KILLED proof of input_ql8qjBHEJU.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 428 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 143 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 967 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 250 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 264 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 107 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 796 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 16 ms] (44) CpxRNTS (45) CompletionProof [UPPER BOUND(ID), 0 ms] (46) CpxTypedWeightedCompleteTrs (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 136 ms] (56) CdtProblem (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms] (58) CdtProblem (59) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 29 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 19 ms] (126) CdtProblem (127) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 4 ms] (128) CdtProblem (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (130) 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: +(x, 0) -> x +(0, x) -> x +(s(x), s(y)) -> s(s(+(x, y))) *(x, 0) -> 0 *(0, x) -> 0 *(s(x), s(y)) -> s(+(*(x, y), +(x, y))) sum(nil) -> 0 sum(cons(x, l)) -> +(x, sum(l)) prod(nil) -> s(0) prod(cons(x, l)) -> *(x, prod(l)) 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: +'(x, 0') -> x +'(0', x) -> x +'(s(x), s(y)) -> s(s(+'(x, y))) *'(x, 0') -> 0' *'(0', x) -> 0' *'(s(x), s(y)) -> s(+'(*'(x, y), +'(x, y))) sum(nil) -> 0' sum(cons(x, l)) -> +'(x, sum(l)) prod(nil) -> s(0') prod(cons(x, l)) -> *'(x, prod(l)) 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: +(x, 0) -> x +(0, x) -> x +(s(x), s(y)) -> s(s(+(x, y))) *(x, 0) -> 0 *(0, x) -> 0 *(s(x), s(y)) -> s(+(*(x, y), +(x, y))) sum(nil) -> 0 sum(cons(x, l)) -> +(x, sum(l)) prod(nil) -> s(0) prod(cons(x, l)) -> *(x, prod(l)) 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: +(x, 0) -> x [1] +(0, x) -> x [1] +(s(x), s(y)) -> s(s(+(x, y))) [1] *(x, 0) -> 0 [1] *(0, x) -> 0 [1] *(s(x), s(y)) -> s(+(*(x, y), +(x, y))) [1] sum(nil) -> 0 [1] sum(cons(x, l)) -> +(x, sum(l)) [1] prod(nil) -> s(0) [1] prod(cons(x, l)) -> *(x, prod(l)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: + => plus * => times ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: plus(x, 0) -> x [1] plus(0, x) -> x [1] plus(s(x), s(y)) -> s(s(plus(x, y))) [1] times(x, 0) -> 0 [1] times(0, x) -> 0 [1] times(s(x), s(y)) -> s(plus(times(x, y), plus(x, y))) [1] sum(nil) -> 0 [1] sum(cons(x, l)) -> plus(x, sum(l)) [1] prod(nil) -> s(0) [1] prod(cons(x, l)) -> times(x, prod(l)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: plus(x, 0) -> x [1] plus(0, x) -> x [1] plus(s(x), s(y)) -> s(s(plus(x, y))) [1] times(x, 0) -> 0 [1] times(0, x) -> 0 [1] times(s(x), s(y)) -> s(plus(times(x, y), plus(x, y))) [1] sum(nil) -> 0 [1] sum(cons(x, l)) -> plus(x, sum(l)) [1] prod(nil) -> s(0) [1] prod(cons(x, l)) -> times(x, prod(l)) [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s sum :: nil:cons -> 0:s nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons prod :: nil:cons -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (11) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: times_2 plus_2 prod_1 sum_1 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (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(x, 0) -> x [1] plus(0, x) -> x [1] plus(s(x), s(y)) -> s(s(plus(x, y))) [1] times(x, 0) -> 0 [1] times(0, x) -> 0 [1] times(s(x), s(y)) -> s(plus(times(x, y), plus(x, y))) [1] sum(nil) -> 0 [1] sum(cons(x, l)) -> plus(x, sum(l)) [1] prod(nil) -> s(0) [1] prod(cons(x, l)) -> times(x, prod(l)) [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s sum :: nil:cons -> 0:s nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons prod :: nil:cons -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (13) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (14) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: plus(x, 0) -> x [1] plus(0, x) -> x [1] plus(s(x), s(y)) -> s(s(plus(x, y))) [1] times(x, 0) -> 0 [1] times(0, x) -> 0 [1] times(s(x), s(0)) -> s(plus(0, x)) [3] times(s(0), s(0)) -> s(plus(0, 0)) [3] times(s(0), s(0)) -> s(plus(0, 0)) [3] times(s(0), s(y)) -> s(plus(0, y)) [3] times(s(s(x')), s(s(y'))) -> s(plus(s(plus(times(x', y'), plus(x', y'))), s(s(plus(x', y'))))) [3] sum(nil) -> 0 [1] sum(cons(x, nil)) -> plus(x, 0) [2] sum(cons(x, cons(x'', l'))) -> plus(x, plus(x'', sum(l'))) [2] prod(nil) -> s(0) [1] prod(cons(x, nil)) -> times(x, s(0)) [2] prod(cons(x, cons(x1, l''))) -> times(x, times(x1, prod(l''))) [2] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s sum :: nil:cons -> 0:s nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons prod :: nil:cons -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (15) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 nil => 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + (1 + plus(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0 prod(z) -{ 2 }-> times(x, 1 + 0) :|: x >= 0, z = 1 + x + 0 prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l') sum(z) -{ 2 }-> plus(x, 0) :|: x >= 0, z = 1 + x + 0 sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0 times(z, z') -{ 3 }-> 1 + plus(0, x) :|: x >= 0, z' = 1 + 0, z = 1 + x times(z, z') -{ 3 }-> 1 + plus(0, y) :|: z' = 1 + y, z = 1 + 0, y >= 0 times(z, z') -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0, z' = 1 + 0 times(z, z') -{ 3 }-> 1 + plus(1 + plus(times(x', y'), plus(x', y')), 1 + (1 + plus(x', y'))) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x') ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + (1 + plus(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0 prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0 prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l') sum(z) -{ 2 }-> plus(z - 1, 0) :|: z - 1 >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0, z' = 1 + 0 times(z, z') -{ 3 }-> 1 + plus(0, z - 1) :|: z - 1 >= 0, z' = 1 + 0 times(z, z') -{ 3 }-> 1 + plus(0, z' - 1) :|: z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 3 }-> 1 + plus(1 + plus(times(z - 2, z' - 2), plus(z - 2, z' - 2)), 1 + (1 + plus(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { plus } { times } { sum } { prod } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + (1 + plus(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0 prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0 prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l') sum(z) -{ 2 }-> plus(z - 1, 0) :|: z - 1 >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0, z' = 1 + 0 times(z, z') -{ 3 }-> 1 + plus(0, z - 1) :|: z - 1 >= 0, z' = 1 + 0 times(z, z') -{ 3 }-> 1 + plus(0, z' - 1) :|: z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 3 }-> 1 + plus(1 + plus(times(z - 2, z' - 2), plus(z - 2, z' - 2)), 1 + (1 + plus(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {plus}, {times}, {sum}, {prod} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + (1 + plus(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0 prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0 prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l') sum(z) -{ 2 }-> plus(z - 1, 0) :|: z - 1 >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0, z' = 1 + 0 times(z, z') -{ 3 }-> 1 + plus(0, z - 1) :|: z - 1 >= 0, z' = 1 + 0 times(z, z') -{ 3 }-> 1 + plus(0, z' - 1) :|: z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 3 }-> 1 + plus(1 + plus(times(z - 2, z' - 2), plus(z - 2, z' - 2)), 1 + (1 + plus(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {plus}, {times}, {sum}, {prod} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + (1 + plus(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0 prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0 prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l') sum(z) -{ 2 }-> plus(z - 1, 0) :|: z - 1 >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0, z' = 1 + 0 times(z, z') -{ 3 }-> 1 + plus(0, z - 1) :|: z - 1 >= 0, z' = 1 + 0 times(z, z') -{ 3 }-> 1 + plus(0, z' - 1) :|: z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 3 }-> 1 + plus(1 + plus(times(z - 2, z' - 2), plus(z - 2, z' - 2)), 1 + (1 + plus(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {plus}, {times}, {sum}, {prod} Previous analysis results are: plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + (1 + plus(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0 prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0 prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l') sum(z) -{ 2 }-> plus(z - 1, 0) :|: z - 1 >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0, z' = 1 + 0 times(z, z') -{ 3 }-> 1 + plus(0, z - 1) :|: z - 1 >= 0, z' = 1 + 0 times(z, z') -{ 3 }-> 1 + plus(0, z' - 1) :|: z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 3 }-> 1 + plus(1 + plus(times(z - 2, z' - 2), plus(z - 2, z' - 2)), 1 + (1 + plus(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {times}, {sum}, {prod} Previous analysis results are: plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0 prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0 prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0 prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0 sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l') sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0 times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0 times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 3 + 2*z' }-> 1 + plus(1 + plus(times(z - 2, z' - 2), s2), 1 + (1 + s3)) :|: s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {times}, {sum}, {prod} Previous analysis results are: plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 3 + 5*z + 2*z*z' + 2*z^2 + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0 prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0 prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0 prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0 sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l') sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0 times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0 times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 3 + 2*z' }-> 1 + plus(1 + plus(times(z - 2, z' - 2), s2), 1 + (1 + s3)) :|: s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {times}, {sum}, {prod} Previous analysis results are: plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] times: runtime: ?, size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + z'] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 15 + 2*z + 4*z*z' + 2*z^2 + z' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0 prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0 prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0 prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0 sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l') sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0 times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0 