KILLED proof of input_9yM2F4QxEd.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) CompletionProof [UPPER BOUND(ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxTypedWeightedCompleteTrs (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 282 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 101 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 85 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 74 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 304 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 73 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 438 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 6 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 862 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 134 ms] (52) CpxRNTS (53) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (54) CdtProblem (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 114 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 18 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 14 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 24 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 33 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 1 ms] (130) CdtProblem (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (174) 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: terms(N) -> cons(recip(sqr(N)), terms(s(N))) sqr(0) -> 0 sqr(s(X)) -> s(add(sqr(X), dbl(X))) dbl(0) -> 0 dbl(s(X)) -> s(s(dbl(X))) add(0, X) -> X add(s(X), Y) -> s(add(X, Y)) first(0, X) -> nil first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: terms(N) -> cons(recip(sqr(N)), terms(s(N))) sqr(0') -> 0' sqr(s(X)) -> s(add(sqr(X), dbl(X))) dbl(0') -> 0' dbl(s(X)) -> s(s(dbl(X))) add(0', X) -> X add(s(X), Y) -> s(add(X, Y)) first(0', X) -> nil first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: terms(N) -> cons(recip(sqr(N)), terms(s(N))) sqr(0) -> 0 sqr(s(X)) -> s(add(sqr(X), dbl(X))) dbl(0) -> 0 dbl(s(X)) -> s(s(dbl(X))) add(0, X) -> X add(s(X), Y) -> s(add(X, Y)) first(0, X) -> nil first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: terms(N) -> cons(recip(sqr(N)), terms(s(N))) [1] sqr(0) -> 0 [1] sqr(s(X)) -> s(add(sqr(X), dbl(X))) [1] dbl(0) -> 0 [1] dbl(s(X)) -> s(s(dbl(X))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] first(0, X) -> nil [1] first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: terms(N) -> cons(recip(sqr(N)), terms(s(N))) [1] sqr(0) -> 0 [1] sqr(s(X)) -> s(add(sqr(X), dbl(X))) [1] dbl(0) -> 0 [1] dbl(s(X)) -> s(s(dbl(X))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] first(0, X) -> nil [1] first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1] The TRS has the following type information: terms :: s:0 -> cons:nil cons :: recip -> cons:nil -> cons:nil recip :: s:0 -> recip sqr :: s:0 -> s:0 s :: s:0 -> s:0 0 :: s:0 add :: s:0 -> s:0 -> s:0 dbl :: s:0 -> s:0 first :: s:0 -> cons:nil -> cons:nil nil :: cons:nil Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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: first(v0, v1) -> null_first [0] And the following fresh constants: null_first, const ---------------------------------------- (10) 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: terms(N) -> cons(recip(sqr(N)), terms(s(N))) [1] sqr(0) -> 0 [1] sqr(s(X)) -> s(add(sqr(X), dbl(X))) [1] dbl(0) -> 0 [1] dbl(s(X)) -> s(s(dbl(X))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] first(0, X) -> nil [1] first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1] first(v0, v1) -> null_first [0] The TRS has the following type information: terms :: s:0 -> cons:nil:null_first cons :: recip -> cons:nil:null_first -> cons:nil:null_first recip :: s:0 -> recip sqr :: s:0 -> s:0 s :: s:0 -> s:0 0 :: s:0 add :: s:0 -> s:0 -> s:0 dbl :: s:0 -> s:0 first :: s:0 -> cons:nil:null_first -> cons:nil:null_first nil :: cons:nil:null_first null_first :: cons:nil:null_first const :: recip Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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 null_first => 0 const => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(X)) :|: z = 1 + X, X >= 0 first(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0 first(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 first(z, z') -{ 1 }-> 1 + Y + first(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 1 }-> 1 + add(sqr(X), dbl(X)) :|: z = 1 + X, X >= 0 terms(z) -{ 1 }-> 1 + (1 + sqr(N)) + terms(1 + N) :|: z = N, N >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (13) 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: terms_1 first_2 (c) The following functions are completely defined: sqr_1 dbl_1 add_2 Due to the following rules being added: none And the following fresh constants: const ---------------------------------------- (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: terms(N) -> cons(recip(sqr(N)), terms(s(N))) [1] sqr(0) -> 0 [1] sqr(s(X)) -> s(add(sqr(X), dbl(X))) [1] dbl(0) -> 0 [1] dbl(s(X)) -> s(s(dbl(X))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] first(0, X) -> nil [1] first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1] The TRS has the following type information: terms :: s:0 -> cons:nil cons :: recip -> cons:nil -> cons:nil recip :: s:0 -> recip sqr :: s:0 -> s:0 s :: s:0 -> s:0 0 :: s:0 add :: s:0 -> s:0 -> s:0 dbl :: s:0 -> s:0 first :: s:0 -> cons:nil -> cons:nil