KILLED proof of input_QQxqDe2JZM.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) InliningProof [UPPER BOUND(ID), 143 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 11 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 4 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 216 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 35 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 254 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 87 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 257 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 146 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 1370 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 38 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 1520 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 128 ms] (50) CpxRNTS (51) CompletionProof [UPPER BOUND(ID), 0 ms] (52) CpxTypedWeightedCompleteTrs (53) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (54) CpxRNTS (55) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (56) CdtProblem (57) CdtLeafRemovalProof [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)), 174 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 42 ms] (64) CdtProblem (65) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 79 ms] (66) CdtProblem (67) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 23 ms] (92) CdtProblem (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 2 ms] (112) CdtProblem (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 1 ms] (128) CdtProblem (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 10 ms] (130) CdtProblem (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 4 ms] (132) CdtProblem (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 5 ms] (142) CdtProblem (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 29 ms] (146) CdtProblem (147) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 36 ms] (148) CdtProblem (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 13 ms] (150) CdtProblem (151) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 41 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (158) 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: from(X) -> cons(X, n__from(s(X))) sel(0, cons(X, XS)) -> X sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) minus(X, 0) -> 0 minus(s(X), s(Y)) -> minus(X, Y) quot(0, s(Y)) -> 0 quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y))) zWquot(XS, nil) -> nil zWquot(nil, XS) -> nil zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS))) from(X) -> n__from(X) zWquot(X1, X2) -> n__zWquot(X1, X2) activate(n__from(X)) -> from(X) activate(n__zWquot(X1, X2)) -> zWquot(X1, X2) activate(X) -> X 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: from(X) -> cons(X, n__from(s(X))) sel(0', cons(X, XS)) -> X sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) minus(X, 0') -> 0' minus(s(X), s(Y)) -> minus(X, Y) quot(0', s(Y)) -> 0' quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y))) zWquot(XS, nil) -> nil zWquot(nil, XS) -> nil zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS))) from(X) -> n__from(X) zWquot(X1, X2) -> n__zWquot(X1, X2) activate(n__from(X)) -> from(X) activate(n__zWquot(X1, X2)) -> zWquot(X1, X2) activate(X) -> X 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: from(X) -> cons(X, n__from(s(X))) sel(0, cons(X, XS)) -> X sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) minus(X, 0) -> 0 minus(s(X), s(Y)) -> minus(X, Y) quot(0, s(Y)) -> 0 quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y))) zWquot(XS, nil) -> nil zWquot(nil, XS) -> nil zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS))) from(X) -> n__from(X) zWquot(X1, X2) -> n__zWquot(X1, X2) activate(n__from(X)) -> from(X) activate(n__zWquot(X1, X2)) -> zWquot(X1, X2) activate(X) -> X 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: from(X) -> cons(X, n__from(s(X))) [1] sel(0, cons(X, XS)) -> X [1] sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) [1] minus(X, 0) -> 0 [1] minus(s(X), s(Y)) -> minus(X, Y) [1] quot(0, s(Y)) -> 0 [1] quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y))) [1] zWquot(XS, nil) -> nil [1] zWquot(nil, XS) -> nil [1] zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS))) [1] from(X) -> n__from(X) [1] zWquot(X1, X2) -> n__zWquot(X1, X2) [1] activate(n__from(X)) -> from(X) [1] activate(n__zWquot(X1, X2)) -> zWquot(X1, X2) [1] activate(X) -> X [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: from(X) -> cons(X, n__from(s(X))) [1] sel(0, cons(X, XS)) -> X [1] sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) [1] minus(X, 0) -> 0 [1] minus(s(X), s(Y)) -> minus(X, Y) [1] quot(0, s(Y)) -> 0 [1] quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y))) [1] zWquot(XS, nil) -> nil [1] zWquot(nil, XS) -> nil [1] zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS))) [1] from(X) -> n__from(X) [1] zWquot(X1, X2) -> n__zWquot(X1, X2) [1] activate(n__from(X)) -> from(X) [1] activate(n__zWquot(X1, X2)) -> zWquot(X1, X2) [1] activate(X) -> X [1] The TRS has the following type information: from :: s:0 -> n__from:cons:nil:n__zWquot cons :: s:0 -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot n__from :: s:0 -> n__from:cons:nil:n__zWquot s :: s:0 -> s:0 sel :: s:0 -> n__from:cons:nil:n__zWquot -> s:0 0 :: s:0 activate :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot minus :: s:0 -> s:0 -> s:0 quot :: s:0 -> s:0 -> s:0 zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot nil :: n__from:cons:nil:n__zWquot n__zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: sel_2 (c) The following functions are completely defined: minus_2 activate_1 from_1 zWquot_2 quot_2 Due to the following rules being added: minus(v0, v1) -> 0 [0] quot(v0, v1) -> 0 [0] And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: from(X) -> cons(X, n__from(s(X))) [1] sel(0, cons(X, XS)) -> X [1] sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) [1] minus(X, 0) -> 0 [1] minus(s(X), s(Y)) -> minus(X, Y) [1] quot(0, s(Y)) -> 0 [1] quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y))) [1] zWquot(XS, nil) -> nil [1] zWquot(nil, XS) -> nil [1] zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS))) [1] from(X) -> n__from(X) [1] zWquot(X1, X2) -> n__zWquot(X1, X2) [1] activate(n__from(X)) -> from(X) [1] activate(n__zWquot(X1, X2)) -> zWquot(X1, X2) [1] activate(X) -> X [1] minus(v0, v1) -> 0 [0] quot(v0, v1) -> 0 [0] The TRS has the following type information: from :: s:0 -> n__from:cons:nil:n__zWquot cons :: s:0 -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot n__from :: s:0 -> n__from:cons:nil:n__zWquot s :: s:0 -> s:0 sel :: s:0 -> n__from:cons:nil:n__zWquot -> s:0 0 :: s:0 activate :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot minus :: s:0 -> s:0 -> s:0 quot :: s:0 -> s:0 -> s:0 zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot nil :: n__from:cons:nil:n__zWquot n__zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: from(X) -> cons(X, n__from(s(X))) [1] sel(0, cons(X, XS)) -> X [1] sel(s(N), cons(X, n__from(X'))) -> sel(N, from(X')) [2] sel(s(N), cons(X, n__zWquot(X1', X2'))) -> sel(N, zWquot(X1', X2')) [2] sel(s(N), cons(X, XS)) -> sel(N, XS) [2] minus(X, 0) -> 0 [1] minus(s(X), s(Y)) -> minus(X, Y) [1] quot(0, s(Y)) -> 0 [1] quot(s(X), s(0)) -> s(quot(0, s(0))) [2] quot(s(s(X'')), s(s(Y'))) -> s(quot(minus(X'', Y'), s(s(Y')))) [2] quot(s(X), s(Y)) -> s(quot(0, s(Y))) [1] zWquot(XS, nil) -> nil [1] zWquot(nil, XS) -> nil [1] zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS))) [1] from(X) -> n__from(X) [1] zWquot(X1, X2) -> n__zWquot(X1, X2) [1] activate(n__from(X)) -> from(X) [1] activate(n__zWquot(X1, X2)) -> zWquot(X1, X2) [1] activate(X) -> X [1] minus(v0, v1) -> 0 [0] quot(v0, v1) -> 0 [0] The TRS has the following type information: from :: s:0 -> n__from:cons:nil:n__zWquot cons :: s:0 -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot n__from :: s:0 -> n__from:cons:nil:n__zWquot s :: s:0 -> s:0 sel :: s:0 -> n__from:cons:nil:n__zWquot -> s:0 0 :: s:0 activate :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot minus :: s:0 -> s:0 -> s:0 quot :: s:0 -> s:0 -> s:0 zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot nil :: n__from:cons:nil:n__zWquot n__zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 nil => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> from(X) :|: z = 1 + X, X >= 0 from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X