times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 3 + 2*z' }-> 1 + plus(1 + plus(times(z - 2, z' - 2), s2), 1 + (1 + s3)) :|: s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {sum}, {prod} Previous analysis results are: plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] times: runtime: O(n^2) [15 + 2*z + 4*z*z' + 2*z^2 + z'], size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + z'] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0 prod(z) -{ 14 + 2*z + 2*z^2 }-> s8 :|: s8 >= 0, s8 <= 5 * (z - 1) + (1 + 0) + 2 * ((1 + 0) * (z - 1)) + 2 * ((z - 1) * (z - 1)) + 3, z - 1 >= 0 prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0 prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0 sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l') sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0 times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0 times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 42 + s2 + s3 + -14*z + 4*z*z' + 2*z^2 + -5*z' }-> 1 + s7 :|: s5 >= 0, s5 <= 5 * (z - 2) + (z' - 2) + 2 * ((z' - 2) * (z - 2)) + 2 * ((z - 2) * (z - 2)) + 3, s6 >= 0, s6 <= s5 + s2, s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s3)), s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {sum}, {prod} Previous analysis results are: plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] times: runtime: O(n^2) [15 + 2*z + 4*z*z' + 2*z^2 + z'], size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + z'] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: sum after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0 prod(z) -{ 14 + 2*z + 2*z^2 }-> s8 :|: s8 >= 0, s8 <= 5 * (z - 1) + (1 + 0) + 2 * ((1 + 0) * (z - 1)) + 2 * ((z - 1) * (z - 1)) + 3, z - 1 >= 0 prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0 prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0 sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l') sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0 times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0 times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 42 + s2 + s3 + -14*z + 4*z*z' + 2*z^2 + -5*z' }-> 1 + s7 :|: s5 >= 0, s5 <= 5 * (z - 2) + (z' - 2) + 2 * ((z' - 2) * (z - 2)) + 2 * ((z - 2) * (z - 2)) + 3, s6 >= 0, s6 <= s5 + s2, s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s3)), s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {sum}, {prod} Previous analysis results are: plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] times: runtime: O(n^2) [15 + 2*z + 4*z*z' + 2*z^2 + z'], size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + z'] sum: runtime: ?, size: O(n^1) [z] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: sum after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 6 + 4*z + 2*z^2 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0 prod(z) -{ 14 + 2*z + 2*z^2 }-> s8 :|: s8 >= 0, s8 <= 5 * (z - 1) + (1 + 0) + 2 * ((1 + 0) * (z - 1)) + 2 * ((z - 1) * (z - 1)) + 3, z - 1 >= 0 prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0 prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0 sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l') sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0 times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0 times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 42 + s2 + s3 + -14*z + 4*z*z' + 2*z^2 + -5*z' }-> 1 + s7 :|: s5 >= 0, s5 <= 5 * (z - 2) + (z' - 2) + 2 * ((z' - 2) * (z - 2)) + 2 * ((z - 2) * (z - 2)) + 3, s6 >= 0, s6 <= s5 + s2, s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s3)), s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {prod} Previous analysis results are: plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] times: runtime: O(n^2) [15 + 2*z + 4*z*z' + 2*z^2 + z'], size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + z'] sum: runtime: O(n^2) [6 + 4*z + 2*z^2], size: O(n^1) [z] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0 prod(z) -{ 14 + 2*z + 2*z^2 }-> s8 :|: s8 >= 0, s8 <= 5 * (z - 1) + (1 + 0) + 2 * ((1 + 0) * (z - 1)) + 2 * ((z - 1) * (z - 1)) + 3, z - 1 >= 0 prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0 prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 12 + 4*l' + 2*l'^2 + s10 + s9 }-> s11 :|: s9 >= 0, s9 <= l', s10 >= 0, s10 <= x'' + s9, s11 >= 0, s11 <= x + s10, x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l') sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0 times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0 times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 42 + s2 + s3 + -14*z + 4*z*z' + 2*z^2 + -5*z' }-> 1 + s7 :|: s5 >= 0, s5 <= 5 * (z - 2) + (z' - 2) + 2 * ((z' - 2) * (z - 2)) + 2 * ((z - 2) * (z - 2)) + 3, s6 >= 0, s6 <= s5 + s2, s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s3)), s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {prod} Previous analysis results are: plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] times: runtime: O(n^2) [15 + 2*z + 4*z*z' + 2*z^2 + z'], size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + z'] sum: runtime: O(n^2) [6 + 4*z + 2*z^2], size: O(n^1) [z] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: prod after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0 prod(z) -{ 14 + 2*z + 2*z^2 }-> s8 :|: s8 >= 0, s8 <= 5 * (z - 1) + (1 + 0) + 2 * ((1 + 0) * (z - 1)) + 2 * ((z - 1) * (z - 1)) + 3, z - 1 >= 0 prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0 prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 12 + 4*l' + 2*l'^2 + s10 + s9 }-> s11 :|: s9 >= 0, s9 <= l', s10 >= 0, s10 <= x'' + s9, s11 >= 0, s11 <= x + s10, x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l') sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0 times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0 times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 42 + s2 + s3 + -14*z + 4*z*z' + 2*z^2 + -5*z' }-> 1 + s7 :|: s5 >= 0, s5 <= 5 * (z - 2) + (z' - 2) + 2 * ((z' - 2) * (z - 2)) + 2 * ((z - 2) * (z - 2)) + 3, s6 >= 0, s6 <= s5 + s2, s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s3)), s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {prod} Previous analysis results are: plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] times: runtime: O(n^2) [15 + 2*z + 4*z*z' + 2*z^2 + z'], size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + z'] sum: runtime: O(n^2) [6 + 4*z + 2*z^2], size: O(n^1) [z] prod: runtime: ?, size: INF ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: prod after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0 prod(z) -{ 14 + 2*z + 2*z^2 }-> s8 :|: s8 >= 0, s8 <= 5 * (z - 1) + (1 + 0) + 2 * ((1 + 0) * (z - 1)) + 2 * ((z - 1) * (z - 1)) + 3, z - 1 >= 0 prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0 prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 12 + 4*l' + 2*l'^2 + s10 + s9 }-> s11 :|: s9 >= 0, s9 <= l', s10 >= 0, s10 <= x'' + s9, s11 >= 0, s11 <= x + s10, x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l') sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0 sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0 times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0 times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 42 + s2 + s3 + -14*z + 4*z*z' + 2*z^2 + -5*z' }-> 1 + s7 :|: s5 >= 0, s5 <= 5 * (z - 2) + (z' - 2) + 2 * ((z' - 2) * (z - 2)) + 2 * ((z - 2) * (z - 2)) + 3, s6 >= 0, s6 <= s5 + s2, s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s3)), s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0 Function symbols to be analyzed: {prod} Previous analysis results are: plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z'] times: runtime: O(n^2) [15 + 2*z + 4*z*z' + 2*z^2 + z'], size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + z'] sum: runtime: O(n^2) [6 + 4*z + 2*z^2], size: O(n^1) [z] prod: runtime: INF, size: INF ---------------------------------------- (45) 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: none And the following fresh constants: none ---------------------------------------- (46) 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(x, 0) -> x [1] plus(0, x) -> x [1] plus(s(x), s(y)) -> s(s(plus(x, y))) [1] times(x, 0) -> 0 [1] times(0, x) -> 0 [1] times(s(x), s(y)) -> s(plus(times(x, y), plus(x, y))) [1] sum(nil) -> 0 [1] sum(cons(x, l)) -> plus(x, sum(l)) [1] prod(nil) -> s(0) [1] prod(cons(x, l)) -> times(x, prod(l)) [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s sum :: nil:cons -> 0:s nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons prod :: nil:cons -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (47) 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 nil => 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + (1 + plus(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x prod(z) -{ 1 }-> times(x, prod(l)) :|: x >= 0, l >= 0, z = 1 + x + l prod(z) -{ 1 }-> 1 + 0 :|: z = 0 sum(z) -{ 1 }-> plus(x, sum(l)) :|: x >= 0, l >= 0, z = 1 + x + l sum(z) -{ 1 }-> 0 :|: z = 0 times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0 times(z, z') -{ 1 }-> 1 + plus(times(x, y), plus(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (49) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(z0, 0) -> c +'(0, z0) -> c1 +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(z0, 0) -> c3 *'(0, z0) -> c4 *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) *'(s(z0), s(z1)) -> c6(+'(*(z0, z1), +(z0, z1)), +'(z0, z1)) SUM(nil) -> c7 SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) PROD(nil) -> c9 PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) S tuples: +'(z0, 0) -> c +'(0, z0) -> c1 +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(z0, 0) -> c3 *'(0, z0) -> c4 *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) *'(s(z0), s(z1)) -> c6(+'(*(z0, z1), +(z0, z1)), +'(z0, z1)) SUM(nil) -> c7 SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) PROD(nil) -> c9 PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) K tuples:none Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c, c1, c2_1, c3, c4, c5_2, c6_2, c7, c8_2, c9, c10_2 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: SUM(nil) -> c7 +'(0, z0) -> c1 *'(z0, 0) -> c3 +'(z0, 0) -> c *'(0, z0) -> c4 PROD(nil) -> c9 ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) *'(s(z0), s(z1)) -> c6(+'(*(z0, z1), +(z0, z1)), +'(z0, z1)) SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) *'(s(z0), s(z1)) -> c6(+'(*(z0, z1), +(z0, z1)), +'(z0, z1)) SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) K tuples:none Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c5_2, c6_2, c8_2, c10_2 ---------------------------------------- (53) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) K tuples:none Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c5_2, c8_2, c10_2, c_1 ---------------------------------------- (55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) We considered the (Usable) Rules:none And the Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [1] + x_1 + x_2 POL(*'(x_1, x_2)) = [1] POL(+(x_1, x_2)) = [1] + x_1 