KILLED proof of input_bobfuNKmV1.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 11 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 4 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 243 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 50 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 1582 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 349 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (38) CdtProblem (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms] (46) CdtProblem (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 46 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 44 ms] (78) CdtProblem (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 24 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 54 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 55 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 4 ms] (114) CdtProblem (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 11 ms] (116) CdtProblem (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (120) 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: plus(0, x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0, y) -> 0 times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0) -> s(0) exp(x, s(y)) -> times(x, exp(x, y)) p(s(0)) -> 0 p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0)) towerIter(0, y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) 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: plus(0', x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0', y) -> 0' times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0') -> s(0') exp(x, s(y)) -> times(x, exp(x, y)) p(s(0')) -> 0' p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0')) towerIter(0', y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) 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: plus(0, x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0, y) -> 0 times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0) -> s(0) exp(x, s(y)) -> times(x, exp(x, y)) p(s(0)) -> 0 p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0)) towerIter(0, y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) 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: plus(0, x) -> x [1] plus(s(x), y) -> s(plus(p(s(x)), y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(p(s(x)), y)) [1] exp(x, 0) -> s(0) [1] exp(x, s(y)) -> times(x, exp(x, y)) [1] p(s(0)) -> 0 [1] p(s(s(x))) -> s(p(s(x))) [1] tower(x, y) -> towerIter(x, y, s(0)) [1] towerIter(0, y, z) -> z [1] towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: plus(0, x) -> x [1] plus(s(x), y) -> s(plus(p(s(x)), y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(p(s(x)), y)) [1] exp(x, 0) -> s(0) [1] exp(x, s(y)) -> times(x, exp(x, y)) [1] p(s(0)) -> 0 [1] p(s(s(x))) -> s(p(s(x))) [1] tower(x, y) -> towerIter(x, y, s(0)) [1] towerIter(0, y, z) -> z [1] towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s p :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s exp :: 0:s -> 0:s -> 0:s tower :: 0:s -> 0:s -> 0:s towerIter :: 0:s -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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: tower_2 towerIter_3 (c) The following functions are completely defined: p_1 times_2 exp_2 plus_2 Due to the following rules being added: p(v0) -> 0 [0] And the following fresh constants: none ---------------------------------------- (10) 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: plus(0, x) -> x [1] plus(s(x), y) -> s(plus(p(s(x)), y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(p(s(x)), y)) [1] exp(x, 0) -> s(0) [1] exp(x, s(y)) -> times(x, exp(x, y)) [1] p(s(0)) -> 0 [1] p(s(s(x))) -> s(p(s(x))) [1] tower(x, y) -> towerIter(x, y, s(0)) [1] towerIter(0, y, z) -> z [1] towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) [1] p(v0) -> 0 [0] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s p :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s exp :: 0:s -> 0:s -> 0:s tower :: 0:s -> 0:s -> 0:s towerIter :: 0:s -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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: plus(0, x) -> x [1] plus(s(0), y) -> s(plus(0, y)) [2] plus(s(s(x')), y) -> s(plus(s(p(s(x'))), y)) [2] plus(s(x), y) -> s(plus(0, y)) [1] times(0, y) -> 0 [1] times(s(0), y) -> plus(y, times(0, y)) [2] times(s(s(x'')), y) -> plus(y, times(s(p(s(x''))), y)) [2] times(s(x), y) -> plus(y, times(0, y)) [1] exp(x, 0) -> s(0) [1] exp(x, s(0)) -> times(x, s(0)) [2] exp(x, s(s(y'))) -> times(x, times(x, exp(x, y'))) [2] p(s(0)) -> 0 [1] p(s(s(x))) -> s(p(s(x))) [1] tower(x, y) -> towerIter(x, y, s(0)) [1] towerIter(0, y, z) -> z [1] towerIter(s(0), y, 0) -> towerIter(0, y, s(0)) [3] towerIter(s(0), y, s(y'')) -> towerIter(0, y, times(y, exp(y, y''))) [3] towerIter(s(s(x1)), y, 