nil :: cons:nil const :: recip Rewrite Strategy: INNERMOST ---------------------------------------- (15) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (16) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: terms(N) -> cons(recip(sqr(N)), terms(s(N))) [1] sqr(0) -> 0 [1] sqr(s(0)) -> s(add(0, 0)) [3] sqr(s(s(X'))) -> s(add(s(add(sqr(X'), dbl(X'))), s(s(dbl(X'))))) [3] dbl(0) -> 0 [1] dbl(s(X)) -> s(s(dbl(X))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] first(0, X) -> nil [1] first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1] The TRS has the following type information: terms :: s:0 -> cons:nil cons :: recip -> cons:nil -> cons:nil recip :: s:0 -> recip sqr :: s:0 -> s:0 s :: s:0 -> s:0 0 :: s:0 add :: s:0 -> s:0 -> s:0 dbl :: s:0 -> s:0 first :: s:0 -> cons:nil -> cons:nil nil :: cons:nil const :: recip Rewrite Strategy: INNERMOST ---------------------------------------- (17) 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 const => 0 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(X)) :|: z = 1 + X, X >= 0 first(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0 sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(X'), dbl(X')), 1 + (1 + dbl(X'))) :|: X' >= 0, z = 1 + (1 + X') terms(z) -{ 1 }-> 1 + (1 + sqr(N)) + terms(1 + N) :|: z = N, N >= 0 ---------------------------------------- (19) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0 sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0 terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0 ---------------------------------------- (21) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { first } { dbl } { add } { sqr } { terms } ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0 sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0 terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0 Function symbols to be analyzed: {first}, {dbl}, {add}, {sqr}, {terms} ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0 sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0 terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0 Function symbols to be analyzed: {first}, {dbl}, {add}, {sqr}, {terms} ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: first after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0 sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0 terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0 Function symbols to be analyzed: {first}, {dbl}, {add}, {sqr}, {terms} Previous analysis results are: first: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: first after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0 sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0 terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0 Function symbols to be analyzed: {dbl}, {add}, {sqr}, {terms} Previous analysis results are: first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0 sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0 terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0 Function symbols to be analyzed: {dbl}, {add}, {sqr}, {terms} Previous analysis results are: first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: dbl after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0 sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0 terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0 Function symbols to be analyzed: {dbl}, {add}, {sqr}, {terms} Previous analysis results are: first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] dbl: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: dbl after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0 sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0 terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0 Function symbols to be analyzed: {add}, {sqr}, {terms} Previous analysis results are: first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (35) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0 sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0 terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0 Function symbols to be analyzed: {add}, {sqr}, {terms} Previous analysis results are: first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: add after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0 sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0 terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0 Function symbols to be analyzed: {add}, {sqr}, {terms} Previous analysis results are: first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] add: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: add after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0 sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0 terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0 Function symbols to be analyzed: {sqr}, {terms} Previous analysis results are: first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (41) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 4 }-> 1 + s2 :|: s2 >= 0, s2 <= 0 + 0, z = 1 + 0 sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0 terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0 Function symbols to be analyzed: {sqr}, {terms} Previous analysis results are: first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: sqr after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + 4*z + 4*z^2 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 4 }-> 1 + s2 :|: s2 >= 