KILLED proof of input_Td8le5XExs.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CompletionProof [UPPER BOUND(ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (22) CdtProblem (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 2 ms] (62) CdtProblem (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 168 ms] (80) CdtProblem (81) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 115 ms] (88) CdtProblem (89) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 127 ms] (90) 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: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(x, nil) -> x foldf(x, cons(y, z)) -> f(foldf(x, z), y) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b)) f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) 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: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(x, nil) -> x foldf(x, cons(y, z)) -> f(foldf(x, z), y) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b)) f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) 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: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(x, nil) -> x foldf(x, cons(y, z)) -> f(foldf(x, z), y) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b)) f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) 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: g(A) -> A [1] g(B) -> A [1] g(B) -> B [1] g(C) -> A [1] g(C) -> B [1] g(C) -> C [1] foldf(x, nil) -> x [1] foldf(x, cons(y, z)) -> f(foldf(x, z), y) [1] f(t, x) -> f'(t, g(x)) [1] f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) [1] f'(triple(a, b, c), B) -> f(triple(a, b, c), A) [1] f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b)) [1] f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) [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: g(A) -> A [1] g(B) -> A [1] g(B) -> B [1] g(C) -> A [1] g(C) -> B [1] g(C) -> C [1] foldf(x, nil) -> x [1] foldf(x, cons(y, z)) -> f(foldf(x, z), y) [1] f(t, x) -> f'(t, g(x)) [1] f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) [1] f'(triple(a, b, c), B) -> f(triple(a, b, c), A) [1] f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b)) [1] f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) [1] The TRS has the following type information: g :: A:B:C -> A:B:C A :: A:B:C B :: A:B:C C :: A:B:C foldf :: triple -> nil:cons -> triple nil :: nil:cons cons :: A:B:C -> nil:cons -> nil:cons f :: triple -> A:B:C -> triple f' :: triple -> A:B:C -> triple triple :: nil:cons -> nil:cons -> nil:cons -> triple f'' :: triple -> triple 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: none (c) The following functions are completely defined: g_1 foldf_2 f_2 f'_2 f''_1 Due to the following rules being added: f'(v0, v1) -> const [0] f''(v0) -> const [0] And the following fresh constants: const ---------------------------------------- (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: g(A) -> A [1] g(B) -> A [1] g(B) -> B [1] g(C) -> A [1] g(C) -> B [1] g(C) -> C [1] foldf(x, nil) -> x [1] foldf(x, cons(y, z)) -> f(foldf(x, z), y) [1] f(t, x) -> f'(t, g(x)) [1] f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) [1] f'(triple(a, b, c), B) -> f(triple(a, b, c), A) [1] f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b)) [1] f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) [1] f'(v0, v1) -> const [0] f''(v0) -> const [0] The TRS has the following type information: g :: A:B:C -> A:B:C A :: A:B:C B :: A:B:C C :: A:B:C foldf :: triple:const -> nil:cons -> triple:const nil :: nil:cons cons :: A:B:C -> nil:cons -> nil:cons f :: triple:const -> A:B:C -> triple:const f' :: triple:const -> A:B:C -> triple:const triple :: nil:cons -> nil:cons -> nil:cons -> triple:const