POL(+'(x_1, x_2)) = 0 POL(0) = [1] POL(PROD(x_1)) = x_1 POL(SUM(x_1)) = x_1 POL(c(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = [1] POL(prod(x_1)) = [1] + x_1 POL(s(x_1)) = [1] POL(sum(x_1)) = [1] + x_1 ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) K tuples: SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c5_2, c8_2, c10_2, c_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. PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) We considered the (Usable) Rules:none And the Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [1] + x_1 + x_2 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = [1] + x_1 POL(+'(x_1, x_2)) = 0 POL(0) = [1] POL(PROD(x_1)) = x_1 POL(SUM(x_1)) = x_1 POL(c(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = [1] POL(prod(x_1)) = [1] + x_1 POL(s(x_1)) = [1] POL(sum(x_1)) = [1] + x_1 ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) K tuples: SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c5_2, c8_2, c10_2, c_1 ---------------------------------------- (59) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) We considered the (Usable) Rules:none And the Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [1] + x_1 + x_2 POL(*'(x_1, x_2)) = [1] + x_1 POL(+(x_1, x_2)) = x_1 + x_2 POL(+'(x_1, x_2)) = 0 POL(0) = [1] POL(PROD(x_1)) = x_1 POL(SUM(x_1)) = x_1 POL(c(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = [1] + x_1 + x_2 POL(nil) = [1] POL(prod(x_1)) = [1] + x_1 POL(s(x_1)) = [1] + x_1 POL(sum(x_1)) = [1] + x_1 ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c5_2, c8_2, c10_2, c_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) by *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0), *'(z0, 0)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0), *'(0, z0)) *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(z0), s(0)) -> c5(+'(0, +(z0, 0)), *'(z0, 0)) *'(s(0), s(z0)) -> c5(+'(0, +(0, z0)), *'(0, z0)) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0), *'(z0, 0)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0), *'(0, z0)) *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(z0), s(0)) -> c5(+'(0, +(z0, 0)), *'(z0, 0)) *'(s(0), s(z0)) -> c5(+'(0, +(0, z0)), *'(0, z0)) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, SUM_1, PROD_1, *'_2 Compound Symbols: c2_1, c8_2, c10_2, c_1, c5_2 ---------------------------------------- (63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: *'(s(0), s(z0)) -> c5(+'(0, +(0, z0)), *'(0, z0)) *'(s(z0), s(0)) -> c5(+'(0, +(z0, 0)), *'(z0, 0)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0), *'(z0, 0)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0), *'(0, z0)) *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, SUM_1, PROD_1, *'_2 Compound Symbols: c2_1, c8_2, c10_2, c_1, c5_2 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1))) *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0)) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, SUM_1, PROD_1, *'_2 Compound Symbols: c2_1, c8_2, c10_2, c_1, c5_2, c5_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) by SUM(cons(x0, nil)) -> c8(+'(x0, 0), SUM(nil)) SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1))) *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0)) SUM(cons(x0, nil)) -> c8(+'(x0, 0), SUM(nil)) SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, PROD_1, *'_2, SUM_1 Compound Symbols: c2_1, c10_2, c_1, c5_2, c5_1, c8_2 ---------------------------------------- (69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SUM(cons(x0, nil)) -> c8(+'(x0, 0), SUM(nil)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1))) *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0)) SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, PROD_1, *'_2, SUM_1 Compound Symbols: c2_1, c10_2, c_1, c5_2, c5_1, c8_2 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) by PROD(cons(x0, nil)) -> c10(*'(x0, s(0)), PROD(nil)) PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1))) *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0)) SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) PROD(cons(x0, nil)) -> c10(*'(x0, s(0)), PROD(nil)) PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c_1, c5_2, c5_1, c8_2, c10_2 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1))) *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0)) SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c_1, c5_2, c5_1, c8_2, c10_2, c10_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) by *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(z0), s(0)) -> c(+'(0, +(z0, 0))) *'(s(0), s(z0)) -> c(+'(0, +(0, z0))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1))) *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0)) SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(z0), s(0)) -> c(+'(0, +(z0, 0))) *'(s(0), s(z0)) -> c(+'(0, +(0, z0))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c_1, c5_2, c5_1, c8_2, c10_2, c10_1 ---------------------------------------- (77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: *'(s(z0), s(0)) -> c(+'(0, +(z0, 0))) *'(s(0), s(z0)) -> c(+'(0, +(0, z0))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1))) *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0)) SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c_1, c5_2, c5_1, c8_2, c10_2, c10_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1))) by *'(s(s(z0)), s(s(0))) -> c5(+'(*(s(z0), s(0)), s(s(z0))), *'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c5(+'(*(s(0), s(z0)), s(s(z0))), *'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1))) *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0)) SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(z0)), s(s(0))) -> c5(+'(*(s(z0), s(0)), s(s(z0))), *'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c5(+'(*(s(0), s(z0)), s(s(z0))), *'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c_1, c5_2, c5_1, c8_2, c10_2, c10_1 ---------------------------------------- (81) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1))) *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0)) SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c_1, c5_2, c5_1, c8_2, c10_2, c10_1, c1_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1))) by *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c5(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))), *'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c5(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))), *'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c5(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))), *'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c5(+'(s(+(0, +(0, z0))), +(s(0), s(z0))), *'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0)) SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c5(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))), *'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c5(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))), *'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c5(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))), *'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c5(+'(s(+(0, +(0, z0))), +(s(0), s(z0))), *'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c_1, c5_1, c8_2, c10_2, c10_1, c5_2, c1_1 ---------------------------------------- (85) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0)) SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c_1, c5_1, c8_2, c10_2, c10_1, c5_2, c1_1, c3_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0)) by *'(s(z0), s(0)) -> c5(+'(0, z0)) *'(s(0), s(0)) -> c5(+'(0, 0)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0)) SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) *'(s(z0), s(0)) -> c5(+'(0, z0)) *'(s(0), s(0)) -> c5(+'(0, 0)) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c_1, c5_1, c8_2, c10_2, c10_1, c5_2, c1_1, c3_1 ---------------------------------------- (89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: *'(s(0), s(0)) -> c5(+'(0, 0)) *'(s(z0), s(0)) -> c5(+'(0, z0)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0)) SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c_1, c5_1, c8_2, c10_2, c10_1, c5_2, c1_1, c3_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0)) by *'(s(0), s(0)) -> c5(+'(0, 0)) *'(s(0), s(z0)) -> c5(+'(0, z0)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) *'(s(0), s(0)) -> c5(+'(0, 0)) *'(s(0), s(z0)) -> c5(+'(0, z0)) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c_1, c8_2, c10_2, c10_1, c5_2, c1_1, c3_1, c5_1 ---------------------------------------- (93) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: *'(s(0), s(z0)) -> c5(+'(0, z0)) *'(s(0), s(0)) -> c5(+'(0, 0)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, SUM_1, PROD_1 Compound Symbols: c2_1, c_1, c8_2, c10_2, c10_1, c5_2, c1_1, c3_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) by SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)), SUM(cons(x1, nil))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)), SUM(cons(x1, nil))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, PROD_1, SUM_1 Compound Symbols: c2_1, c_1, c10_2, c10_1, c5_2, c1_1, c3_1, c8_2 ---------------------------------------- (97) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, PROD_1, SUM_1 Compound Symbols: c2_1, c_1, c10_2, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) by PROD(cons(x0, cons(0, x2))) -> c10(*'(x0, 0), PROD(cons(0, x2))) PROD(cons(x0, cons(x1, nil))) -> c10(*'(x0, *(x1, s(0))), PROD(cons(x1, nil))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(0, x2))) -> c10(*'(x0, 0), PROD(cons(0, x2))) PROD(cons(x0, cons(x1, nil))) -> c10(*'(x0, *(x1, s(0))), PROD(cons(x1, nil))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, PROD_1, SUM_1 Compound Symbols: c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2 ---------------------------------------- (101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(x1, nil))) -> c10(*'(x0, *(x1, s(0))), PROD(cons(x1, nil))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, PROD_1, SUM_1 Compound Symbols: c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2 ---------------------------------------- (103) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2))) PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0)))) PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, PROD_1, SUM_1 Compound Symbols: c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) by *'(s(z0), s(0)) -> c(+'(0, z0)) *'(s(0), s(0)) -> c(+'(0, 0)) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2))) PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0)))) PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil))) *'(s(z0), s(0)) -> c(+'(0, z0)) *'(s(0), s(0)) -> c(+'(0, 0)) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, PROD_1, SUM_1 Compound Symbols: c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1 ---------------------------------------- (107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: *'(s(z0), s(0)) -> c(+'(0, z0)) *'(s(0), s(0)) -> c(+'(0, 0)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2))) PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0)))) PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, PROD_1, SUM_1 Compound Symbols: c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) by *'(s(0), s(0)) -> c(+'(0, 0)) *'(s(0), s(z0)) -> c(+'(0, z0)) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2))) PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0)))) PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil))) *'(s(0), s(0)) -> c(+'(0, 0)) *'(s(0), s(z0)) -> c(+'(0, z0)) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, PROD_1, SUM_1 Compound Symbols: c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1 ---------------------------------------- (111) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: *'(s(0), s(z0)) -> c(+'(0, z0)) *'(s(0), s(0)) -> c(+'(0, 0)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2))) PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0)))) PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, PROD_1, SUM_1 Compound Symbols: c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) by *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1))))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1))))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2))) PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0)))) PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil))) *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1))))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1))))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, PROD_1, SUM_1 Compound Symbols: c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) by *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1))))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2))) PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0)))) PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil))) *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1))))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1))))) S tuples: +'(s(z0), s(z1)) -> c2(+'(z0, z1)) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: +'_2, *'_2, PROD_1, SUM_1 Compound Symbols: c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1 ---------------------------------------- (117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(s(z0), s(z1)) -> c2(+'(z0, z1)) by +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2))) PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0)))) PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil))) *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1))))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1))))) +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1))) S tuples: +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1))) K tuples: *'(s(z0), s(z1)) -> c(+'(z0, z1)) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: *'_2, PROD_1, SUM_1, +'_2 Compound Symbols: c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1, c2_1 ---------------------------------------- (119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace *'(s(z0), s(z1)) -> c(+'(z0, z1)) by *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1)))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2))) PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0)))) PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil))) *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1))))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1))))) +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1))) *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1)))) S tuples: +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1))) K tuples: *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1)))) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: PROD_1, *'_2, SUM_1, +'_2 Compound Symbols: c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1, c_1, c2_1 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: PROD(cons(x0, nil)) -> c10(*'(x0, s(0))) *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0))) *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0))) *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0))) PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil))) *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2))) PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0)))) *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1))))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1))))) +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1))) *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1)))) S tuples: +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1))) K tuples: *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1)))) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: *'_2, SUM_1, PROD_1, +'_2 Compound Symbols: c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c10_1, c4_1, c_1, c2_1 ---------------------------------------- (123) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2))) by PROD(cons(z0, cons(0, cons(y1, cons(y2, y3))))) -> c10(PROD(cons(0, cons(y1, cons(y2, y3))))) PROD(cons(z0, cons(0, cons(0, y1)))) -> c10(PROD(cons(0, cons(0, y1)))) PROD(cons(z0, cons(0, cons(y1, nil)))) -> c10(PROD(cons(0, cons(y1, nil)))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0)))) *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1))))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1))))) +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1))) *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1)))) PROD(cons(z0, cons(0, cons(y1, cons(y2, y3))))) -> c10(PROD(cons(0, cons(y1, cons(y2, y3))))) PROD(cons(z0, cons(0, cons(0, y1)))) -> c10(PROD(cons(0, cons(0, y1)))) PROD(cons(z0, cons(0, cons(y1, nil)))) -> c10(PROD(cons(0, cons(y1, nil)))) S