0, s2 <= 0 + 0, z = 1 + 0 sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0 terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0 Function symbols to be analyzed: {sqr}, {terms} Previous analysis results are: first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] sqr: runtime: ?, size: O(n^2) [1 + 4*z + 4*z^2] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: sqr after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 5 + 22*z + 8*z^3 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 4 }-> 1 + s2 :|: s2 >= 0, s2 <= 0 + 0, z = 1 + 0 sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s'), 1 + (1 + s'')) :|: s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0 terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + terms(1 + z) :|: z >= 0 Function symbols to be analyzed: {terms} Previous analysis results are: first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2] ---------------------------------------- (47) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 4 }-> 1 + s2 :|: s2 >= 0, s2 <= 0 + 0, z = 1 + 0 sqr(z) -{ -99 + s5 + s6 + 120*z + -48*z^2 + 8*z^3 }-> 1 + s7 :|: s5 >= 0, s5 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s6 >= 0, s6 <= s5 + s', s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s'')), s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0 terms(z) -{ 6 + 22*z + 8*z^3 }-> 1 + (1 + s4) + terms(1 + z) :|: s4 >= 0, s4 <= 4 * (z * z) + 4 * z + 1, z >= 0 Function symbols to be analyzed: {terms} Previous analysis results are: first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: terms after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 4 }-> 1 + s2 :|: s2 >= 0, s2 <= 0 + 0, z = 1 + 0 sqr(z) -{ -99 + s5 + s6 + 120*z + -48*z^2 + 8*z^3 }-> 1 + s7 :|: s5 >= 0, s5 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s6 >= 0, s6 <= s5 + s', s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s'')), s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0 terms(z) -{ 6 + 22*z + 8*z^3 }-> 1 + (1 + s4) + terms(1 + z) :|: s4 >= 0, s4 <= 4 * (z * z) + 4 * z + 1, z >= 0 Function symbols to be analyzed: {terms} Previous analysis results are: first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2] terms: runtime: ?, size: O(1) [0] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: terms after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z' >= 0, z - 1 >= 0 dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0 first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 + z }-> 1 + Y + s :|: s >= 0, s <= Z, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z sqr(z) -{ 1 }-> 0 :|: z = 0 sqr(z) -{ 4 }-> 1 + s2 :|: s2 >= 0, s2 <= 0 + 0, z = 1 + 0 sqr(z) -{ -99 + s5 + s6 + 120*z + -48*z^2 + 8*z^3 }-> 1 + s7 :|: s5 >= 0, s5 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s6 >= 0, s6 <= s5 + s', s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s'')), s' >= 0, s' <= 2 * (z - 2), s'' >= 0, s'' <= 2 * (z - 2), z - 2 >= 0 terms(z) -{ 6 + 22*z + 8*z^3 }-> 1 + (1 + s4) + terms(1 + z) :|: s4 >= 0, s4 <= 4 * (z * z) + 4 * z + 1, z >= 0 Function symbols to be analyzed: {terms} Previous analysis results are: first: runtime: O(n^1) [1 + z], size: O(n^1) [z'] dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2] terms: runtime: INF, size: O(1) [0] ---------------------------------------- (53) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: terms(z0) -> cons(recip(sqr(z0)), terms(s(z0))) sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) first(0, z0) -> nil first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2)) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) SQR(0) -> c2 SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0)) DBL(0) -> c5 DBL(s(z0)) -> c6(DBL(z0)) ADD(0, z0) -> c7 ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(0, z0) -> c9 FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) S tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) SQR(0) -> c2 SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0)) DBL(0) -> c5 DBL(s(z0)) -> c6(DBL(z0)) ADD(0, z0) -> c7 ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(0, z0) -> c9 FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) K tuples:none Defined Rule Symbols: terms_1, sqr_1, dbl_1, add_2, first_2 Defined Pair Symbols: TERMS_1, SQR_1, DBL_1, ADD_2, FIRST_2 Compound Symbols: c_1, c1_1, c2, c3_2, c4_2, c5, c6_1, c7, c8_1, c9, c10_1 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: ADD(0, z0) -> c7 SQR(0) -> c2 FIRST(0, z0) -> c9 DBL(0) -> c5 ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: terms(z0) -> cons(recip(sqr(z0)), terms(s(z0))) sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) first(0, z0) -> nil first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2)) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0)) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) S tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) SQR(s(z0)) -> c4(ADD(sqr(z0), dbl(z0)), DBL(z0)) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) K tuples:none Defined Rule Symbols: terms_1, sqr_1, dbl_1, add_2, first_2 Defined Pair Symbols: TERMS_1, SQR_1, DBL_1, ADD_2, FIRST_2 Compound Symbols: c_1, c1_1, c3_2, c4_2, c6_1, c8_1, c10_1 ---------------------------------------- (57) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: terms(z0) -> cons(recip(sqr(z0)), terms(s(z0))) sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) first(0, z0) -> nil