0) -> towerIter(s(p(s(x1))), y, s(0)) [3] towerIter(s(s(x1)), y, s(y1)) -> towerIter(s(p(s(x1))), y, times(y, exp(y, y1))) [3] towerIter(s(x), y, 0) -> towerIter(0, y, s(0)) [2] towerIter(s(x), y, s(y2)) -> towerIter(0, y, times(y, exp(y, y2))) [2] p(v0) -> 0 [0] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s p :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s exp :: 0:s -> 0:s -> 0:s tower :: 0:s -> 0:s -> 0:s towerIter :: 0:s -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: exp(z', z'') -{ 2 }-> times(x, times(x, exp(x, y'))) :|: z' = x, x >= 0, z'' = 1 + (1 + y'), y' >= 0 exp(z', z'') -{ 2 }-> times(x, 1 + 0) :|: z' = x, x >= 0, z'' = 1 + 0 exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' = x, x >= 0 p(z') -{ 1 }-> 0 :|: z' = 1 + 0 p(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 p(z') -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z' = 1 + (1 + x) plus(z', z'') -{ 1 }-> x :|: x >= 0, z'' = x, z' = 0 plus(z', z'') -{ 2 }-> 1 + plus(0, y) :|: z'' = y, y >= 0, z' = 1 + 0 plus(z', z'') -{ 1 }-> 1 + plus(0, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0 plus(z', z'') -{ 2 }-> 1 + plus(1 + p(1 + x'), y) :|: z' = 1 + (1 + x'), z'' = y, x' >= 0, y >= 0 times(z', z'') -{ 2 }-> plus(y, times(0, y)) :|: z'' = y, y >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(y, times(0, y)) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0 times(z', z'') -{ 2 }-> plus(y, times(1 + p(1 + x''), y)) :|: z' = 1 + (1 + x''), z'' = y, y >= 0, x'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0 tower(z', z'') -{ 1 }-> towerIter(x, y, 1 + 0) :|: z' = x, z'' = y, x >= 0, y >= 0 towerIter(z', z'', z1) -{ 1 }-> z :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, y, times(y, exp(y, y''))) :|: z'' = y, y >= 0, z1 = 1 + y'', y'' >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, y, times(y, exp(y, y2))) :|: z' = 1 + x, z1 = 1 + y2, z'' = y, x >= 0, y >= 0, y2 >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, y, 1 + 0) :|: z1 = 0, z'' = y, y >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, y, 1 + 0) :|: z' = 1 + x, z1 = 0, z'' = y, x >= 0, y >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + x1), y, times(y, exp(y, y1))) :|: y1 >= 0, x1 >= 0, z' = 1 + (1 + x1), z1 = 1 + y1, z'' = y, y >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + x1), y, 1 + 0) :|: z1 = 0, x1 >= 0, z' = 1 + (1 + x1), z'' = y, y >= 0 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0 exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0 p(z') -{ 1 }-> 0 :|: z' = 1 + 0 p(z') -{ 0 }-> 0 :|: z' >= 0 p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0 plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0 plus(z', z'') -{ 2 }-> 1 + plus(1 + p(1 + (z' - 2)), z'') :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', times(1 + p(1 + (z' - 2)), z'')) :|: z'' >= 0, z' - 2 >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', times(z'', exp(z'', z1 - 1))) :|: z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', 1 + 0) :|: z1 = 0, z' - 2 >= 0, z'' >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { p } { plus } { times } { exp } { towerIter } { tower } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0 exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0 p(z') -{ 1 }-> 0 :|: z' = 1 + 0 p(z') -{ 0 }-> 0 :|: z' >= 0 p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0 plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0 plus(z', z'') -{ 2 }-> 1 + plus(1 + p(1 + (z' - 2)), z'') :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', times(1 + p(1 + (z' - 2)), z'')) :|: z'' >= 0, z' - 2 >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', times(z'', exp(z'', z1 - 1))) :|: z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', 1 + 0) :|: z1 = 0, z' - 2 >= 0, z'' >= 0 Function symbols to be analyzed: {p}, {plus}, {times}, {exp}, {towerIter}, {tower} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0 exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0 p(z') -{ 1 }-> 0 :|: z' = 1 + 0 p(z') -{ 0 }-> 0 :|: z' >= 0 p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0 plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0 plus(z', z'') -{ 2 }-> 1 + plus(1 + p(1 + (z' - 2)), z'') :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', times(1 + p(1 + (z' - 2)), z'')) :|: z'' >= 0, z' - 2 >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', times(z'', exp(z'', z1 - 1))) :|: z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', 1 + 0) :|: z1 = 0, z' - 2 >= 0, z'' >= 0 Function symbols to be analyzed: {p}, {plus}, {times}, {exp}, {towerIter}, {tower} ---------------------------------------- (21) 