s10 >= 0, s10 <= 1 + (z - 1) + s9, s2 >= 0, s2 <= z - 1, s3 >= 0, s3 <= 1 + s2, s' >= 0, s' <= z' - 1 + (z - 1), z - 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 + s1 :|: s1 >= 0, s1 <= 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 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s5 :|: s5 >= 0, s5 <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s8 :|: s6 >= 0, s6 <= z + 2 * ((z' - 2) * z), s7 >= 0, s7 <= z + s6, s8 >= 0, s8 <= z + s7, 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'] p: runtime: O(1) [1], size: O(n^1) [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] ---------------------------------------- (63) 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] p(v0) -> null_p [0] f(v0, v1) -> 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_p, null_f, null_min, null_max, null_plus, null_times ---------------------------------------- (64) 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] p(s(x)) -> x [1] f(s(x), s(y)) -> f(minus(min(s(x), s(y)), max(s(x), s(y))), times(s(x), s(y))) [1] minus(v0, v1) -> null_minus [0] p(v0) -> null_p [0] f(v0, v1) -> 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_p:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times 0 :: 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times s :: 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times max :: 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times plus :: 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times minus :: 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times times :: 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times p :: 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times f :: 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times -> 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times -> null_f null_minus :: 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times null_p :: 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times null_f :: null_f null_min :: 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times null_max :: 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times null_plus :: 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times null_times :: 0:s:null_minus:null_p:null_min:null_max:null_plus:null_times Rewrite Strategy: INNERMOST ---------------------------------------- (65) 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_p => 0 null_f => 0 null_min => 0 null_max => 0 null_plus => 0 null_times => 0 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 1 }-> f(minus(min(1 + x, 1 + y), max(1 + x, 1 + y)), times(1 + x, 1 + y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 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 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 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. ---------------------------------------- (67) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (68) 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)) p(s(z0)) -> z0 f(s(z0), s(z1)) -> f(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))) 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)) P(s(z0)) -> c12 F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) 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)) P(s(z0)) -> c12 F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) K tuples:none Defined Rule Symbols: min_2, max_2, +_2, -_2, *_2, p_1, f_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, P_1, F_2 Compound Symbols: c, c1, c2_1, c3, c4, c5_1, c6, c7_1, c8, c9_1, c10, c11_2, c12, c13_3, c14_3, c15_2 ---------------------------------------- (69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 8 trailing nodes: *'(z0, 0) -> c10 MIN(z0, 0) -> c1 P(s(z0)) -> c12 -'(z0, 0) -> c8 +'(0, z0) -> c6 MAX(z0, 0) -> c4 MIN(0, z0) -> c MAX(0, z0) -> c3 ---------------------------------------- (70) 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)) p(s(z0)) -> z0 f(s(z0), s(z1)) -> f(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(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), s(z1)) -> c13(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) 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), s(z1)) -> c13(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) K tuples:none Defined Rule Symbols: min_2, max_2, +_2, -_2, *_2, p_1, f_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_3, c14_3, c15_2 ---------------------------------------- (71) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: p(s(z0)) -> z0 f(s(z0), s(z1)) -> f(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(z1)) -> c13(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) 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), s(z1)) -> c13(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_3, c14_3, c15_2 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(x0), s(z1)) -> c13(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MIN(s(x0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), 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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(z1)) -> c14(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(x0), s(z1)) -> c13(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MIN(s(x0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) 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), s(z1)) -> c14(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(x0), s(z1)) -> c13(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MIN(s(x0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c14_3, c15_2, c13_3 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) by F(s(x0), s(z1)) -> c14(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(z1)) -> c15(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(x0), s(z1)) -> c13(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MIN(s(x0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(x0), s(z1)) -> c14(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) 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), s(z1)) -> c15(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(x0), s(z1)) -> c13(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MIN(s(x0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(x0), s(z1)) -> c14(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c15_2, c13_3, c14_3 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) by F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(z1)) -> c13(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MIN(s(x0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(x0), s(z1)) -> c14(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) 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(x0), s(z1)) -> c13(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MIN(s(x0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(x0), s(z1)) -> c14(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_3, c14_3, c15_2 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), s(z1)) -> c13(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MIN(s(x0), s(z1))) by F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(x0), s(z1)) -> c14(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) 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), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(x0), s(z1)) -> c14(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_3, c14_3, c15_2 