tuples: +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1))) K tuples: *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1)))) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: *'_2, SUM_1, PROD_1, +'_2 Compound Symbols: c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1, c_1, c2_1, c10_1 ---------------------------------------- (125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1))) by +'(s(s(s(y0))), s(s(s(y1)))) -> c2(+'(s(s(y0)), s(s(y1)))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0)))) *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1))))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1))))) *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1)))) PROD(cons(z0, cons(0, cons(y1, cons(y2, y3))))) -> c10(PROD(cons(0, cons(y1, cons(y2, y3))))) PROD(cons(z0, cons(0, cons(0, y1)))) -> c10(PROD(cons(0, cons(0, y1)))) PROD(cons(z0, cons(0, cons(y1, nil)))) -> c10(PROD(cons(0, cons(y1, nil)))) +'(s(s(s(y0))), s(s(s(y1)))) -> c2(+'(s(s(y0)), s(s(y1)))) S tuples: +'(s(s(s(y0))), s(s(s(y1)))) -> c2(+'(s(s(y0)), s(s(y1)))) K tuples: *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1)))) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: *'_2, SUM_1, PROD_1, +'_2 Compound Symbols: c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1, c_1, c10_1, c2_1 ---------------------------------------- (127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1)))) by *'(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c(+'(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0)))) *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1))))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1))))) PROD(cons(z0, cons(0, cons(y1, cons(y2, y3))))) -> c10(PROD(cons(0, cons(y1, cons(y2, y3))))) PROD(cons(z0, cons(0, cons(0, y1)))) -> c10(PROD(cons(0, cons(0, y1)))) PROD(cons(z0, cons(0, cons(y1, nil)))) -> c10(PROD(cons(0, cons(y1, nil)))) +'(s(s(s(y0))), s(s(s(y1)))) -> c2(+'(s(s(y0)), s(s(y1)))) *'(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c(+'(s(s(s(y0))), s(s(s(y1))))) S tuples: +'(s(s(s(y0))), s(s(s(y1)))) -> c2(+'(s(s(y0)), s(s(y1)))) K tuples: *'(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c(+'(s(s(s(y0))), s(s(s(y1))))) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: *'_2, SUM_1, PROD_1, +'_2 Compound Symbols: c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1, c_1, c10_1, c2_1 ---------------------------------------- (129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PROD(cons(z0, cons(0, cons(0, y1)))) -> c10(PROD(cons(0, cons(0, y1)))) by PROD(cons(z0, cons(0, cons(0, cons(y2, y3))))) -> c10(PROD(cons(0, cons(0, cons(y2, y3))))) PROD(cons(z0, cons(0, cons(0, nil)))) -> c10(PROD(cons(0, cons(0, nil)))) PROD(cons(z0, cons(0, cons(0, cons(y1, cons(y2, y3)))))) -> c10(PROD(cons(0, cons(0, cons(y1, cons(y2, y3)))))) PROD(cons(z0, cons(0, cons(0, cons(0, y1))))) -> c10(PROD(cons(0, cons(0, cons(0, y1))))) PROD(cons(z0, cons(0, cons(0, cons(y1, nil))))) -> c10(PROD(cons(0, cons(0, cons(y1, nil))))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(0, z0) -> z0 +(s(z0), s(z1)) -> s(s(+(z0, z1))) *(z0, 0) -> 0 *(0, z0) -> 0 *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1))) sum(nil) -> 0 sum(cons(z0, z1)) -> +(z0, sum(z1)) prod(nil) -> s(0) prod(cons(z0, z1)) -> *(z0, prod(z1)) Tuples: *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1))) *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2))) SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1)))) SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0))) PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1)))) PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0)))) *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0)))) *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1))))))) *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1))))) *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0)))) *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0)))) *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1))))) PROD(cons(z0, cons(0, cons(y1, cons(y2, y3))))) -> c10(PROD(cons(0, cons(y1, cons(y2, y3))))) PROD(cons(z0, cons(0, cons(y1, nil)))) -> c10(PROD(cons(0, cons(y1, nil)))) +'(s(s(s(y0))), s(s(s(y1)))) -> c2(+'(s(s(y0)), s(s(y1)))) *'(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c(+'(s(s(s(y0))), s(s(s(y1))))) PROD(cons(z0, cons(0, cons(0, cons(y2, y3))))) -> c10(PROD(cons(0, cons(0, cons(y2, y3))))) PROD(cons(z0, cons(0, cons(0, nil)))) -> c10(PROD(cons(0, cons(0, nil)))) PROD(cons(z0, cons(0, cons(0, cons(y1, cons(y2, y3)))))) -> c10(PROD(cons(0, cons(0, cons(y1, cons(y2, y3)))))) PROD(cons(z0, cons(0, cons(0, cons(0, y1))))) -> c10(PROD(cons(0, cons(0, cons(0, y1))))) PROD(cons(z0, cons(0, cons(0, cons(y1, nil))))) -> c10(PROD(cons(0, cons(0, cons(y1, nil))))) S tuples: +'(s(s(s(y0))), s(s(s(y1)))) -> c2(+'(s(s(y0)), s(s(y1)))) K tuples: *'(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c(+'(s(s(s(y0))), s(s(s(y1))))) Defined Rule Symbols: +_2, *_2, sum_1, prod_1 Defined Pair Symbols: *'_2, SUM_1, PROD_1, +'_2 Compound Symbols: c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1, c_1, c10_1, c2_1