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' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0 exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0 p(z') -{ 1 }-> 0 :|: z' = 1 + 0 p(z') -{ 0 }-> 0 :|: z' >= 0 p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0 plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0 plus(z', z'') -{ 2 }-> 1 + plus(1 + p(1 + (z' - 2)), z'') :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', times(1 + p(1 + (z' - 2)), z'')) :|: z'' >= 0, z' - 2 >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', times(z'', exp(z'', z1 - 1))) :|: z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', 1 + 0) :|: z1 = 0, z' - 2 >= 0, z'' >= 0 Function symbols to be analyzed: {p}, {plus}, {times}, {exp}, {towerIter}, {tower} Previous analysis results are: p: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0 exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0 p(z') -{ 1 }-> 0 :|: z' = 1 + 0 p(z') -{ 0 }-> 0 :|: z' >= 0 p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0 plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0 plus(z', z'') -{ 2 }-> 1 + plus(1 + p(1 + (z' - 2)), z'') :|: z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', times(1 + p(1 + (z' - 2)), z'')) :|: z'' >= 0, z' - 2 >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', times(z'', exp(z'', z1 - 1))) :|: z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', 1 + 0) :|: z1 = 0, z' - 2 >= 0, z'' >= 0 Function symbols to be analyzed: {plus}, {times}, {exp}, {towerIter}, {tower} Previous analysis results are: p: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0 exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0 p(z') -{ 1 }-> 0 :|: z' = 1 + 0 p(z') -{ 0 }-> 0 :|: z' >= 0 p(z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= 1 + (z' - 2), z' - 2 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0 plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0 plus(z', z'') -{ 2 + z' }-> 1 + plus(1 + s, z'') :|: s >= 0, s <= 1 + (z' - 2), z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 + z' }-> plus(z'', times(1 + s', z'')) :|: s' >= 0, s' <= 1 + (z' - 2), z'' >= 0, z' - 2 >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 + z' }-> towerIter(1 + s1, z'', 1 + 0) :|: s1 >= 0, s1 <= 1 + (z' - 2), z1 = 0, z' - 2 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 + z' }-> towerIter(1 + s2, z'', times(z'', exp(z'', z1 - 1))) :|: s2 >= 0, s2 <= 1 + (z' - 2), z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0 Function symbols to be analyzed: {plus}, {times}, {exp}, {towerIter}, {tower} Previous analysis results are: p: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0 exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0 p(z') -{ 1 }-> 0 :|: z' = 1 + 0 p(z') -{ 0 }-> 0 :|: z' >= 0 p(z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= 1 + (z' - 2), z' - 2 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0 plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0 plus(z', z'') -{ 2 + z' }-> 1 + plus(1 + s, z'') :|: s >= 0, s <= 1 + (z' - 2), z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 + z' }-> plus(z'', times(1 + s', z'')) :|: s' >= 0, s' <= 1 + (z' - 2), z'' >= 0, z' - 2 >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 + z' }-> towerIter(1 + s1, z'', 1 + 0) :|: s1 >= 0, s1 <= 1 + (z' - 2), z1 = 0, z' - 2 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 + z' }-> towerIter(1 + s2, z'', times(z'', exp(z'', z1 - 1))) :|: s2 >= 0, s2 <= 1 + (z' - 2), z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0 Function symbols to be analyzed: {plus}, {times}, {exp}, {towerIter}, {tower} Previous analysis results are: p: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] plus: runtime: ?, size: INF ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0 exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0 exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0 p(z') -{ 1 }-> 0 :|: z' = 1 + 0 p(z') -{ 0 }-> 0 :|: z' >= 0 p(z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= 1 + (z' - 2), z' - 2 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0 plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0 plus(z', z'') -{ 2 + z' }-> 1 + plus(1 + s, z'') :|: s >= 0, s <= 1 + (z' - 2), z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 + z' }-> plus(z'', times(1 + s', z'')) :|: s' >= 0, s' <= 1 + (z' - 2), z'' >= 0, z' - 2 >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0 towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0 towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 + z' }-> towerIter(1 + s1, z'', 1 + 0) :|: s1 >= 0, s1 <= 1 + (z' - 2), z1 = 0, z' - 2 >= 0, z'' >= 0 towerIter(z', z'', z1) -{ 3 + z' }-> towerIter(1 + s2, z'', times(z'', exp(z'', z1 - 1))) :|: s2 >= 0, s2 <= 1 + (z' - 2), z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0 