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(x0), s(z1)) -> c13(F(-(min(s(x0), s(z1)), s(max(x0, z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MIN(s(x0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), 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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(x0), s(z1)) -> c14(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) 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), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(x0), s(z1)) -> c14(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_3, c14_3, c15_2 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(x0), s(z1)) -> c13(F(-(s(min(x0, z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MIN(s(x0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), 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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(z1)) -> c14(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), 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(x0), s(z1)) -> c14(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c14_3, c15_2, c13_3 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), s(z1)) -> c14(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) by F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) 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), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c14_3, c15_2, c13_3 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) by F(s(x0), s(z1)) -> c14(F(-(min(s(x0), s(z1)), s(max(x0, z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) 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), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c14_3, c15_2, c13_3 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) by F(s(x0), s(z1)) -> c14(F(-(s(min(x0, z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), 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(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c15_2, c13_3, c14_3 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) by F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) 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), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c15_2, c13_3, c14_3, c15_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), s(x1)) -> c15(*'(s(x0), s(x1))) 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), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [1] + x_1 + x_2 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, x_2)) = [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(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c14(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 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] + x_1 + x_2 POL(min(x_1, x_2)) = 0 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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) 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), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c15_2, c13_3, c14_3, c15_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) by F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), s(max(x0, z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) 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), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c15_2, c13_3, c14_3, c15_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) by F(s(x0), s(z1)) -> c15(F(-(s(min(x0, z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), 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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_3, c14_3, c15_2, c15_1 ---------------------------------------- (99) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(0)) -> c13(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MIN(s(x0), s(0))) by F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_3, c14_3, c15_2, c15_1 ---------------------------------------- (101) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(s(z1))) -> c13(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MIN(s(x0), s(s(z1)))) by F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_3, c14_3, c15_2, c15_1 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MIN_2, MAX_2, +'_2, -'_2, *'_2, F_2 Compound Symbols: c2_1, c5_1, c7_1, c9_1, c11_2, c13_3, c14_3, c15_2, c15_1 ---------------------------------------- (105) 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))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MIN(s(0), s(z0))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MIN(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MAX_2, +'_2, -'_2, *'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_2, c13_3, c14_3, c15_2, c15_1, c2_1 ---------------------------------------- (107) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: MAX_2, +'_2, -'_2, *'_2, F_2, MIN_2 Compound Symbols: c5_1, c7_1, c9_1, c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2 ---------------------------------------- (109) 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))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) S 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0))), MAX(s(x0), s(0))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0))), MAX(s(0), s(z0))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0))), MAX(s(z0), s(0))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: +'_2, -'_2, *'_2, F_2, MIN_2, MAX_2 Compound Symbols: c7_1, c9_1, c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1 ---------------------------------------- (111) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) S 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: +'_2, -'_2, *'_2, F_2, MIN_2, MAX_2 Compound Symbols: c7_1, c9_1, c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) S 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: +'_2, -'_2, *'_2, F_2, MIN_2, MAX_2 Compound Symbols: c7_1, c9_1, c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(z1)) -> c13(F(-(min(s(x0), s(z1)), s(max(x0, z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MIN(s(x0), s(z1))) by F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) S 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), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: +'_2, -'_2, *'_2, F_2, MIN_2, MAX_2 Compound Symbols: c7_1, c9_1, c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2 ---------------------------------------- (117) 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)) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) S tuples: -'(s(z0), s(z1)) -> c9(-'(z0, z1)) *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: -'_2, *'_2, F_2, MIN_2, MAX_2, +'_2 Compound Symbols: c9_1, c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1 ---------------------------------------- (119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace -'(s(z0), s(z1)) -> c9(-'(z0, z1)) by -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) by F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(z1)) -> c13(F(-(s(min(x0, z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MIN(s(x0), s(z1))) by F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) by F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) by F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (131) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(s(z1))) -> c14(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), -'(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), MAX(s(x0), s(s(z1)))) by