from(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X minus(z, z') -{ 1 }-> minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 minus(z, z') -{ 1 }-> 0 :|: X >= 0, z = X, z' = 0 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quot(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0 quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quot(z, z') -{ 2 }-> 1 + quot(minus(X'', Y'), 1 + (1 + Y')) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + (1 + X''), X'' >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 quot(z, z') -{ 2 }-> 1 + quot(0, 1 + 0) :|: z = 1 + X, X >= 0, z' = 1 + 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(N, XS) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0 sel(z, z') -{ 2 }-> sel(N, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, z = 1 + N, X >= 0, z' = 1 + X + (1 + X1' + X2'), N >= 0 sel(z, z') -{ 2 }-> sel(N, from(X')) :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0 zWquot(z, z') -{ 1 }-> 0 :|: z = XS, XS >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z' = XS, z = 0, XS >= 0 zWquot(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 zWquot(z, z') -{ 1 }-> 1 + quot(X, Y) + (1 + activate(XS) + activate(YS)) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X from(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z = 1 + X, X >= 0, X' >= 0, X = X' from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X from(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X minus(z, z') -{ 1 }-> minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 minus(z, z') -{ 1 }-> 0 :|: X >= 0, z = X, z' = 0 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quot(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0 quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quot(z, z') -{ 2 }-> 1 + quot(minus(X'', Y'), 1 + (1 + Y')) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + (1 + X''), X'' >= 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 quot(z, z') -{ 2 }-> 1 + quot(0, 1 + 0) :|: z = 1 + X, X >= 0, z' = 1 + 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(N, XS) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0 sel(z, z') -{ 2 }-> sel(N, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, z = 1 + N, X >= 0, z' = 1 + X + (1 + X1' + X2'), N >= 0 sel(z, z') -{ 3 }-> sel(N, 1 + X'') :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(N, 1 + X'' + (1 + (1 + X''))) :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z = XS, XS >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z' = XS, z = 0, XS >= 0 zWquot(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 zWquot(z, z') -{ 1 }-> 1 + quot(X, Y) + (1 + activate(XS) + activate(YS)) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 2 }-> 1 + quot(0, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + quot(X, Y) + (1 + activate(XS) + activate(YS)) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { from } { minus } { quot } { zWquot, activate } { sel } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 2 }-> 1 + quot(0, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + quot(X, Y) + (1 + activate(XS) + activate(YS)) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 Function symbols to be analyzed: {from}, {minus}, {quot}, {zWquot,activate}, {sel} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 2 }-> 1 + quot(0, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + quot(X, Y) + (1 + activate(XS) + activate(YS)) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 Function symbols to be analyzed: {from}, {minus}, {quot}, {zWquot,activate}, {sel} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: from after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 2*z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 2 }-> 1 + quot(0, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + quot(X, Y) + (1 + activate(XS) + activate(YS)) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 Function symbols to be analyzed: {from}, {minus}, {quot}, {zWquot,activate}, {sel} Previous analysis results are: from: runtime: ?, size: O(n^1) [3 + 2*z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: from after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 2 }-> 1 + quot(0, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + quot(X, Y) + (1 + activate(XS) + activate(YS)) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 Function symbols to be analyzed: {minus}, {quot}, {zWquot,activate}, {sel} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [3 + 2*z] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 2 }-> 1 + quot(0, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + quot(X, Y) + (1 + activate(XS) + activate(YS)) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 Function symbols to be analyzed: {minus}, {quot}, {zWquot,activate}, {sel} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [3 + 2*z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 2 }-> 1 + quot(0, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + quot(X, Y) + (1 + activate(XS) + activate(YS)) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 Function symbols to be analyzed: {minus}, {quot}, {zWquot,activate}, {sel} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [3 + 2*z] minus: runtime: ?, size: O(1) [0] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 2 }-> 1 + quot(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 2 }-> 1 + quot(0, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + quot(X, Y) + (1 + activate(XS) + activate(YS)) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 Function symbols to be analyzed: {quot}, {zWquot,activate}, {sel} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [3 + 2*z] minus: runtime: O(n^1) [1 + z'], size: O(1) [0] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 2 }-> 1 + quot(0, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + quot(X, Y) + (1 + activate(XS) + activate(YS)) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 Function symbols to be analyzed: {quot}, {zWquot,activate}, {sel} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [3 + 2*z] minus: runtime: O(n^1) [1 + z'], size: O(1) [0] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: quot after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 2 }-> 1 + quot(0, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + quot(X, Y) + (1 + activate(XS) + activate(YS)) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 Function symbols to be analyzed: {quot}, {zWquot,activate}, {sel} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [3 + 2*z] minus: runtime: O(n^1) [1 + z'], size: O(1) [0] quot: runtime: ?, size: O(1) [1] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: quot after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z' ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 + z' }-> 1 + quot(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 2 }-> 1 + quot(0, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 1 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 zWquot(z, z') -{ 1 }-> 1 + quot(X, Y) + (1 + activate(XS) + activate(YS)) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 Function symbols to be analyzed: {zWquot,activate}, {sel} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [3 + 2*z] minus: runtime: O(n^1) [1 + z'], size: O(1) [0] quot: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 6 }-> 1 + s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 4 + 2*z' }-> 1 + s1 :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 4 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 4 + Y }-> 1 + s3 + (1 + activate(XS) + activate(YS)) :|: s3 >= 0, s3 <= 1, z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 Function symbols to be analyzed: {zWquot,activate}, {sel} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [3 + 2*z] minus: runtime: O(n^1) [1 + z'], size: O(1) [0] quot: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: zWquot after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 8 + 14*z + 12*z*z' + 6*z^2 + 14*z' + 6*z'^2 Computed SIZE bound using KoAT for: activate after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 12 + 32*z + 24*z^2 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 6 }-> 1 + s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 4 + 2*z' }-> 1 + s1 :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 4 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 4 + Y }-> 