Function symbols to be analyzed: {plus}, {times}, {exp}, {towerIter}, {tower} Previous analysis results are: p: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] plus: runtime: INF, size: INF ---------------------------------------- (31) 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: p(v0) -> null_p [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] exp(v0, v1) -> null_exp [0] towerIter(v0, v1, v2) -> null_towerIter [0] And the following fresh constants: null_p, null_plus, null_times, null_exp, null_towerIter ---------------------------------------- (32) 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: plus(0, x) -> x [1] plus(s(x), y) -> s(plus(p(s(x)), y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(p(s(x)), y)) [1] exp(x, 0) -> s(0) [1] exp(x, s(y)) -> times(x, exp(x, y)) [1] p(s(0)) -> 0 [1] p(s(s(x))) -> s(p(s(x))) [1] tower(x, y) -> towerIter(x, y, s(0)) [1] towerIter(0, y, z) -> z [1] towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) [1] p(v0) -> null_p [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] exp(v0, v1) -> null_exp [0] towerIter(v0, v1, v2) -> null_towerIter [0] The TRS has the following type information: plus :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter 0 :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter s :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter p :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter times :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter exp :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter tower :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter towerIter :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter null_p :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter null_plus :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter null_times :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter null_exp :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter null_towerIter :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter Rewrite Strategy: INNERMOST ---------------------------------------- (33) 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_p => 0 null_plus => 0 null_times => 0 null_exp => 0 null_towerIter => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: exp(z', z'') -{ 1 }-> times(x, exp(x, y)) :|: z' = x, x >= 0, y >= 0, z'' = 1 + y exp(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' = x, x >= 0 p(z') -{ 1 }-> 0 :|: z' = 1 + 0 p(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 p(z') -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z' = 1 + (1 + x) plus(z', z'') -{ 1 }-> x :|: x >= 0, z'' = x, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 plus(z', z'') -{ 1 }-> 1 + plus(p(1 + x), y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0 times(z', z'') -{ 1 }-> plus(y, times(p(1 + x), y)) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0 times(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 tower(z', z'') -{ 1 }-> towerIter(x, y, 1 + 0) :|: z' = x, z'' = y, x >= 0, y >= 0 towerIter(z', z'', z1) -{ 1 }-> z :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 0 towerIter(z', z'', z1) -{ 1 }-> towerIter(p(1 + x), y, exp(y, z)) :|: z' = 1 + x, z1 = z, z >= 0, z'' = y, x >= 0, y >= 0 towerIter(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) tower(z0, z1) -> towerIter(z0, z1, s(0)) towerIter(0, z0, z1) -> z1 towerIter(s(z0), z1, z2) -> towerIter(p(s(z0)), z1, exp(z1, z2)) Tuples: PLUS(0, z0) -> c PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0))) TIMES(0, z0) -> c2 TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) EXP(z0, 0) -> c4 EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(0)) -> c6 P(s(s(z0))) -> c7(P(s(z0))) TOWER(z0, z1) -> c8(TOWERITER(z0, z1, s(0))) TOWERITER(0, z0, z1) -> c9 TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) S tuples: PLUS(0, z0) -> c PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0))) TIMES(0, z0) -> c2 TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) EXP(z0, 0) -> c4 EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(0)) -> c6 P(s(s(z0))) -> c7(P(s(z0))) TOWER(z0, z1) -> c8(TOWERITER(z0, z1, s(0))) TOWERITER(0, z0, z1) -> c9 TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) K tuples:none Defined Rule Symbols: plus_2, times_2, exp_2, p_1, tower_2, towerIter_3 Defined Pair Symbols: PLUS_2, TIMES_2, EXP_2, P_1, TOWER_2, TOWERITER_3 Compound Symbols: c, c1_2, c2, c3_3, c4, c5_2, c6, c7_1, c8_1, c9, c10_2, c11_2 ---------------------------------------- (37) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: TOWER(z0, z1) -> c8(TOWERITER(z0, z1, s(0))) Removed 5 trailing nodes: TIMES(0, z0) -> c2 P(s(0)) -> c6 EXP(z0, 0) -> c4 TOWERITER(0, z0, z1) -> c9 