first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2)) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) S tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) K tuples:none Defined Rule Symbols: terms_1, sqr_1, dbl_1, add_2, first_2 Defined Pair Symbols: TERMS_1, SQR_1, DBL_1, ADD_2, FIRST_2 Compound Symbols: c_1, c1_1, c3_2, c6_1, c8_1, c10_1, c2_1 ---------------------------------------- (59) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: terms(z0) -> cons(recip(sqr(z0)), terms(s(z0))) first(0, z0) -> nil first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) S tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) K tuples:none Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, SQR_1, DBL_1, ADD_2, FIRST_2 Compound Symbols: c_1, c1_1, c3_2, c6_1, c8_1, c10_1, c2_1 ---------------------------------------- (61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) We considered the (Usable) Rules:none And the Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ADD(x_1, x_2)) = 0 POL(DBL(x_1)) = 0 POL(FIRST(x_1, x_2)) = x_2 POL(SQR(x_1)) = 0 POL(TERMS(x_1)) = [1] POL(add(x_1, x_2)) = [1] + [3]x_1 + [2]x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(dbl(x_1)) = 0 POL(s(x_1)) = [3] + x_1 POL(sqr(x_1)) = [3] + [3]x_1 ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, SQR_1, DBL_1, ADD_2, FIRST_2 Compound Symbols: c_1, c1_1, c3_2, c6_1, c8_1, c10_1, c2_1 ---------------------------------------- (63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) We considered the (Usable) Rules:none And the Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ADD(x_1, x_2)) = 0 POL(DBL(x_1)) = 0 POL(FIRST(x_1, x_2)) = x_1 + x_2 POL(SQR(x_1)) = [1] POL(TERMS(x_1)) = [1] POL(add(x_1, x_2)) = x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = x_2 POL(dbl(x_1)) = [1] + x_1 POL(s(x_1)) = [1] + x_1 POL(sqr(x_1)) = [1] + x_1 ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, SQR_1, DBL_1, ADD_2, FIRST_2 Compound Symbols: c_1, c1_1, c3_2, c6_1, c8_1, c10_1, c2_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQR(s(z0)) -> c3(ADD(sqr(z0), dbl(z0)), SQR(z0)) by SQR(s(0)) -> c3(ADD(sqr(0), 0), SQR(0)) SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(0)) -> c3(ADD(0, dbl(0)), SQR(0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0), SQR(0)) SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(0)) -> c3(ADD(0, dbl(0)), SQR(0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(0)) -> c3(ADD(sqr(0), 0), SQR(0)) SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(0)) -> c3(ADD(0, dbl(0)), SQR(0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SQR(s(0)) -> c3(ADD(0, dbl(0)), SQR(0)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0), SQR(0)) SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(0)) -> c3(ADD(sqr(0), 0), SQR(0)) SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) SQR(s(0)) -> c3(ADD(sqr(0), 0)) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) SQR(s(0)) -> c3(ADD(sqr(0), 0)) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c3_1 ---------------------------------------- (71) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SQR(s(0)) -> c3(ADD(sqr(0), 0)) We considered the (Usable) Rules: dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) And the Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) SQR(s(0)) -> c3(ADD(sqr(0), 0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ADD(x_1, x_2)) = x_2 POL(DBL(x_1)) = [1] + x_1 POL(FIRST(x_1, x_2)) = x_1 POL(SQR(x_1)) = [1] + x_1 POL(TERMS(x_1)) = [1] + x_1 POL(add(x_1, x_2)) = x_1 + x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = x_2 POL(dbl(x_1)) = 0 POL(s(x_1)) = x_1 POL(sqr(x_1)) = [1] + x_1 ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) SQR(s(0)) -> c3(ADD(sqr(0), 0)) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0)) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c3_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) by SQR(s(0)) -> c2(ADD(sqr(0), 0)) SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) SQR(s(0)) -> c2(ADD(0, dbl(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) SQR(s(0)) -> c3(ADD(sqr(0), 0)) SQR(s(0)) -> c2(ADD(sqr(0), 0)) SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) SQR(s(0)) -> c2(ADD(0, dbl(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(ADD(sqr(z0), dbl(z0))) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0)) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c3_1 ---------------------------------------- (75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SQR(s(0)) -> c2(ADD(0, dbl(0))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) SQR(s(0)) -> c3(ADD(sqr(0), 0)) SQR(s(0)) -> c2(ADD(sqr(0), 0)) SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0)) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c3_1 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQR(s(s(z0))) -> c3(ADD(sqr(s(z0)), s(s(dbl(z0)))), SQR(s(z0))) by SQR(s(s(0))) -> c3(ADD(sqr(s(0)), s(s(0))), SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) SQR(s(0)) -> c3(ADD(sqr(0), 0)) SQR(s(0)) -> c2(ADD(sqr(0), 0)) SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) SQR(s(s(0))) -> c3(ADD(sqr(s(0)), s(s(0))), SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) SQR(s(s(0))) -> c3(ADD(sqr(s(0)), s(s(0))), SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0)) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c3_1 ---------------------------------------- (79) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) SQR(s(0)) -> c3(ADD(sqr(0), 0)) SQR(s(0)) -> c2(ADD(sqr(0), 0)) SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c4(SQR(s(0))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c4(SQR(s(0))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0)) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c3_1, c4_1 ---------------------------------------- (81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) We considered the (Usable) Rules:none And the Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) SQR(s(0)) -> c3(ADD(sqr(0), 0)) SQR(s(0)) -> c2(ADD(sqr(0), 0)) SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c4(SQR(s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ADD(x_1, x_2)) = 0 POL(DBL(x_1)) = [1] POL(FIRST(x_1, x_2)) = 0 POL(SQR(x_1)) = [1] POL(TERMS(x_1)) = [1] POL(add(x_1, x_2)) = [1] + x_1 + x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = x_2 POL(dbl(x_1)) = [1] + x_1 POL(s(x_1)) = [1] POL(sqr(x_1)) = [1] + x_1 ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) SQR(s(0)) -> c3(ADD(sqr(0), 0)) SQR(s(0)) -> c2(ADD(sqr(0), 0)) SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c4(SQR(s(0))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c3_1, c4_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0))), SQR(s(z0))) by SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c3(ADD(s(add(sqr(0), 0)), dbl(s(0))), SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c3(ADD(s(add(0, dbl(0))), dbl(s(0))), SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0)) SQR(s(0)) -> c2(ADD(sqr(0), 0)) SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c3(ADD(s(add(sqr(0), 0)), dbl(s(0))), SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c3(ADD(s(add(0, dbl(0))), dbl(s(0))), SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c3(ADD(s(add(sqr(0), 0)), dbl(s(0))), SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c3(ADD(s(add(0, dbl(0))), dbl(s(0))), SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_1, c3_2, c4_1 ---------------------------------------- (85) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0)) SQR(s(0)) -> c2(ADD(sqr(0), 0)) SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_1, c3_2, c4_1, c5_1 ---------------------------------------- (87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) We considered the (Usable) Rules:none And the Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0)) SQR(s(0)) -> c2(ADD(sqr(0), 0)) SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ADD(x_1, x_2)) = 0 POL(DBL(x_1)) = [1] POL(FIRST(x_1, x_2)) = x_2 POL(SQR(x_1)) = [1] POL(TERMS(x_1)) = [1] POL(add(x_1, x_2)) = [1] + x_1 + x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = x_2 POL(dbl(x_1)) = [1] + x_1 POL(s(x_1)) = [1] POL(sqr(x_1)) = [1] + x_1 ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0)) SQR(s(0)) -> c2(ADD(sqr(0), 0)) SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(SQR(s(0))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_1, c3_2, c4_1, c5_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQR(s(0)) -> c3(ADD(sqr(0), 0)) by SQR(s(0)) -> c3(ADD(0, 0)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c2(ADD(sqr(0), 0)) SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(0)) -> c3(ADD(0, 0)) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(SQR(s(0))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c3(ADD(sqr(0), 0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c3_1 ---------------------------------------- (91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SQR(s(0)) -> c3(ADD(0, 0)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(0)) -> c2(ADD(sqr(0), 0)) SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(SQR(s(0))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQR(s(0)) -> c2(ADD(sqr(0), 0)) by SQR(s(0)) -> c2(ADD(0, 0)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(0)) -> c2(ADD(0, 0)) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(SQR(s(0))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1 ---------------------------------------- (95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SQR(s(0)) -> c2(ADD(0, 0)) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(SQR(s(0))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQR(s(s(z0))) -> c2(ADD(sqr(s(z0)), s(s(dbl(z0))))) by SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0))))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0))))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(SQR(s(0))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))) by SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0))))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(SQR(s(0))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1 ---------------------------------------- (101) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(s(z0)))) -> c3(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) by SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0))))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) by SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0))))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0)))), SQR(s(s(z0)))) by SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0))))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1 ---------------------------------------- (107) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0)))), SQR(s(s(z0)))) by SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0))))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) by SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0))))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1 ---------------------------------------- (111) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) by SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c(SQR(z0)) TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0))))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) K tuples: TERMS(z0) -> c(SQR(z0)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c_1, c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1 ---------------------------------------- (113) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace TERMS(z0) -> c(SQR(z0)) by TERMS(s(x0)) -> c(SQR(s(x0))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0))))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) TERMS(s(x0)) -> c(SQR(s(x0))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(x0)) -> c(SQR(s(x0))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(0))) -> c2(ADD(sqr(s(0)), s(s(0)))) by SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0))))))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) TERMS(s(x0)) -> c(SQR(s(x0))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(x0)) -> c(SQR(s(x0))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(s(z0)))) -> c2(ADD(sqr(s(s(z0))), s(s(s(s(dbl(z0))))))) by SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) TERMS(s(x0)) -> c(SQR(s(x0))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) S tuples: TERMS(z0) -> c1(TERMS(s(z0))) DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(x0)) -> c(SQR(s(x0))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: TERMS_1, DBL_1, ADD_2, FIRST_2, SQR_1 Compound Symbols: c1_1, c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1 ---------------------------------------- (119) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace TERMS(z0) -> c1(TERMS(s(z0))) by TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) TERMS(s(x0)) -> c(SQR(s(x0))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) S tuples: DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(x0)) -> c(SQR(s(x0))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: DBL_1, ADD_2, FIRST_2, SQR_1, TERMS_1 Compound Symbols: c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1, c1_1 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), dbl(s(0)))) by SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) TERMS(s(x0)) -> c(SQR(s(x0))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) S tuples: DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(x0)) -> c(SQR(s(x0))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: DBL_1, ADD_2, FIRST_2, SQR_1, TERMS_1 Compound Symbols: c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1, c1_1 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), dbl(s(s(z0))))) by SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) TERMS(s(x0)) -> c(SQR(s(x0))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) S tuples: DBL(s(z0)) -> c6(DBL(z0)) ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(x0)) -> c(SQR(s(x0))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: DBL_1, ADD_2, FIRST_2, SQR_1, TERMS_1 Compound Symbols: c6_1, c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1, c1_1 ---------------------------------------- (125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DBL(s(z0)) -> c6(DBL(z0)) by DBL(s(s(y0))) -> c6(DBL(s(y0))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) TERMS(s(x0)) -> c(SQR(s(x0))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) S tuples: ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) DBL(s(s(y0))) -> c6(DBL(s(y0))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(x0)) -> c(SQR(s(x0))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: ADD_2, FIRST_2, SQR_1, TERMS_1, DBL_1 Compound Symbols: c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1, c1_1, c6_1 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), dbl(s(0)))) by SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) TERMS(s(x0)) -> c(SQR(s(x0))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) S tuples: ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) DBL(s(s(y0))) -> c6(DBL(s(y0))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(x0)) -> c(SQR(s(x0))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: ADD_2, FIRST_2, SQR_1, TERMS_1, DBL_1 Compound Symbols: c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1, c1_1, c6_1 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), dbl(s(s(z0))))) by SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) TERMS(s(x0)) -> c(SQR(s(x0))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) S tuples: ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) DBL(s(s(y0))) -> c6(DBL(s(y0))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(x0)) -> c(SQR(s(x0))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: ADD_2, FIRST_2, SQR_1, TERMS_1, DBL_1 Compound Symbols: c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1, c1_1, c6_1 ---------------------------------------- (131) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) by SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) TERMS(s(x0)) -> c(SQR(s(x0))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) S tuples: ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(x0)) -> c(SQR(s(x0))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: ADD_2, FIRST_2, SQR_1, TERMS_1, DBL_1 Compound Symbols: c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c_1, c1_1, c6_1 ---------------------------------------- (133) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace TERMS(s(x0)) -> c(SQR(s(x0))) by TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) S tuples: ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: ADD_2, FIRST_2, SQR_1, TERMS_1, DBL_1 Compound Symbols: c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c1_1, c6_1, c_1 ---------------------------------------- (135) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace TERMS(s(x0)) -> c1(TERMS(s(s(x0)))) by TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: ADD(s(z0), z1) -> c8(ADD(z0, z1)) FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) S tuples: ADD(s(z0), z1) -> c8(ADD(z0, z1)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: ADD_2, FIRST_2, SQR_1, DBL_1, TERMS_1 Compound Symbols: c8_1, c10_1, c2_1, c3_2, c4_1, c5_1, c6_1, c_1, c1_1 ---------------------------------------- (137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ADD(s(z0), z1) -> c8(ADD(z0, z1)) by ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) K tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: FIRST_2, SQR_1, DBL_1, TERMS_1, ADD_2 Compound Symbols: c10_1, c2_1, c3_2, c4_1, c5_1, c6_1, c_1, c1_1, c8_1 ---------------------------------------- (139) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) by SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) K tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: FIRST_2, SQR_1, DBL_1, TERMS_1, ADD_2 Compound Symbols: c10_1, c2_1, c3_2, c4_1, c5_1, c6_1, c_1, c1_1, c8_1 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) by SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) K tuples: FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: FIRST_2, SQR_1, DBL_1, TERMS_1, ADD_2 Compound Symbols: c10_1, c2_1, c3_2, c4_1, c5_1, c6_1, c_1, c1_1, c8_1 ---------------------------------------- (143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FIRST(s(z0), cons(z1, z2)) -> c10(FIRST(z0, z2)) by FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) K tuples: SQR(s(z0)) -> c2(DBL(z0)) SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2 Compound Symbols: c2_1, c3_2, c4_1, c5_1, c6_1, c_1, c1_1, c8_1, c10_1 ---------------------------------------- (145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SQR(s(z0)) -> c2(DBL(z0)) by SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(0))) -> c4(SQR(s(0))) SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) K tuples: SQR(s(s(0))) -> c4(ADD(sqr(s(0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), dbl(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), dbl(s(0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2 Compound Symbols: c3_2, c4_1, c5_1, c2_1, c6_1, c_1, c1_1, c8_1, c10_1 ---------------------------------------- (147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: SQR(s(s(0))) -> c5(SQR(s(0))) SQR(s(s(0))) -> c4(SQR(s(0))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) K tuples: TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2 Compound Symbols: c3_2, c2_1, c5_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0))))), SQR(s(s(z0)))) by SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) K tuples: TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2 Compound Symbols: c3_2, c2_1, c5_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) by SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0)))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0)))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) K tuples: TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2 Compound Symbols: c3_2, c2_1, c5_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1 ---------------------------------------- (153) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) by SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0)))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0)))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) K tuples: TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2 Compound Symbols: c3_2, c2_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1 ---------------------------------------- (155) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), dbl(0))), s(s(0)))) by SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0)))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0)))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) K tuples: TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2 Compound Symbols: c3_2, c2_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) by SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0))))))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0))))))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) K tuples: TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2 Compound Symbols: c3_2, c2_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1 ---------------------------------------- (159) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(dbl(0))))) by SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0)))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0))))))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) K tuples: TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2 Compound Symbols: c3_2, c2_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(dbl(s(z0)))))) by SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0))))))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0))))))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) K tuples: TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2 Compound Symbols: c3_2, c2_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1 ---------------------------------------- (163) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(dbl(0))))) by SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(0)))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(0)))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) K tuples: TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2 Compound Symbols: c3_2, c2_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1 ---------------------------------------- (165) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(dbl(s(z0)))))) by SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) DBL(s(s(y0))) -> c6(DBL(s(y0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) K tuples: TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, DBL_1, TERMS_1, ADD_2, FIRST_2 Compound Symbols: c3_2, c2_1, c6_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1 ---------------------------------------- (167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DBL(s(s(y0))) -> c6(DBL(s(y0))) by DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0)))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0)))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0)))) K tuples: TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, TERMS_1, ADD_2, FIRST_2, DBL_1 Compound Symbols: c3_2, c2_1, c_1, c1_1, c8_1, c4_1, c10_1, c5_1, c6_1 ---------------------------------------- (169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ADD(s(s(y0)), z1) -> c8(ADD(s(y0), z1)) by ADD(s(s(s(y0))), z1) -> c8(ADD(s(s(y0)), z1)) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0)))) ADD(s(s(s(y0))), z1) -> c8(ADD(s(s(y0)), z1)) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0)))) ADD(s(s(s(y0))), z1) -> c8(ADD(s(s(y0)), z1)) K tuples: TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, TERMS_1, FIRST_2, DBL_1, ADD_2 Compound Symbols: c3_2, c2_1, c_1, c1_1, c4_1, c10_1, c5_1, c6_1, c8_1 ---------------------------------------- (171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c10(FIRST(s(y0), cons(y1, y2))) by FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c10(FIRST(s(s(y0)), cons(z2, cons(y2, y3)))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0)))) ADD(s(s(s(y0))), z1) -> c8(ADD(s(s(y0)), z1)) FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c10(FIRST(s(s(y0)), cons(z2, cons(y2, y3)))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0)))) ADD(s(s(s(y0))), z1) -> c8(ADD(s(s(y0)), z1)) K tuples: TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c10(FIRST(s(s(y0)), cons(z2, cons(y2, y3)))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, TERMS_1, DBL_1, ADD_2, FIRST_2 Compound Symbols: c3_2, c2_1, c_1, c1_1, c4_1, c5_1, c6_1, c8_1, c10_1 ---------------------------------------- (173) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SQR(s(s(s(y0)))) -> c2(DBL(s(s(y0)))) by SQR(s(s(s(s(y0))))) -> c2(DBL(s(s(s(y0))))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: sqr(0) -> 0 sqr(s(z0)) -> s(add(sqr(z0), dbl(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) Tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(z0))) -> c2(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) SQR(s(s(0))) -> c4(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) SQR(s(s(0))) -> c5(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(0))) -> c5(ADD(s(add(0, dbl(0))), s(s(0)))) SQR(s(s(0))) -> c2(ADD(s(add(sqr(0), 0)), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0))))))) SQR(s(s(0))) -> c2(ADD(s(add(0, dbl(0))), s(s(0)))) SQR(s(s(s(z0)))) -> c2(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0))))))) DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0)))) ADD(s(s(s(y0))), z1) -> c8(ADD(s(s(y0)), z1)) FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c10(FIRST(s(s(y0)), cons(z2, cons(y2, y3)))) SQR(s(s(s(s(y0))))) -> c2(DBL(s(s(s(y0))))) S tuples: SQR(s(s(z0))) -> c3(ADD(s(add(sqr(z0), dbl(z0))), s(s(dbl(z0)))), SQR(s(z0))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(sqr(s(z0)), s(s(dbl(z0))))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) TERMS(s(s(x0))) -> c1(TERMS(s(s(s(x0))))) SQR(s(s(s(z0)))) -> c3(ADD(s(add(s(add(sqr(z0), dbl(z0))), dbl(s(z0)))), s(s(s(s(dbl(z0)))))), SQR(s(s(z0)))) DBL(s(s(s(y0)))) -> c6(DBL(s(s(y0)))) ADD(s(s(s(y0))), z1) -> c8(ADD(s(s(y0)), z1)) K tuples: TERMS(s(s(x0))) -> c(SQR(s(s(x0)))) FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c10(FIRST(s(s(y0)), cons(z2, cons(y2, y3)))) SQR(s(s(s(s(y0))))) -> c2(DBL(s(s(s(y0))))) Defined Rule Symbols: sqr_1, add_2, dbl_1 Defined Pair Symbols: SQR_1, TERMS_1, DBL_1, ADD_2, FIRST_2 Compound Symbols: c3_2, c2_1, c_1, c1_1, c4_1, c5_1, c6_1, c8_1, c10_1