f'' :: triple:const -> triple:const const :: triple:const 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: g(A) -> A [1] g(B) -> A [1] g(B) -> B [1] g(C) -> A [1] g(C) -> B [1] g(C) -> C [1] foldf(x, nil) -> x [1] foldf(x, cons(y, nil)) -> f(x, y) [2] foldf(x, cons(y, cons(y', z'))) -> f(f(foldf(x, z'), y'), y) [2] f(t, A) -> f'(t, A) [2] f(t, B) -> f'(t, A) [2] f(t, B) -> f'(t, B) [2] f(t, C) -> f'(t, A) [2] f(t, C) -> f'(t, B) [2] f(t, C) -> f'(t, C) [2] f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) [1] f'(triple(a, b, c), B) -> f(triple(a, b, c), A) [1] f'(triple(a, nil, c), A) -> f''(triple(cons(A, a), nil, c)) [2] f'(triple(a, cons(y'', z''), c), A) -> f''(f(foldf(triple(cons(A, a), nil, c), z''), y'')) [2] f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) [1] f'(v0, v1) -> const [0] f''(v0) -> const [0] The TRS has the following type information: g :: A:B:C -> A:B:C A :: A:B:C B :: A:B:C C :: A:B:C foldf :: triple:const -> nil:cons -> triple:const nil :: nil:cons cons :: A:B:C -> nil:cons -> nil:cons f :: triple:const -> A:B:C -> triple:const f' :: triple:const -> A:B:C -> triple:const triple :: nil:cons -> nil:cons -> nil:cons -> triple:const f'' :: triple:const -> triple:const const :: triple:const 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: A => 0 B => 1 C => 2 nil => 0 const => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: f(z, z1) -{ 2 }-> f'(t, 2) :|: z1 = 2, z = t, t >= 0 f(z, z1) -{ 2 }-> f'(t, 1) :|: z = t, z1 = 1, t >= 0 f(z, z1) -{ 2 }-> f'(t, 1) :|: z1 = 2, z = t, t >= 0 f(z, z1) -{ 2 }-> f'(t, 0) :|: z1 = 0, z = t, t >= 0 f(z, z1) -{ 2 }-> f'(t, 0) :|: z = t, z1 = 1, t >= 0 f(z, z1) -{ 2 }-> f'(t, 0) :|: z1 = 2, z = t, t >= 0 f'(z, z1) -{ 2 }-> f''(f(foldf(1 + (1 + 0 + a) + 0 + c, z''), y'')) :|: z1 = 0, a >= 0, c >= 0, z'' >= 0, z = 1 + a + (1 + y'' + z'') + c, y'' >= 0 f'(z, z1) -{ 2 }-> f''(1 + (1 + 0 + a) + 0 + c) :|: z1 = 0, a >= 0, c >= 0, z = 1 + a + 0 + c f'(z, z1) -{ 1 }-> f(1 + a + b + c, 0) :|: b >= 0, a >= 0, c >= 0, z1 = 1, z = 1 + a + b + c f'(z, z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v1, v1 >= 0, z = v0 f'(z, z1) -{ 1 }-> 1 + a + b + (1 + 2 + c) :|: b >= 0, z1 = 2, a >= 0, c >= 0, z = 1 + a + b + c f''(z) -{ 1 }-> foldf(1 + a + b + 0, c) :|: b >= 0, a >= 0, c >= 0, z = 1 + a + b + c f''(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 foldf(z, z1) -{ 1 }-> x :|: z1 = 0, x >= 0, z = x foldf(z, z1) -{ 2 }-> f(x, y) :|: x >= 0, y >= 0, z = x, z1 = 1 + y + 0 foldf(z, z1) -{ 2 }-> f(f(foldf(x, z'), y'), y) :|: z1 = 1 + y + (1 + y' + z'), x >= 0, y >= 0, z' >= 0, y' >= 0, z = x g(z) -{ 1 }-> 2 :|: z = 2 g(z) -{ 1 }-> 1 :|: z = 1 g(z) -{ 1 }-> 1 :|: z = 2 g(z) -{ 1 }-> 0 :|: z = 0 g(z) -{ 1 }-> 0 :|: z = 1 g(z) -{ 1 }-> 0 :|: z = 2 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z, z1) -{ 2 }-> f'(z, 2) :|: z1 = 2, z >= 0 f(z, z1) -{ 2 }-> f'(z, 1) :|: z1 = 1, z >= 0 f(z, z1) -{ 2 }-> f'(z, 1) :|: z1 = 2, z >= 0 f(z, z1) -{ 2 }-> f'(z, 0) :|: z1 = 0, z >= 0 f(z, z1) -{ 2 }-> f'(z, 0) :|: z1 = 1, z >= 0 f(z, z1) -{ 2 }-> f'(z, 0) :|: z1 = 2, z >= 0 f'(z, z1) -{ 2 }-> f''(f(foldf(1 + (1 + 0 + a) + 0 + c, z''), y'')) :|: z1 = 0, a >= 0, c >= 0, z'' >= 0, z = 1 + a + (1 + y'' + z'') + c, y'' >= 0 f'(z, z1) -{ 2 }-> f''(1 + (1 + 0 + a) + 0 + c) :|: z1 = 0, a >= 0, c >= 0, z = 1 + a + 0 + c f'(z, z1) -{ 1 }-> f(1 + a + b + c, 0) :|: b >= 0, a >= 0, c >= 0, z1 = 1, z = 1 + a + b + c f'(z, z1) -{ 0 }-> 0 :|: z >= 0, z1 >= 0 f'(z, z1) -{ 1 }-> 1 + a + b + (1 + 2 + c) :|: b >= 0, z1 = 2, a >= 0, c >= 0, z = 1 + a + b + c f''(z) -{ 1 }-> foldf(1 + a + b + 0, c) :|: b >= 0, a >= 0, c >= 0, z = 1 + a + b + c f''(z) -{ 0 }-> 0 :|: z >= 0 foldf(z, z1) -{ 1 }-> z :|: z1 = 0, z >= 0 foldf(z, z1) -{ 