PLUS(0, z0) -> c ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) tower(z0, z1) -> towerIter(z0, z1, s(0)) towerIter(0, z0, z1) -> z1 towerIter(s(z0), z1, z2) -> towerIter(p(s(z0)), z1, exp(z1, z2)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0))) TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0))) TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) K tuples:none Defined Rule Symbols: plus_2, times_2, exp_2, p_1, tower_2, towerIter_3 Defined Pair Symbols: PLUS_2, TIMES_2, EXP_2, P_1, TOWERITER_3 Compound Symbols: c1_2, c3_3, c5_2, c7_1, c10_2, c11_2 ---------------------------------------- (39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: tower(z0, z1) -> towerIter(z0, z1, s(0)) towerIter(0, z0, z1) -> z1 towerIter(s(z0), z1, z2) -> towerIter(p(s(z0)), z1, exp(z1, z2)) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0))) TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0))) TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: PLUS_2, TIMES_2, EXP_2, P_1, TOWERITER_3 Compound Symbols: c1_2, c3_3, c5_2, c7_1, c10_2, c11_2 ---------------------------------------- (41) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0))) by PLUS(s(0), x1) -> c1(PLUS(0, x1), P(s(0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) PLUS(s(0), x1) -> c1(PLUS(0, x1), P(s(0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) PLUS(s(0), x1) -> c1(PLUS(0, x1), P(s(0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: TIMES_2, EXP_2, P_1, TOWERITER_3, PLUS_2 Compound Symbols: c3_3, c5_2, c7_1, c10_2, c11_2, c1_2 ---------------------------------------- (43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PLUS(s(0), x1) -> c1(PLUS(0, x1), P(s(0))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) S tuples: TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: TIMES_2, EXP_2, P_1, TOWERITER_3, PLUS_2 Compound Symbols: c3_3, c5_2, c7_1, c10_2, c11_2, c1_2 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) by TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)), TIMES(0, x1), P(s(0))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)), TIMES(0, x1), P(s(0))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)), TIMES(0, x1), P(s(0))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TOWERITER_3, PLUS_2, TIMES_2 Compound Symbols: c5_2, c7_1, c10_2, c11_2, c1_2, c3_3, c3_1 ---------------------------------------- (47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TOWERITER_3, PLUS_2, TIMES_2 Compound Symbols: c5_2, c7_1, c10_2, c11_2, c1_2, c3_3, c3_1 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) by TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(0), x1, x2) -> c10(TOWERITER(0, x1, exp(x1, x2)), P(s(0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(0), x1, x2) -> c10(TOWERITER(0, x1, exp(x1, x2)), P(s(0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(0), x1, x2) -> c10(TOWERITER(0, x1, exp(x1, x2)), P(s(0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TOWERITER_3, PLUS_2, TIMES_2 Compound Symbols: c5_2, c7_1, c11_2, c1_2, c3_3, c3_1, c10_2 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: TOWERITER(s(0), x1, x2) -> c10(TOWERITER(0, x1, exp(x1, x2)), P(s(0))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TOWERITER_3, PLUS_2, TIMES_2 Compound Symbols: c5_2, c7_1, c11_2, c1_2, c3_3, c3_1, c10_2 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) by TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)), EXP(z0, 0)) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(0), x1, x2) -> c11(TOWERITER(0, x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)), EXP(z0, 0)) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(0), x1, x2) -> c11(TOWERITER(0, x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)), EXP(z0, 0)) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(0), x1, x2) -> c11(TOWERITER(0, x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, PLUS_2, TIMES_2, TOWERITER_3 Compound Symbols: c5_2, c7_1, c1_2, c3_3, c3_1, c10_2, c11_2 ---------------------------------------- (55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) K tuples:none Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, PLUS_2, TIMES_2, TOWERITER_3 Compound Symbols: c5_2, c7_1, c1_2, c3_3, c3_1, c10_2, c11_2, c11_1 ---------------------------------------- (57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) We considered the (Usable) Rules:none And the Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EXP(x_1, x_2)) = 0 POL(P(x_1)) = x_1 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(TOWERITER(x_1, x_2, x_3)) = [1] POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c7(x_1)) = x_1 POL(exp(x_1, x_2)) = x_1 + x_2 POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = [1] + x_2 