F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) by F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) by F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(z1)) -> c14(F(-(min(s(x0), s(z1)), s(max(x0, z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) by F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (139) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) by F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) by F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (143) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(z1)) -> c14(F(-(s(min(x0, z1)), max(s(x0), s(z1))), +(s(x0), *(s(x0), z1))), -'(min(s(x0), s(z1)), max(s(x0), s(z1))), MAX(s(x0), s(z1))) by F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) by F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) by F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(0)) -> c15(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), *'(s(x0), s(0))) by F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(s(z1))) -> c15(F(-(min(s(x0), s(s(z1))), max(s(x0), s(s(z1)))), +(s(x0), +(s(x0), *(s(x0), z1)))), *'(s(x0), s(s(z1)))) by F(s(z0), s(s(z1))) -> c15(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), *'(s(z0), s(s(z1)))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c15(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), *'(s(z0), s(s(z1)))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c15(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), *'(s(z0), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (153) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) by F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c15(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c15(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (155) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) by F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c15(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c15(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0), s(z1)) -> c15(F(-(min(s(x0), s(z1)), s(max(x0, z1))), +(s(x0), *(s(x0), z1))), *'(s(x0), s(z1))) by F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c15(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c15(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (159) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) by F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), *'(s(s(z0)), s(s(z1)))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c15(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), *'(s(s(z0)), s(s(z1)))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c15(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), *'(s(s(z0)), s(s(z1)))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), *'(s(0), s(z0))) by F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), +(s(0), *(s(0), z0))), *'(s(0), s(z0))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c15(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), +(s(0), *(s(0), z0))), *'(s(0), s(z0))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c15(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), +(s(0), *(s(0), z0))), *'(s(0), s(z0))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1 ---------------------------------------- (163) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), *'(s(z0), s(0))) by F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), +(s(z0), *(s(z0), 0))), *'(s(z0), s(0))) ---------------------------------------- (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) min(s(z0), s(z1)) -> s(min(z0, z1)) min(0, z0) -> 0 min(z0, 0) -> 0 max(s(z0), s(z1)) -> s(max(z0, z1)) max(0, z0) -> z0 max(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), +(s(z0), *(s(z0), z1))), *'(s(z0), s(z1))) F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c15(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), +(s(0), *(s(0), z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), +(s(z0), *(s(z0), 0))), *'(s(z0), s(0))) S tuples: *'(z0, s(z1)) -> c11(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0), s(x1)) -> c13(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MIN(s(z0), s(x1))) F(s(z0), s(x1)) -> c14(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), -'(min(s(z0), s(x1)), max(s(z0), s(x1))), MAX(s(z0), s(x1))) F(s(z0), s(x1)) -> c15(F(-(min(s(z0), s(x1)), max(s(z0), s(x1))), s(+(z0, *(s(z0), x1)))), *'(s(z0), s(x1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), s(max(z0, z1))), *(s(z0), s(z1))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), *(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c13(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MIN(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) F(s(0), s(z0)) -> c13(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c13(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c13(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(0)) -> c13(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) F(s(x0), s(0)) -> c14(F(-(min(s(x0), s(0)), max(s(x0), s(0))), +(s(x0), 0)), -'(min(s(x0), s(0)), max(s(x0), s(0)))) F(s(0), s(z0)) -> c14(F(-(min(s(0), s(z0)), s(z0)), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(min(s(z0), s(0)), s(z0)), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(0), s(z0)) -> c14(F(-(s(0), max(s(0), s(z0))), *(s(0), s(z0))), -'(min(s(0), s(z0)), max(s(0), s(z0)))) F(s(z0), s(0)) -> c14(F(-(s(0), max(s(z0), s(0))), *(s(z0), s(0))), -'(min(s(z0), s(0)), max(s(z0), s(0)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) +'(s(s(y0)), z1) -> c7(+'(s(y0), z1)) -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c13(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MIN(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) F(s(z0), s(s(z1))) -> c14(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), -'(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), MAX(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(z1)) -> c14(F(-(s(min(z0, z1)), s(max(z0, z1))), +(s(z0), *(s(z0), z1))), -'(min(s(z0), s(z1)), max(s(z0), s(z1))), MAX(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c14(F(-(s(s(min(z0, z1))), max(s(s(z0)), s(s(z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), -'(min(s(s(z0)), s(s(z1))), max(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), max(s(z0), s(0))), s(+(z0, 0))), *'(s(z0), s(0))) F(s(z0), s(s(z1))) -> c15(F(-(min(s(z0), s(s(z1))), max(s(z0), s(s(z1)))), s(+(z0, +(s(z0), *(s(z0), z1))))), *'(s(z0), s(s(z1)))) F(s(z0), s(z1)) -> c15(F(-(min(s(z0), s(z1)), s(max(z0, z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(z0), s(z1)) -> c15(F(-(s(min(z0, z1)), max(s(z0), s(z1))), s(+(z0, *(s(z0), z1)))), *'(s(z0), s(z1))) F(s(s(z0)), s(s(z1))) -> c15(F(-(min(s(s(z0)), s(s(z1))), s(s(max(z0, z1)))), +(s(s(z0)), *(s(s(z0)), s(z1)))), *'(s(s(z0)), s(s(z1)))) F(s(0), s(z0)) -> c15(F(-(min(s(0), s(z0)), s(z0)), +(s(0), *(s(0), z0))), *'(s(0), s(z0))) F(s(z0), s(0)) -> c15(F(-(min(s(z0), s(0)), s(z0)), +(s(z0), *(s(z0), 0))), *'(s(z0), s(0))) K tuples: F(s(x0), s(x1)) -> c15(*'(s(x0), s(x1))) Defined Rule Symbols: *_2, +_2, -_2, min_2, max_2 Defined Pair Symbols: *'_2, F_2, MIN_2, MAX_2, +'_2, -'_2 Compound Symbols: c11_2, c13_3, c14_3, c15_2, c15_1, c2_1, c13_2, c5_1, c14_2, c7_1, c9_1