1 + s3 + (1 + activate(XS) + activate(YS)) :|: s3 >= 0, s3 <= 1, z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 Function symbols to be analyzed: {zWquot,activate}, {sel} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [3 + 2*z] minus: runtime: O(n^1) [1 + z'], size: O(1) [0] quot: runtime: O(n^1) [3 + z'], size: O(1) [1] zWquot: runtime: ?, size: O(n^2) [8 + 14*z + 12*z*z' + 6*z^2 + 14*z' + 6*z'^2] activate: runtime: ?, size: O(n^2) [12 + 32*z + 24*z^2] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: zWquot after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 11 + 12*z + 2*z*z' + z^2 + 12*z' + z'^2 Computed RUNTIME bound using KoAT for: activate after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 15 + 24*z + 4*z^2 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 6 }-> 1 + s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 4 + 2*z' }-> 1 + s1 :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 4 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, zWquot(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 4 + Y }-> 1 + s3 + (1 + activate(XS) + activate(YS)) :|: s3 >= 0, s3 <= 1, z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 Function symbols to be analyzed: {sel} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [3 + 2*z] minus: runtime: O(n^1) [1 + z'], size: O(1) [0] quot: runtime: O(n^1) [3 + z'], size: O(1) [1] zWquot: runtime: O(n^2) [11 + 12*z + 2*z*z' + z^2 + 12*z' + z'^2], size: O(n^2) [8 + 14*z + 12*z*z' + 6*z^2 + 14*z' + 6*z'^2] activate: runtime: O(n^2) [15 + 24*z + 4*z^2], size: O(n^2) [12 + 32*z + 24*z^2] ---------------------------------------- (45) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 12 + 12*X1 + 2*X1*X2 + X1^2 + 12*X2 + X2^2 }-> s7 :|: s7 >= 0, s7 <= 14 * X1 + 14 * X2 + 6 * (X1 * X1) + 12 * (X2 * X1) + 6 * (X2 * X2) + 8, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 6 }-> 1 + s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 4 + 2*z' }-> 1 + s1 :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 4 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 13 + 12*X1' + 2*X1'*X2' + X1'^2 + 12*X2' + X2'^2 }-> sel(z - 1, s4) :|: s4 >= 0, s4 <= 14 * X1' + 14 * X2' + 6 * (X1' * X1') + 12 * (X2' * X1') + 6 * (X2' * X2') + 8, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 34 + 24*XS + 4*XS^2 + Y + 24*YS + 4*YS^2 }-> 1 + s3 + (1 + s5 + s6) :|: s5 >= 0, s5 <= 32 * XS + 24 * (XS * XS) + 12, s6 >= 0, s6 <= 32 * YS + 24 * (YS * YS) + 12, s3 >= 0, s3 <= 1, z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 Function symbols to be analyzed: {sel} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [3 + 2*z] minus: runtime: O(n^1) [1 + z'], size: O(1) [0] quot: runtime: O(n^1) [3 + z'], size: O(1) [1] zWquot: runtime: O(n^2) [11 + 12*z + 2*z*z' + z^2 + 12*z' + z'^2], size: O(n^2) [8 + 14*z + 12*z*z' + 6*z^2 + 14*z' + 6*z'^2] activate: runtime: O(n^2) [15 + 24*z + 4*z^2], size: O(n^2) [12 + 32*z + 24*z^2] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: sel after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 12 + 12*X1 + 2*X1*X2 + X1^2 + 12*X2 + X2^2 }-> s7 :|: s7 >= 0, s7 <= 14 * X1 + 14 * X2 + 6 * (X1 * X1) + 12 * (X2 * X1) + 6 * (X2 * X2) + 8, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 6 }-> 1 + s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 4 + 2*z' }-> 1 + s1 :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 4 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 13 + 12*X1' + 2*X1'*X2' + X1'^2 + 12*X2' + X2'^2 }-> sel(z - 1, s4) :|: s4 >= 0, s4 <= 14 * X1' + 14 * X2' + 6 * (X1' * X1') + 12 * (X2' * X1') + 6 * (X2' * X2') + 8, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 34 + 24*XS + 4*XS^2 + Y + 24*YS + 4*YS^2 }-> 1 + s3 + (1 + s5 + s6) :|: s5 >= 0, s5 <= 32 * XS + 24 * (XS * XS) + 12, s6 >= 0, s6 <= 32 * YS + 24 * (YS * YS) + 12, s3 >= 0, s3 <= 1, z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 Function symbols to be analyzed: {sel} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [3 + 2*z] minus: runtime: O(n^1) [1 + z'], size: O(1) [0] quot: runtime: O(n^1) [3 + z'], size: O(1) [1] zWquot: runtime: O(n^2) [11 + 12*z + 2*z*z' + z^2 + 12*z' + z'^2], size: O(n^2) [8 + 14*z + 12*z*z' + 6*z^2 + 14*z' + 6*z'^2] activate: runtime: O(n^2) [15 + 24*z + 4*z^2], size: O(n^2) [12 + 32*z + 24*z^2] sel: runtime: ?, size: INF ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: sel after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 12 + 12*X1 + 2*X1*X2 + X1^2 + 12*X2 + X2^2 }-> s7 :|: s7 >= 0, s7 <= 14 * X1 + 14 * X2 + 6 * (X1 * X1) + 12 * (X2 * X1) + 6 * (X2 * X2) + 8, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X' from(z) -{ 1 }-> 1 + z :|: z >= 0 from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quot(z, z') -{ 6 }-> 1 + s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' = 1 + 0 quot(z, z') -{ 4 + 2*z' }-> 1 + s1 :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 quot(z, z') -{ 4 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 13 + 12*X1' + 2*X1'*X2' + X1'^2 + 12*X2' + X2'^2 }-> sel(z - 1, s4) :|: s4 >= 0, s4 <= 14 * X1' + 14 * X2' + 6 * (X1' * X1') + 12 * (X2' * X1') + 6 * (X2' * X2') + 8, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X'' zWquot(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0 zWquot(z, z') -{ 34 + 24*XS + 4*XS^2 + Y + 24*YS + 4*YS^2 }-> 1 + s3 + (1 + s5 + s6) :|: s5 >= 0, s5 <= 32 * XS + 24 * (XS * XS) + 12, s6 >= 0, s6 <= 32 * YS + 24 * (YS * YS) + 12, s3 >= 0, s3 <= 1, z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 zWquot(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 Function symbols to be analyzed: {sel} Previous analysis results are: from: runtime: O(1) [1], size: O(n^1) [3 + 2*z] minus: runtime: O(n^1) [1 + z'], size: O(1) [0] quot: runtime: O(n^1) [3 + z'], size: O(1) [1] zWquot: runtime: O(n^2) [11 + 12*z + 2*z*z' + z^2 + 12*z' + z'^2], size: O(n^2) [8 + 14*z + 12*z*z' + 6*z^2 + 14*z' + 6*z'^2] activate: runtime: O(n^2) [15 + 24*z + 4*z^2], size: O(n^2) [12 + 32*z + 24*z^2] sel: runtime: INF, size: INF ---------------------------------------- (51) 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: sel(v0, v1) -> null_sel [0] minus(v0, v1) -> null_minus [0] quot(v0, v1) -> null_quot [0] And the following fresh constants: null_sel, null_minus, null_quot ---------------------------------------- (52) 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: from(X) -> cons(X, n__from(s(X))) [1] sel(0, cons(X, XS)) -> X [1] sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) [1] minus(X, 0) -> 0 [1] minus(s(X), s(Y)) -> minus(X, Y) [1] quot(0, s(Y)) -> 0 [1] quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y))) [1] zWquot(XS, nil) -> nil [1] zWquot(nil, XS) -> nil [1] zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS))) [1] from(X) -> n__from(X) [1] zWquot(X1, X2) -> n__zWquot(X1, X2) [1] activate(n__from(X)) -> from(X) [1] activate(n__zWquot(X1, X2)) -> zWquot(X1, X2) [1] activate(X) -> X [1] sel(v0, v1) -> null_sel [0] minus(v0, v1) -> null_minus [0] quot(v0, v1) -> null_quot [0] The TRS has the following type information: from :: s:0:null_sel:null_minus:null_quot -> n__from:cons:nil:n__zWquot cons :: s:0:null_sel:null_minus:null_quot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot n__from :: s:0:null_sel:null_minus:null_quot -> n__from:cons:nil:n__zWquot s :: s:0:null_sel:null_minus:null_quot -> s:0:null_sel:null_minus:null_quot sel :: s:0:null_sel:null_minus:null_quot -> n__from:cons:nil:n__zWquot -> s:0:null_sel:null_minus:null_quot 0 :: s:0:null_sel:null_minus:null_quot activate :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot minus :: s:0:null_sel:null_minus:null_quot -> s:0:null_sel:null_minus:null_quot -> s:0:null_sel:null_minus:null_quot quot :: s:0:null_sel:null_minus:null_quot -> s:0:null_sel:null_minus:null_quot -> s:0:null_sel:null_minus:null_quot zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot nil :: n__from:cons:nil:n__zWquot n__zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot null_sel :: s:0:null_sel:null_minus:null_quot null_minus :: s:0:null_sel:null_minus:null_quot null_quot :: s:0:null_sel:null_minus:null_quot Rewrite Strategy: INNERMOST ---------------------------------------- (53) 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_sel => 0 null_minus => 0 null_quot => 0 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> zWquot(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> from(X) :|: z = 1 + X, X >= 0 from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X from(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X minus(z, z') -{ 1 }-> minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 minus(z, z') -{ 1 }-> 0 :|: X >= 0, z = X, z' = 0 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quot(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0 quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quot(z, z') -{ 1 }-> 1 + quot(minus(X, Y), 1 + Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 1 }-> sel(N, activate(XS)) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0 sel(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 zWquot(z, z') -{ 1 }-> 0 :|: z = XS, XS >= 0, z' = 0 zWquot(z, z') -{ 1 }-> 0 :|: z' = XS, z = 0, XS >= 0 zWquot(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 zWquot(z, z') -{ 1 }-> 1 + quot(X, Y) + (1 + activate(XS) + activate(YS)) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (55) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 Tuples: FROM(z0) -> c FROM(z0) -> c1 SEL(0, cons(z0, z1)) -> c2 SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) MINUS(z0, 0) -> c4 MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(0, s(z0)) -> c6 QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(z0, nil) -> c8 ZWQUOT(nil, z0) -> c9 ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ZWQUOT(z0, z1) -> c13 ACTIVATE(n__from(z0)) -> c14(FROM(z0)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ACTIVATE(z0) -> c16 S tuples: FROM(z0) -> c FROM(z0) -> c1 SEL(0, cons(z0, z1)) -> c2 SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) MINUS(z0, 0) -> c4 MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(0, s(z0)) -> c6 QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(z0, nil) -> c8 ZWQUOT(nil, z0) -> c9 ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ZWQUOT(z0, z1) -> c13 ACTIVATE(n__from(z0)) -> c14(FROM(z0)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ACTIVATE(z0) -> c16 K tuples:none Defined Rule Symbols: from_1, sel_2, minus_2, quot_2, zWquot_2, activate_1 Defined Pair Symbols: FROM_1, SEL_2, MINUS_2, QUOT_2, ZWQUOT_2, ACTIVATE_1 Compound Symbols: c, c1, c2, c3_2, c4, c5_1, c6, c7_2, c8, c9, c10_1, c11_1, c12_1, c13, c14_1, c15_1, c16 ---------------------------------------- (57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: QUOT(0, s(z0)) -> c6 ZWQUOT(z0, nil) -> c8 SEL(0, cons(z0, z1)) -> c2 FROM(z0) -> c1 MINUS(z0, 0) -> c4 ZWQUOT(z0, z1) -> c13 ACTIVATE(z0) -> c16 ZWQUOT(nil, z0) -> c9 ACTIVATE(n__from(z0)) -> c14(FROM(z0)) FROM(z0) -> c ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 Tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) S tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) K tuples:none Defined Rule Symbols: from_1, sel_2, minus_2, quot_2, zWquot_2, activate_1 Defined Pair Symbols: SEL_2, MINUS_2, QUOT_2, ZWQUOT_2, ACTIVATE_1 Compound Symbols: c3_2, c5_1, c7_2, c10_1, c11_1, c12_1, c15_1 ---------------------------------------- (59) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) S tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) K tuples:none Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, MINUS_2, QUOT_2, ZWQUOT_2, ACTIVATE_1 Compound Symbols: c3_2, c5_1, c7_2, c10_1, c11_1, c12_1, c15_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. SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) We considered the (Usable) Rules:none And the Tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ACTIVATE(x_1)) = 0 POL(MINUS(x_1, x_2)) = 0 POL(QUOT(x_1, x_2)) = 0 POL(SEL(x_1, x_2)) = [2]x_1 POL(ZWQUOT(x_1, x_2)) = 0 POL(activate(x_1)) = [3] + [2]x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = 0 POL(from(x_1)) = [3] + [3]x_1 POL(minus(x_1, x_2)) = [3]x_1 + [3]x_2 POL(n__from(x_1)) = [3] + x_1 POL(n__zWquot(x_1, x_2)) = 0 POL(nil) = [3] POL(quot(x_1, x_2)) = [3] + [3]x_2 POL(s(x_1)) = [1] + x_1 POL(zWquot(x_1, x_2)) = [3] ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) K tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, MINUS_2, QUOT_2, ZWQUOT_2, ACTIVATE_1 Compound Symbols: c3_2, c5_1, c7_2, c10_1, c11_1, c12_1, c15_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. ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) We considered the (Usable) Rules:none And the Tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ACTIVATE(x_1)) = [1] POL(MINUS(x_1, x_2)) = 0 POL(QUOT(x_1, x_2)) = 0 POL(SEL(x_1, x_2)) = x_1 POL(ZWQUOT(x_1, x_2)) = [1] POL(activate(x_1)) = [1] + x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = x_1 POL(from(x_1)) = [1] + x_1 POL(minus(x_1, x_2)) = [1] + x_1 + x_2 POL(n__from(x_1)) = [1] + x_1 POL(n__zWquot(x_1, x_2)) = [1] + x_1 POL(nil) = [1] POL(quot(x_1, x_2)) = [1] + x_1 + x_2 POL(s(x_1)) = [1] + x_1 POL(zWquot(x_1, x_2)) = [1] + x_2 ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) K tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, MINUS_2, QUOT_2, ZWQUOT_2, ACTIVATE_1 Compound Symbols: c3_2, c5_1, c7_2, c10_1, c11_1, c12_1, c15_1 ---------------------------------------- (65) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) We considered the (Usable) Rules: from(z0) -> cons(z0, n__from(s(z0))) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) from(z0) -> n__from(z0) activate(n__from(z0)) -> from(z0) activate(z0) -> z0 zWquot(z0, z1) -> n__zWquot(z0, z1) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) And the Tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ACTIVATE(x_1)) = [2]x_1 POL(MINUS(x_1, x_2)) = 0 POL(QUOT(x_1, x_2)) = 0 POL(SEL(x_1, x_2)) = x_1*x_2 POL(ZWQUOT(x_1, x_2)) = [2]x_1 + [2]x_2 POL(activate(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = x_2 POL(from(x_1)) = 0 POL(minus(x_1, x_2)) = [1] POL(n__from(x_1)) = 0 POL(n__zWquot(x_1, x_2)) = [2] + x_1 + x_2 POL(nil) = [2] POL(quot(x_1, x_2)) = [2]x_1 + [2]x_1*x_2 + [2]x_1^2 POL(s(x_1)) = [2] + x_1 POL(zWquot(x_1, x_2)) = [2] + x_1 + x_2 ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) K tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, MINUS_2, QUOT_2, ZWQUOT_2, ACTIVATE_1 Compound Symbols: c3_2, c5_1, c7_2, c10_1, c11_1, c12_1, c15_1 ---------------------------------------- (67) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) K tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, MINUS_2, QUOT_2, ZWQUOT_2, ACTIVATE_1 Compound Symbols: c3_2, c5_1, c7_2, c10_1, c11_1, c12_1, c15_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) by SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, from(z0)), ACTIVATE(n__from(z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(SEL(x0, zWquot(z0, z1)), ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, from(z0)), ACTIVATE(n__from(z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(SEL(x0, zWquot(z0, z1)), ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) K tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: MINUS_2, QUOT_2, ZWQUOT_2, ACTIVATE_1, SEL_2 Compound Symbols: c5_1, c7_2, c10_1, c11_1, c12_1, c15_1, c3_2 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(SEL(x0, zWquot(z0, z1)), ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, from(z0))) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) K tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: MINUS_2, QUOT_2, ZWQUOT_2, ACTIVATE_1, SEL_2 Compound Symbols: c5_1, c7_2, c10_1, c11_1, c12_1, c15_1, c3_2, c3_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUOT(s(z0), s(z1)) -> c7(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1)) by QUOT(s(z0), s(0)) -> c7(QUOT(0, s(0)), MINUS(z0, 0)) QUOT(s(s(z0)), s(s(z1))) -> c7(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(SEL(x0, zWquot(z0, z1)), ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, from(z0))) QUOT(s(z0), s(0)) -> c7(QUOT(0, s(0)), MINUS(z0, 0)) QUOT(s(s(z0)), s(s(z1))) -> c7(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(z0), s(0)) -> c7(QUOT(0, s(0)), MINUS(z0, 0)) QUOT(s(s(z0)), s(s(z1))) -> c7(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) K tuples: SEL(s(z0), cons(z1, z2)) -> c3(SEL(z0, activate(z2)), ACTIVATE(z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: MINUS_2, ZWQUOT_2, ACTIVATE_1, SEL_2, QUOT_2 Compound Symbols: c5_1, c10_1, c11_1, c12_1, c15_1, c3_2, c3_1, c7_2 ---------------------------------------- (75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: QUOT(s(z0), s(0)) -> c7(QUOT(0, s(0)), MINUS(z0, 