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = x_2 ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, PLUS_2, TIMES_2, TOWERITER_3 Compound Symbols: c5_2, c7_1, c1_2, c3_3, c3_1, c10_2, c11_2, c11_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) by PLUS(s(s(0)), x1) -> c1(PLUS(s(0), x1), P(s(s(0)))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(0)), x1) -> c1(PLUS(s(0), x1), P(s(s(0)))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(0)), x1) -> c1(PLUS(s(0), x1), P(s(s(0)))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_3, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_3, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) by TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1, c3_3 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) by TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1, c3_3 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) by TOWERITER(s(0), x1, 0) -> c10(TOWERITER(0, x1, s(0)), P(s(0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(0), x1, 0) -> c10(TOWERITER(0, x1, s(0)), P(s(0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(0), x1, 0) -> c10(TOWERITER(0, x1, s(0)), P(s(0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1, c3_3 ---------------------------------------- (69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: TOWERITER(s(0), x1, 0) -> c10(TOWERITER(0, x1, s(0)), P(s(0))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1, c3_3 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) by TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(0), x1, s(x2)) -> c10(TOWERITER(0, x1, times(x1, exp(x1, x2))), P(s(0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(0), x1, s(x2)) -> c10(TOWERITER(0, x1, times(x1, exp(x1, x2))), P(s(0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(0), x1, s(x2)) -> c10(TOWERITER(0, x1, times(x1, exp(x1, x2))), P(s(0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1, c3_3 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: TOWERITER(s(0), x1, s(x2)) -> c10(TOWERITER(0, x1, times(x1, exp(x1, x2))), P(s(0))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1, c3_3 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) by TOWERITER(s(s(x0)), z0, 0) -> c10(TOWERITER(s(p(s(x0))), z0, s(0)), P(s(s(x0)))) TOWERITER(s(s(x0)), z0, s(z1)) -> c10(TOWERITER(s(p(s(x0))), z0, times(z0, exp(z0, z1))), P(s(s(x0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c11_2, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1 ---------------------------------------- (77) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) We considered the (Usable) Rules:none And the Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EXP(x_1, x_2)) = 0 POL(P(x_1)) = x_1 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(TOWERITER(x_1, x_2, x_3)) = [1] POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c7(x_1)) = x_1 POL(exp(x_1, x_2)) = x_1 + x_2 POL(p(x_1)) = [1] POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(times(x_1, x_2)) = 0 ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c11_2, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1 ---------------------------------------- (79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) We considered the (Usable) Rules: p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 And the Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EXP(x_1, x_2)) = 0 POL(P(x_1)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(TOWERITER(x_1, x_2, x_3)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c7(x_1)) = x_1 POL(exp(x_1, x_2)) = x_1 + x_2 POL(p(x_1)) = x_1 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = 0 ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c11_2, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) by TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(0), x1, s(x2)) -> c11(TOWERITER(0, x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(0), x1, s(x2)) -> c11(TOWERITER(0, x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(0), x1, s(x2)) -> c11(TOWERITER(0, x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c11_2, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1 ---------------------------------------- (83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c11_2, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1 ---------------------------------------- (85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) We considered the (Usable) Rules:none And the Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EXP(x_1, x_2)) = 0 POL(P(x_1)) = x_1 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(TOWERITER(x_1, x_2, x_3)) = [1] POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c7(x_1)) = x_1 POL(exp(x_1, x_2)) = x_1 + x_2 POL(p(x_1)) = [1] POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(times(x_1, x_2)) = x_1 ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c11_2, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) by TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)), EXP(z0, 0)) TOWERITER(s(s(x0)), z0, s(z1)) -> c11(TOWERITER(s(p(s(x0))), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)), EXP(z0, 0)) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)), EXP(z0, 0)) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1, c11_2 ---------------------------------------- (89) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1, c11_2 ---------------------------------------- (91) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) We considered the (Usable) Rules: p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 And the Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EXP(x_1, x_2)) = 0 POL(P(x_1)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(TOWERITER(x_1, x_2, x_3)) = x_1 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c7(x_1)) = x_1 POL(exp(x_1, x_2)) = x_1 + x_2 POL(p(x_1)) = x_1 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_1 ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1, c11_2 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) by TOWERITER(s(0), x1, 0) -> c11(TOWERITER(0, x1, s(0))) TOWERITER(s(s(z0)), x1, 0) -> c11(TOWERITER(s(p(s(z0))), x1, s(0))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TOWERITER(s(0), x1, 0) -> c11(TOWERITER(0, x1, s(0))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TOWERITER(s(0), x1, 0) -> c11(TOWERITER(0, x1, s(0))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1, c11_2 ---------------------------------------- (95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: TOWERITER(s(0), x1, 0) -> c11(TOWERITER(0, x1, s(0))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1, c11_2 ---------------------------------------- (97) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) by TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c11_1, c1_2, c1_1, c10_2, c10_1, c11_2, c3_3 ---------------------------------------- (99) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) by TIMES(s(0), z0) -> c3(PLUS(z0, 0)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) P(s(s(z0))) -> c7(P(s(z0))) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2 Compound Symbols: c5_2, c7_1, c3_1, c11_1, c1_2, c1_1, c10_2, c10_1, c11_2, c3_3 ---------------------------------------- (101) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace P(s(s(z0))) -> c7(P(s(z0))) by P(s(s(s(y0)))) -> c7(P(s(s(y0)))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) P(s(s(s(y0)))) -> c7(P(s(s(y0)))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) P(s(s(s(y0)))) -> c7(P(s(s(y0)))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, TIMES_2, TOWERITER_3, PLUS_2, P_1 Compound Symbols: c5_2, c3_1, c11_1, c1_2, c1_1, c10_2, c10_1, c11_2, c3_3, c7_1 ---------------------------------------- (103) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: PLUS(s(s(0)), x1) -> c1(P(s(s(0)))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) P(s(s(s(y0)))) -> c7(P(s(s(y0)))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) P(s(s(s(y0)))) -> c7(P(s(s(y0)))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0)))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, TIMES_2, TOWERITER_3, PLUS_2, P_1 Compound Symbols: c5_2, c3_1, c11_1, c1_2, c1_1, c10_2, c10_1, c11_2, c3_3, c7_1 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) P(s(s(s(y0)))) -> c7(P(s(s(y0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) P(s(s(s(y0)))) -> c7(P(s(s(y0)))) K tuples: TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, TIMES_2, TOWERITER_3, PLUS_2, P_1 Compound Symbols: c5_2, c3_1, c11_1, c1_2, c1_1, c10_2, c10_1, c11_2, c3_3, c7_1 ---------------------------------------- (107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) by TOWERITER(s(0), z0, s(y1)) -> c11(EXP(z0, s(y1))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) P(s(s(s(y0)))) -> c7(P(s(s(y0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) P(s(s(s(y0)))) -> c7(P(s(s(y0)))) K tuples: TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, TIMES_2, PLUS_2, TOWERITER_3, P_1 Compound Symbols: c5_2, c3_1, c1_2, c1_1, c10_2, c10_1, c11_2, c11_1, c3_3, c7_1 ---------------------------------------- (109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) by PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) P(s(s(s(y0)))) -> c7(P(s(s(y0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) P(s(s(s(y0)))) -> c7(P(s(s(y0)))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) K tuples: TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, TIMES_2, PLUS_2, TOWERITER_3, P_1 