0)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(SEL(x0, zWquot(z0, z1)), ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, from(z0))) QUOT(s(s(z0)), s(s(z1))) -> c7(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c7(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) K tuples: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: MINUS_2, ZWQUOT_2, ACTIVATE_1, SEL_2, QUOT_2 Compound Symbols: c5_1, c10_1, c11_1, c12_1, c15_1, c3_2, c3_1, c7_2 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(SEL(x0, zWquot(z0, z1)), ACTIVATE(n__zWquot(z0, z1))) by SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(SEL(x0, nil), ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(SEL(x0, nil), ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(SEL(x0, n__zWquot(z0, z1)), ACTIVATE(n__zWquot(z0, z1))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, from(z0))) QUOT(s(s(z0)), s(s(z1))) -> c7(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(SEL(x0, nil), ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(SEL(x0, nil), ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(SEL(x0, n__zWquot(z0, z1)), ACTIVATE(n__zWquot(z0, z1))) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c7(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) K tuples: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: MINUS_2, ZWQUOT_2, ACTIVATE_1, SEL_2, QUOT_2 Compound Symbols: c5_1, c10_1, c11_1, c12_1, c15_1, c3_2, c3_1, c7_2 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, from(z0))) QUOT(s(s(z0)), s(s(z1))) -> c7(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c7(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) K tuples: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: MINUS_2, ZWQUOT_2, ACTIVATE_1, SEL_2, QUOT_2 Compound Symbols: c5_1, c10_1, c11_1, c12_1, c15_1, c3_2, c3_1, c7_2 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, from(z0))) by SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, n__from(z0))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) QUOT(s(s(z0)), s(s(z1))) -> c7(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, n__from(z0))) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c7(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) K tuples: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: MINUS_2, ZWQUOT_2, ACTIVATE_1, SEL_2, QUOT_2 Compound Symbols: c5_1, c10_1, c11_1, c12_1, c15_1, c3_2, c7_2, c3_1 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, n__from(z0))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) QUOT(s(s(z0)), s(s(z1))) -> c7(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(s(z0)), s(s(z1))) -> c7(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) K tuples: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: MINUS_2, ZWQUOT_2, ACTIVATE_1, SEL_2, QUOT_2 Compound Symbols: c5_1, c10_1, c11_1, c12_1, c15_1, c3_2, c7_2, c3_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QUOT(s(s(z0)), s(s(z1))) -> c7(QUOT(minus(z0, z1), s(s(z1))), MINUS(s(z0), s(z1))) by QUOT(s(s(z0)), s(s(0))) -> c7(QUOT(0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) QUOT(s(s(z0)), s(s(0))) -> c7(QUOT(0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(s(z0)), s(s(0))) -> c7(QUOT(0, s(s(0))), MINUS(s(z0), s(0))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) K tuples: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: MINUS_2, ZWQUOT_2, ACTIVATE_1, SEL_2, QUOT_2 Compound Symbols: c5_1, c10_1, c11_1, c12_1, c15_1, c3_2, c3_1, c7_2, c7_1 ---------------------------------------- (87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) QUOT(s(s(z0)), s(s(0))) -> c7(MINUS(s(z0), s(0))) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) QUOT(s(s(z0)), s(s(0))) -> c7(MINUS(s(z0), s(0))) K tuples: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: MINUS_2, ZWQUOT_2, ACTIVATE_1, SEL_2, QUOT_2 Compound Symbols: c5_1, c10_1, c11_1, c12_1, c15_1, c3_2, c3_1, c7_2, c7_1 ---------------------------------------- (89) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: QUOT(s(s(z0)), s(s(0))) -> c7(MINUS(s(z0), s(0))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) QUOT(s(s(z0)), s(s(0))) -> c7(MINUS(s(z0), s(0))) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) K tuples: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) QUOT(s(s(z0)), s(s(0))) -> c7(MINUS(s(z0), s(0))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: MINUS_2, ZWQUOT_2, ACTIVATE_1, SEL_2, QUOT_2 Compound Symbols: c5_1, c10_1, c11_1, c12_1, c15_1, c3_2, c3_1, c7_2, c7_1 ---------------------------------------- (91) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) We considered the (Usable) Rules:none And the Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) QUOT(s(s(z0)), s(s(0))) -> c7(MINUS(s(z0), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ACTIVATE(x_1)) = [1] POL(MINUS(x_1, x_2)) = 0 POL(QUOT(x_1, x_2)) = [1] POL(SEL(x_1, x_2)) = [1] + x_1 POL(ZWQUOT(x_1, x_2)) = [1] POL(activate(x_1)) = [1] + x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = 0 POL(from(x_1)) = [1] + x_1 POL(minus(x_1, x_2)) = [1] + x_1 + x_2 POL(n__from(x_1)) = [1] + x_1 POL(n__zWquot(x_1, x_2)) = [1] POL(nil) = [1] POL(quot(x_1, x_2)) = x_2 POL(s(x_1)) = [1] + x_1 POL(zWquot(x_1, x_2)) = [1] + x_1 + x_2 ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) QUOT(s(s(z0)), s(s(0))) -> c7(MINUS(s(z0), s(0))) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) K tuples: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) QUOT(s(s(z0)), s(s(0))) -> c7(MINUS(s(z0), s(0))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: MINUS_2, ZWQUOT_2, ACTIVATE_1, SEL_2, QUOT_2 Compound Symbols: c5_1, c10_1, c11_1, c12_1, c15_1, c3_2, c3_1, c7_2, c7_1 ---------------------------------------- (93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) by MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) QUOT(s(s(z0)), s(s(0))) -> c7(MINUS(s(z0), s(0))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) S tuples: QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) K tuples: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) QUOT(s(s(z0)), s(s(0))) -> c7(MINUS(s(z0), s(0))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: ZWQUOT_2, ACTIVATE_1, SEL_2, QUOT_2, MINUS_2 Compound Symbols: c10_1, c11_1, c12_1, c15_1, c3_2, c3_1, c7_2, c7_1, c5_1 ---------------------------------------- (95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: QUOT(s(s(z0)), s(s(0))) -> c7(MINUS(s(z0), s(0))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) S tuples: QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) K tuples: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: ZWQUOT_2, ACTIVATE_1, SEL_2, QUOT_2, MINUS_2 Compound Symbols: c10_1, c11_1, c12_1, c15_1, c3_2, c3_1, c7_2, c7_1, c5_1 ---------------------------------------- (97) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c10(QUOT(z0, z2)) by ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) S tuples: QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) K tuples: ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: ZWQUOT_2, ACTIVATE_1, SEL_2, QUOT_2, MINUS_2 Compound Symbols: c11_1, c12_1, c15_1, c3_2, c3_1, c7_2, c7_1, c5_1, c10_1 ---------------------------------------- (99) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c11(ACTIVATE(z1)) by ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) S tuples: QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) K tuples: ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: ZWQUOT_2, ACTIVATE_1, SEL_2, QUOT_2, MINUS_2 Compound Symbols: c12_1, c15_1, c3_2, c3_1, c7_2, c7_1, c5_1, c10_1, c11_1 ---------------------------------------- (101) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c12(ACTIVATE(z3)) by ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) S tuples: QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) K tuples: ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: ACTIVATE_1, SEL_2, QUOT_2, MINUS_2, ZWQUOT_2 Compound Symbols: c15_1, c3_2, c3_1, c7_2, c7_1, c5_1, c10_1, c11_1, c12_1 ---------------------------------------- (103) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__zWquot(z0, z1)) -> c15(ZWQUOT(z0, z1)) by ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(ACTIVATE(n__zWquot(z0, nil))) SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) S tuples: QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) K tuples: QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, QUOT_2, MINUS_2, ZWQUOT_2, ACTIVATE_1 Compound Symbols: c3_2, c3_1, c7_2, c7_1, c5_1, c10_1, c11_1, c12_1, c15_1 ---------------------------------------- (105) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: SEL(s(x0), cons(x1, n__zWquot(nil, z0))) -> c3(ACTIVATE(n__zWquot(nil, z0))) SEL(s(x0), cons(x1, n__zWquot(z0, nil))) -> c3(ACTIVATE(n__zWquot(z0, nil))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) S tuples: QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) K tuples: QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, QUOT_2, MINUS_2, ZWQUOT_2, ACTIVATE_1 Compound Symbols: c3_2, c3_1, c7_2, c7_1, c5_1, c10_1, c11_1, c12_1, c15_1 ---------------------------------------- (107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(x0), cons(x1, z0)) -> c3(SEL(x0, z0), ACTIVATE(z0)) by SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2)), ACTIVATE(cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5)))), ACTIVATE(cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3))), ACTIVATE(cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2))), ACTIVATE(cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(SEL(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(SEL(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(SEL(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(SEL(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2)), ACTIVATE(cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5)))), ACTIVATE(cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3))), ACTIVATE(cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2))), ACTIVATE(cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(SEL(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(SEL(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(SEL(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(SEL(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) S tuples: QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) K tuples: QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, QUOT_2, MINUS_2, ZWQUOT_2, ACTIVATE_1 Compound Symbols: c3_2, c3_1, c7_2, c7_1, c5_1, c10_1, c11_1, c12_1, c15_1 ---------------------------------------- (109) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 8 trailing tuple parts ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) S tuples: QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) K tuples: QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, QUOT_2, MINUS_2, ZWQUOT_2, ACTIVATE_1 Compound Symbols: c3_2, c3_1, c7_2, c7_1, c5_1, c10_1, c11_1, c12_1, c15_1 ---------------------------------------- (111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(x0), cons(x1, n__zWquot(z0, z1))) -> c3(ACTIVATE(n__zWquot(z0, z1))) by SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) S tuples: QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) K tuples: QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, QUOT_2, MINUS_2, ZWQUOT_2, ACTIVATE_1 Compound Symbols: c3_2, c3_1, c7_2, c7_1, c5_1, c10_1, c11_1, c12_1, c15_1 ---------------------------------------- (113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(x0), cons(x1, n__from(z0))) -> c3(SEL(x0, cons(z0, n__from(s(z0))))) by SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) S tuples: QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) K tuples: QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, QUOT_2, MINUS_2, ZWQUOT_2, ACTIVATE_1 Compound Symbols: c3_2, c7_2, c7_1, c5_1, c10_1, c11_1, c12_1, c15_1, c3_1 ---------------------------------------- (115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace QUOT(s(s(x0)), s(s(x1))) -> c7(MINUS(s(x0), s(x1))) by QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) S tuples: QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, QUOT_2, MINUS_2, ZWQUOT_2, ACTIVATE_1 Compound Symbols: c3_2, c7_2, c5_1, c10_1, c11_1, c12_1, c15_1, c3_1, c7_1 ---------------------------------------- (117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) by MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) S tuples: QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, QUOT_2, ZWQUOT_2, ACTIVATE_1, MINUS_2 Compound Symbols: c3_2, c7_2, c10_1, c11_1, c12_1, c15_1, c3_1, c7_1, c5_1 ---------------------------------------- (119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace QUOT(s(s(s(z0))), s(s(s(z1)))) -> c7(QUOT(minus(z0, z1), s(s(s(z1)))), MINUS(s(s(z0)), s(s(z1)))) by QUOT(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c7(QUOT(minus(s(y0), s(y1)), s(s(s(s(y1))))), MINUS(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c7(QUOT(minus(s(y0), s(y1)), s(s(s(s(y1))))), MINUS(s(s(s(y0))), s(s(s(y1))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c7(QUOT(minus(s(y0), s(y1)), s(s(s(s(y1))))), MINUS(s(s(s(y0))), s(s(s(y1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c11_1, c12_1, c15_1, c3_1, c7_1, c5_1, c7_2 ---------------------------------------- (121) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ZWQUOT(cons(s(s(y0)), z1), cons(s(s(y1)), z3)) -> c10(QUOT(s(s(y0)), s(s(y1)))) by ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c7(QUOT(minus(s(y0), s(y1)), s(s(s(s(y1))))), MINUS(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c7(QUOT(minus(s(y0), s(y1)), s(s(s(s(y1))))), MINUS(s(s(s(y0))), s(s(s(y1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c11_1, c12_1, c15_1, c3_1, c7_1, c5_1, c7_2 ---------------------------------------- (123) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ZWQUOT(cons(z0, n__zWquot(y0, y1)), cons(z2, z3)) -> c11(ACTIVATE(n__zWquot(y0, y1))) by ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c7(QUOT(minus(s(y0), s(y1)), s(s(s(s(y1))))), MINUS(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c7(QUOT(minus(s(y0), s(y1)), s(s(s(s(y1))))), MINUS(s(s(s(y0))), s(s(s(y1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c12_1, c15_1, c3_1, c7_1, c5_1, c7_2, c11_1 ---------------------------------------- (125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(y0, y1))) -> c12(ACTIVATE(n__zWquot(y0, y1))) by ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c7(QUOT(minus(s(y0), s(y1)), s(s(s(s(y1))))), MINUS(s(s(s(y0))), s(s(s(y1))))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c7(QUOT(minus(s(y0), s(y1)), s(s(s(s(y1))))), MINUS(s(s(s(y0))), s(s(s(y1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c7_1, c5_1, c7_2, c11_1, c12_1 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace QUOT(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c7(QUOT(minus(s(y0), s(y1)), s(s(s(s(y1))))), MINUS(s(s(s(y0))), s(s(s(y1))))) by QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c7_1, c5_1, c11_1, c12_1, c7_2 ---------------------------------------- (129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) -> c15(ZWQUOT(cons(s(s(y0)), y1), cons(s(s(y2)), y3))) by ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), z3))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), z3))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c7_1, c5_1, c11_1, c12_1, c7_2 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) -> c15(ZWQUOT(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))) by ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c7_1, c5_1, c11_1, c12_1, c7_2 ---------------------------------------- (133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) -> c15(ZWQUOT(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))) by ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c7_1, c5_1, c11_1, c12_1, c7_2 ---------------------------------------- (135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c3(SEL(s(y0), cons(y1, y2))) by SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(y3, y4))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(y3, y4))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__from(y3))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__from(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__from(y2)))) -> c3(SEL(s(s(y0)), cons(z2, n__from(y2)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3)))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(y3, y4))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(y3, y4))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__from(y3))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__from(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__from(y2)))) -> c3(SEL(s(s(y0)), cons(z2, n__from(y2)))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c7_1, c5_1, c11_1, c12_1, c7_2 ---------------------------------------- (137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(y2, y3)))) -> c3(SEL(s(y0), cons(y1, n__zWquot(y2, y3)))) by SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(y3, y4))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(y3, y4))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__from(y3))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__from(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__from(y2)))) -> c3(SEL(s(s(y0)), cons(z2, n__from(y2)))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c7_1, c5_1, c11_1, c12_1, c7_2 ---------------------------------------- (139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(s(y0)), cons(z1, cons(y1, n__from(y2)))) -> c3(SEL(s(y0), cons(y1, n__from(y2)))) by SEL(s(s(s(y0))), cons(z1, cons(z2, n__from(z3)))) -> c3(SEL(s(s(y0)), cons(z2, n__from(z3)))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(y3, y4))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(y3, y4))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__from(y3))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__from(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__from(y2)))) -> c3(SEL(s(s(y0)), cons(z2, n__from(y2)))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c7_1, c5_1, c11_1, c12_1, c7_2 ---------------------------------------- (141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) by SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(y3, y4))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(y3, y4))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__from(y3))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__from(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__from(y2)))) -> c3(SEL(s(s(y0)), cons(z2, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c7_1, c5_1, c11_1, c12_1, c7_2 ---------------------------------------- (143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) by SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), z6)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), z6)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(y5)), z6)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(y5)), z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(y5)), z6)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(y5)), z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(y6)), z6)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(y6)), z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(y6)), z6)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(y6)), z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(y3, y4))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(y3, y4))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__from(y3))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__from(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__from(y2)))) -> c3(SEL(s(s(y0)), cons(z2, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c7_1, c5_1, c11_1, c12_1, c7_2 ---------------------------------------- (145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) by SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(y5)), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(y5)), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(y5)), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(y5)), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(y6)), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(y6)), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(y6)), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(y6)), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z4, n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z4, n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z4, n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z4, n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z4, n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z4, n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z4, n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z4, n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(y3, y4))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(y3, y4))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__from(y3))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__from(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__from(y2)))) -> c3(SEL(s(s(y0)), cons(z2, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c7_1, c5_1, c11_1, c12_1, c7_2 ---------------------------------------- (147) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) by SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(y3, y4))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(y3, y4))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__from(y3))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__from(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__from(y2)))) -> c3(SEL(s(s(y0)), cons(z2, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c7_1, c5_1, c11_1, c12_1, c7_2 ---------------------------------------- (149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c3(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) by SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), z6)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), z6)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(y5)), z6)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(y5)), z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(y5)), z6)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(y5)), z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(y6)), z6)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(y6)), z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(y6)), z6)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(y6)), z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(z3, z4)), cons(s(s(y2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(y3, y4))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(y3, y4))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__from(y3))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__from(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__from(y2)))) -> c3(SEL(s(s(y0)), cons(z2, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c7_1, c5_1, c11_1, c12_1, c7_2 ---------------------------------------- (151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) by SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(y5)), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(y5)), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(y5)), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(y5)), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(y6)), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(y6)), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(y6)), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(y6)), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(y0)), z3), cons(s(s(y2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z4, n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z4, n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z4, n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z4, n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z4, n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z4, n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z4, n__zWquot(z5, z6))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z4, n__zWquot(z5, z6))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(y1, y2)), cons(z4, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(y3, y4))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(y3, y4))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__from(y3))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__from(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__from(y2)))) -> c3(SEL(s(s(y0)), cons(z2, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c7_1, c5_1, c11_1, c12_1, c7_2 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(s(y0)), cons(z1, n__from(z2))) -> c3(SEL(s(y0), cons(z2, n__from(s(z2))))) by