Compound Symbols: c5_2, c3_1, c1_2, c10_2, c10_1, c11_2, c11_1, c3_3, c7_1, c1_1 ---------------------------------------- (111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) by TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0))))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) P(s(s(s(y0)))) -> c7(P(s(s(y0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0))))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), z0) -> c3(PLUS(z0, 0)) P(s(s(s(y0)))) -> c7(P(s(s(y0)))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) K tuples: TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, TIMES_2, PLUS_2, TOWERITER_3, P_1 Compound Symbols: c5_2, c3_1, c1_2, c10_2, c11_2, c11_1, c3_3, c7_1, c10_1, c1_1 ---------------------------------------- (113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TIMES(s(0), z0) -> c3(PLUS(z0, 0)) by TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) P(s(s(s(y0)))) -> c7(P(s(s(y0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0))))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) P(s(s(s(y0)))) -> c7(P(s(s(y0)))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) K tuples: TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, TIMES_2, PLUS_2, TOWERITER_3, P_1 Compound Symbols: c5_2, c3_1, c1_2, c10_2, c11_2, c11_1, c3_3, c7_1, c10_1, c1_1 ---------------------------------------- (115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace P(s(s(s(y0)))) -> c7(P(s(s(y0)))) by P(s(s(s(s(y0))))) -> c7(P(s(s(s(y0))))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0))))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) P(s(s(s(s(y0))))) -> c7(P(s(s(s(y0))))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) P(s(s(s(s(y0))))) -> c7(P(s(s(s(y0))))) K tuples: TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, TIMES_2, PLUS_2, TOWERITER_3, P_1 Compound Symbols: c5_2, c3_1, c1_2, c10_2, c11_2, c11_1, c3_3, c10_1, c1_1, c7_1 ---------------------------------------- (117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) by PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0))))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) P(s(s(s(s(y0))))) -> c7(P(s(s(s(y0))))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) P(s(s(s(s(y0))))) -> c7(P(s(s(s(y0))))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) K tuples: TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0))))) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, TIMES_2, PLUS_2, TOWERITER_3, P_1 Compound Symbols: c5_2, c3_1, c1_2, c10_2, c11_2, c11_1, c3_3, c10_1, c7_1, c1_1 ---------------------------------------- (119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0))))) by TOWERITER(s(s(s(s(y0)))), z1, z2) -> c10(P(s(s(s(s(y0)))))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(p(s(z0)), z1)) exp(z0, 0) -> s(0) exp(z0, s(z1)) -> times(z0, exp(z0, z1)) Tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) P(s(s(s(s(y0))))) -> c7(P(s(s(s(y0))))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) TOWERITER(s(s(s(s(y0)))), z1, z2) -> c10(P(s(s(s(s(y0)))))) S tuples: EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1)) TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1))) PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0))))) TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1)) TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0)))) TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0))) TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0))) TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0))) TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0))) TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0)))) TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0))))) TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2))) TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2))) TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0))) TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1)))) TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2))) TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0))) TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0)))) TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0)) P(s(s(s(s(y0))))) -> c7(P(s(s(s(y0))))) PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0)))))) K tuples: TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2))) TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2)) TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2))) TOWERITER(s(s(s(s(y0)))), z1, z2) -> c10(P(s(s(s(s(y0)))))) Defined Rule Symbols: p_1, times_2, plus_2, exp_2 Defined Pair Symbols: EXP_2, TIMES_2, PLUS_2, TOWERITER_3, P_1 Compound Symbols: c5_2, c3_1, c1_2, c10_2, c11_2, c11_1, c3_3, c10_1, c7_1, c1_1