SEL(s(s(s(y0))), cons(z1, n__from(z2))) -> c3(SEL(s(s(y0)), cons(z2, n__from(s(z2))))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(y3, y4))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(y3, y4))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__from(y3))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__from(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__from(y2)))) -> c3(SEL(s(s(y0)), cons(z2, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(s(s(y0))), cons(z1, n__from(z2))) -> c3(SEL(s(s(y0)), cons(z2, n__from(s(z2))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, QUOT_2, MINUS_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c7_1, c5_1, c11_1, c12_1, c7_2 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace QUOT(s(s(s(y0))), s(s(s(y1)))) -> c7(MINUS(s(s(y0)), s(s(y1)))) by QUOT(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c7(MINUS(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(y3, y4))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(y3, y4))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__from(y3))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__from(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__from(y2)))) -> c3(SEL(s(s(y0)), cons(z2, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(s(s(y0))), cons(z1, n__from(z2))) -> c3(SEL(s(s(y0)), cons(z2, n__from(s(z2))))) QUOT(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c7(MINUS(s(s(s(y0))), s(s(s(y1))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) QUOT(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c7(MINUS(s(s(s(y0))), s(s(s(y1))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, MINUS_2, QUOT_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c5_1, c11_1, c12_1, c7_2, c7_1 ---------------------------------------- (157) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) by ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__from(z0)) -> from(z0) activate(n__zWquot(z0, z1)) -> zWquot(z0, z1) activate(z0) -> z0 from(z0) -> cons(z0, n__from(s(z0))) from(z0) -> n__from(z0) zWquot(z0, nil) -> nil zWquot(nil, z0) -> nil zWquot(cons(z0, z1), cons(z2, z3)) -> cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3))) zWquot(z0, z1) -> n__zWquot(z0, z1) quot(0, s(z0)) -> 0 quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1))) minus(z0, 0) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) Tuples: SEL(s(x0), cons(x1, n__zWquot(cons(z0, z1), cons(z2, z3)))) -> c3(SEL(x0, cons(quot(z0, z2), n__zWquot(activate(z1), activate(z3)))), ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, z3)))) ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y1))), z3)) -> c10(QUOT(s(s(s(y0))), s(s(s(y1))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) SEL(s(s(y0)), cons(z1, cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) -> c3(SEL(s(y0), cons(y1, n__zWquot(cons(y2, y3), cons(y4, y5))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(cons(y3, y4), cons(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__zWquot(y3, y4))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__zWquot(y3, y4))))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__from(y3))))) -> c3(SEL(s(s(y0)), cons(z2, cons(y2, n__from(y3))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(s(y2))), y3), cons(s(s(s(y4))), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(s(s(y2)), y3), cons(s(s(y4)), y5))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, n__zWquot(y3, y4)), cons(y5, y6))))) SEL(s(s(z0)), cons(z1, cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) -> c3(SEL(s(z0), cons(z2, n__zWquot(cons(y2, y3), cons(y4, n__zWquot(y5, y6)))))) SEL(s(s(s(y0))), cons(z1, cons(z2, n__from(y2)))) -> c3(SEL(s(s(y0)), cons(z2, n__from(y2)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), z3), cons(s(s(z4)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(y1, y2)), cons(s(s(s(y3))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(y1, y2)), cons(s(s(s(s(y3)))), z5)))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(y1, y2)), cons(s(s(z4)), n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), z3), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z3), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z4)), n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(z2)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z4)), n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, z6)))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(s(y4))), y5), cons(s(s(s(y6))), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(s(s(y4)), y5), cons(s(s(y6)), y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, n__zWquot(y5, y6)), cons(y7, y8)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(z3, z4)), cons(z5, n__zWquot(cons(y4, y5), cons(y6, n__zWquot(y7, y8))))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z3, z4)), cons(s(s(s(y2))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) -> c3(ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z3, z4)), cons(s(s(s(s(y2)))), n__zWquot(y3, y4))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z5, n__zWquot(y6, y7))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) -> c3(ACTIVATE(n__zWquot(cons(z2, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z5, n__zWquot(y7, y8))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7)))))) SEL(s(z0), cons(z1, n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) -> c3(ACTIVATE(n__zWquot(cons(z2, z3), cons(z4, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7))))))) SEL(s(s(s(y0))), cons(z1, n__from(z2))) -> c3(SEL(s(s(y0)), cons(z2, n__from(s(z2))))) QUOT(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c7(MINUS(s(s(s(y0))), s(s(s(y1))))) S tuples: MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) QUOT(s(s(s(s(z0)))), s(s(s(s(z1))))) -> c7(QUOT(minus(z0, z1), s(s(s(s(z1))))), MINUS(s(s(s(z0))), s(s(s(z1))))) K tuples: ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) -> c15(ZWQUOT(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))) ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y1)))), z3)) -> c10(QUOT(s(s(s(s(y0)))), s(s(s(s(y1)))))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4)))), cons(z3, z4)) -> c11(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(s(y0))), y1), cons(s(s(s(y2))), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) -> c12(ACTIVATE(n__zWquot(cons(s(s(y0)), y1), cons(s(s(y2)), y3)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) -> c12(ACTIVATE(n__zWquot(cons(y0, n__zWquot(y1, y2)), cons(y3, y4)))) ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) -> c12(ACTIVATE(n__zWquot(cons(y0, y1), cons(y2, n__zWquot(y3, y4))))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) -> c15(ZWQUOT(cons(s(s(z0)), n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(s(s(z2)), z3))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(s(s(z0)), z1), cons(s(s(z2)), n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) -> c15(ZWQUOT(cons(s(s(s(y0))), n__zWquot(z1, z2)), cons(s(s(s(y2))), z4))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), n__zWquot(z1, z2)), cons(s(s(s(s(y2)))), z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z3, z4))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, n__zWquot(z1, z2)), cons(z3, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) ACTIVATE(n__zWquot(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(y0))), z1), cons(s(s(s(y2))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(s(s(s(s(y0)))), z1), cons(s(s(s(s(y2)))), n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(s(y1))), y2), cons(s(s(s(y3))), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(s(s(y1)), y2), cons(s(s(y3)), y4))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, n__zWquot(y2, y3)), cons(y4, y5))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) -> c15(ZWQUOT(cons(z0, n__zWquot(cons(y1, y2), cons(y3, n__zWquot(y4, y5)))), cons(z2, n__zWquot(z3, z4)))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(s(y3))), y4), cons(s(s(s(y5))), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(s(s(y3)), y4), cons(s(s(y5)), y6))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, n__zWquot(y4, y5)), cons(y6, y7))))) ACTIVATE(n__zWquot(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) -> c15(ZWQUOT(cons(z0, z1), cons(z2, n__zWquot(cons(y3, y4), cons(y5, n__zWquot(y6, y7)))))) QUOT(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c7(MINUS(s(s(s(y0))), s(s(s(y1))))) Defined Rule Symbols: activate_1, from_1, zWquot_2, quot_2, minus_2 Defined Pair Symbols: SEL_2, ZWQUOT_2, ACTIVATE_1, MINUS_2, QUOT_2 Compound Symbols: c3_2, c10_1, c15_1, c3_1, c5_1, c11_1, c12_1, c7_2, c7_1