2 }-> f(z, z1 - 1) :|: z >= 0, z1 - 1 >= 0 foldf(z, z1) -{ 2 }-> f(f(foldf(z, z'), y'), y) :|: z1 = 1 + y + (1 + y' + z'), z >= 0, y >= 0, z' >= 0, y' >= 0 g(z) -{ 1 }-> 2 :|: z = 2 g(z) -{ 1 }-> 1 :|: z = 1 g(z) -{ 1 }-> 1 :|: z = 2 g(z) -{ 1 }-> 0 :|: z = 0 g(z) -{ 1 }-> 0 :|: z = 1 g(z) -{ 1 }-> 0 :|: z = 2 ---------------------------------------- (17) 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: f'(v0, v1) -> null_f' [0] f''(v0) -> null_f'' [0] And the following fresh constants: null_f', null_f'' ---------------------------------------- (18) 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: g(A) -> A [1] g(B) -> A [1] g(B) -> B [1] g(C) -> A [1] g(C) -> B [1] g(C) -> C [1] foldf(x, nil) -> x [1] foldf(x, cons(y, z)) -> f(foldf(x, z), y) [1] f(t, x) -> f'(t, g(x)) [1] f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) [1] f'(triple(a, b, c), B) -> f(triple(a, b, c), A) [1] f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b)) [1] f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) [1] f'(v0, v1) -> null_f' [0] f''(v0) -> null_f'' [0] The TRS has the following type information: g :: A:B:C -> A:B:C A :: A:B:C B :: A:B:C C :: A:B:C foldf :: triple:null_f':null_f'' -> nil:cons -> triple:null_f':null_f'' nil :: nil:cons cons :: A:B:C -> nil:cons -> nil:cons f :: triple:null_f':null_f'' -> A:B:C -> triple:null_f':null_f'' f' :: triple:null_f':null_f'' -> A:B:C -> triple:null_f':null_f'' triple :: nil:cons -> nil:cons -> nil:cons -> triple:null_f':null_f'' f'' :: triple:null_f':null_f'' -> triple:null_f':null_f'' null_f' :: triple:null_f':null_f'' null_f'' :: triple:null_f':null_f'' Rewrite Strategy: INNERMOST ---------------------------------------- (19) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: A => 0 B => 1 C => 2 nil => 0 null_f' => 0 null_f'' => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z', z'') -{ 1 }-> f'(t, g(x)) :|: z' = t, x >= 0, z'' = x, t >= 0 f'(z', z'') -{ 1 }-> f''(foldf(1 + (1 + 0 + a) + 0 + c, b)) :|: z'' = 0, b >= 0, a >= 0, c >= 0, z' = 1 + a + b + c f'(z', z'') -{ 1 }-> f(1 + a + b + c, 0) :|: b >= 0, a >= 0, c >= 0, z' = 1 + a + b + c, z'' = 1 f'(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 f'(z', z'') -{ 1 }-> 1 + a + b + (1 + 2 + c) :|: b >= 0, a >= 0, c >= 0, z'' = 2, z' = 1 + a + b + c f''(z') -{ 1 }-> foldf(1 + a + b + 0, c) :|: b >= 0, a >= 0, c >= 0, z' = 1 + a + b + c f''(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 foldf(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0 foldf(z', z'') -{ 1 }-> f(foldf(x, z), y) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z g(z') -{ 1 }-> 2 :|: z' = 2 g(z') -{ 1 }-> 1 :|: z' = 1 g(z') -{ 1 }-> 1 :|: z' = 2 g(z') -{ 1 }-> 0 :|: z' = 0 g(z') -{ 1 }-> 0 :|: z' = 1 g(z') -{ 1 }-> 0 :|: z' = 2 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (21) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: G(A) -> c G(B) -> c1 G(B) -> c2 G(C) -> c3 G(C) -> c4 G(C) -> c5 FOLDF(z0, nil) -> c6 FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F(z0, z1) -> c8(F'(z0, g(z1)), G(z1)) F'(triple(z0, z1, z2), C) -> c9 F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F'(triple(z0, z1, z2), A) -> c11(F''(foldf(triple(cons(A, z0), nil, z2), z1)), FOLDF(triple(cons(A, z0), nil, z2), z1)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) S tuples: G(A) -> c G(B) -> c1 G(B) -> c2 G(C) -> c3 G(C) -> c4 G(C) -> c5 FOLDF(z0, nil) -> c6 FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F(z0, z1) -> c8(F'(z0, g(z1)), G(z1)) F'(triple(z0, z1, z2), C) -> c9 F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F'(triple(z0, z1, z2), A) -> c11(F''(foldf(triple(cons(A, z0), nil, z2), z1)), FOLDF(triple(cons(A, z0), nil, z2), z1)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: G_1, FOLDF_2, F_2, F'_2, F''_1 Compound Symbols: c, c1, c2, c3, c4, c5, c6, c7_2, c8_2, c9, c10_1, c11_2, c12_1 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 8 trailing nodes: G(C) -> c4 F'(triple(z0, z1, z2), C) -> c9 FOLDF(z0, nil) -> c6 G(B) -> c2 G(C) -> c3 G(B) -> c1 G(A) -> c G(C) -> c5 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F(z0, z1) -> c8(F'(z0, g(z1)), G(z1)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F'(triple(z0, z1, z2), A) -> c11(F''(foldf(triple(cons(A, z0), nil, z2), z1)), FOLDF(triple(cons(A, z0), nil, z2), z1)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) S tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F(z0, z1) -> c8(F'(z0, g(z1)), G(z1)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F'(triple(z0, z1, z2), A) -> c11(F''(foldf(triple(cons(A, z0), nil, z2), z1)), FOLDF(triple(cons(A, z0), nil, z2), z1)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: FOLDF_2, F_2, F'_2, F''_1 Compound Symbols: c7_2, c8_2, c10_1, c11_2, c12_1 ---------------------------------------- (25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F'(triple(z0, z1, z2), A) -> c11(F''(foldf(triple(cons(A, z0), nil, z2), z1)), FOLDF(triple(cons(A, z0), nil, z2), z1)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) F(z0, z1) -> c8(F'(z0, g(z1))) S tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F'(triple(z0, z1, z2), A) -> c11(F''(foldf(triple(cons(A, z0), nil, z2), z1)), FOLDF(triple(cons(A, z0), nil, z2), z1)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) F(z0, z1) -> c8(F'(z0, g(z1))) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: FOLDF_2, F'_2, F''_1, F_2 Compound Symbols: c7_2, c10_1, c11_2, c12_1, c8_1 ---------------------------------------- (27) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F'(triple(z0, z1, z2), A) -> c11(F''(foldf(triple(cons(A, z0), nil, z2), z1)), FOLDF(triple(cons(A, z0), nil, z2), z1)) by F'(triple(x0, nil, x2), A) -> c11(F''(triple(cons(A, x0), nil, x2)), FOLDF(triple(cons(A, x0), nil, x2), nil)) F'(triple(x0, cons(z1, z2), x2), A) -> c11(F''(f(foldf(triple(cons(A, x0), nil, x2), z2), z1)), FOLDF(triple(cons(A, x0), nil, x2), cons(z1, z2))) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) F(z0, z1) -> c8(F'(z0, g(z1))) F'(triple(x0, nil, x2), A) -> c11(F''(triple(cons(A, x0), nil, x2)), FOLDF(triple(cons(A, x0), nil, x2), nil)) F'(triple(x0, cons(z1, z2), x2), A) -> c11(F''(f(foldf(triple(cons(A, x0), nil, x2), z2), z1)), FOLDF(triple(cons(A, x0), nil, x2), cons(z1, z2))) S tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) F(z0, z1) -> c8(F'(z0, g(z1))) F'(triple(x0, nil, x2), A) -> c11(F''(triple(cons(A, x0), nil, x2)), FOLDF(triple(cons(A, x0), nil, x2), nil)) F'(triple(x0, cons(z1, z2), x2), A) -> c11(F''(f(foldf(triple(cons(A, x0), nil, x2), z2), z1)), FOLDF(triple(cons(A, x0), nil, x2), cons(z1, z2))) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: FOLDF_2, F'_2, F''_1, F_2 Compound Symbols: c7_2, c10_1, c12_1, c8_1, c11_2 ---------------------------------------- (29) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) F(z0, z1) -> c8(F'(z0, g(z1))) F'(triple(x0, cons(z1, z2), x2), A) -> c11(F''(f(foldf(triple(cons(A, x0), nil, x2), z2), z1)), FOLDF(triple(cons(A, x0), nil, x2), cons(z1, z2))) F'(triple(x0, nil, x2), A) -> c11(F''(triple(cons(A, x0), nil, x2))) S tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) F(z0, z1) -> c8(F'(z0, g(z1))) F'(triple(x0, cons(z1, z2), x2), A) -> c11(F''(f(foldf(triple(cons(A, x0), nil, x2), z2), z1)), FOLDF(triple(cons(A, x0), nil, x2), cons(z1, z2))) F'(triple(x0, nil, x2), A) -> c11(F''(triple(cons(A, x0), nil, x2))) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: FOLDF_2, F'_2, F''_1, F_2 Compound Symbols: c7_2, c10_1, c12_1, c8_1, c11_2, c11_1 ---------------------------------------- (31) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(z0, z1) -> c8(F'(z0, g(z1))) by F(x0, A) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, C)) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) F'(triple(x0, cons(z1, z2), x2), A) -> c11(F''(f(foldf(triple(cons(A, x0), nil, x2), z2), z1)), FOLDF(triple(cons(A, x0), nil, x2), cons(z1, z2))) F'(triple(x0, nil, x2), A) -> c11(F''(triple(cons(A, x0), nil, x2))) F(x0, A) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, C)) S tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) F'(triple(x0, cons(z1, z2), x2), A) -> c11(F''(f(foldf(triple(cons(A, x0), nil, x2), z2), z1)), FOLDF(triple(cons(A, x0), nil, x2), cons(z1, z2))) F'(triple(x0, nil, x2), A) -> c11(F''(triple(cons(A, x0), nil, x2))) F(x0, A) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, C)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: FOLDF_2, F'_2, F''_1, F_2 Compound Symbols: c7_2, c10_1, c12_1, c11_2, c11_1, c8_1 ---------------------------------------- (33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(x0, C) -> c8(F'(x0, C)) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) F'(triple(x0, cons(z1, z2), x2), A) -> c11(F''(f(foldf(triple(cons(A, x0), nil, x2), z2), z1)), FOLDF(triple(cons(A, x0), nil, x2), cons(z1, z2))) F'(triple(x0, nil, x2), A) -> c11(F''(triple(cons(A, x0), nil, x2))) F(x0, A) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) S tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) F'(triple(x0, cons(z1, z2), x2), A) -> c11(F''(f(foldf(triple(cons(A, x0), nil, x2), z2), z1)), FOLDF(triple(cons(A, x0), nil, x2), cons(z1, z2))) F'(triple(x0, nil, x2), A) -> c11(F''(triple(cons(A, x0), nil, x2))) F(x0, A) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: FOLDF_2, F'_2, F''_1, F_2 Compound Symbols: c7_2, c10_1, c12_1, c11_2, c11_1, c8_1 ---------------------------------------- (35) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F'(triple(x0, cons(z1, z2), x2), A) -> c11(F''(f(foldf(triple(cons(A, x0), nil, x2), z2), z1)), FOLDF(triple(cons(A, x0), nil, x2), cons(z1, z2))) by F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) F'(triple(x0, nil, x2), A) -> c11(F''(triple(cons(A, x0), nil, x2))) F(x0, A) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) S tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) F'(triple(x0, nil, x2), A) -> c11(F''(triple(cons(A, x0), nil, x2))) F(x0, A) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: FOLDF_2, F'_2, F''_1, F_2 Compound Symbols: c7_2, c10_1, c12_1, c11_1, c8_1, c11_2 ---------------------------------------- (37) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F''(triple(z0, z1, z2)) -> c12(FOLDF(triple(z0, z1, nil), z2)) by F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F'(triple(x0, nil, x2), A) -> c11(F''(triple(cons(A, x0), nil, x2))) F(x0, A) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) S tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F'(triple(x0, nil, x2), A) -> c11(F''(triple(cons(A, x0), nil, x2))) F(x0, A) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: FOLDF_2, F'_2, F_2, F''_1 Compound Symbols: c7_2, c10_1, c11_1, c8_1, c11_2, c12_1 ---------------------------------------- (39) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F'(triple(x0, nil, x2), A) -> c11(F''(triple(cons(A, x0), nil, x2))) by F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F(x0, A) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) S tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F(x0, A) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: FOLDF_2, F'_2, F_2, F''_1 Compound Symbols: c7_2, c10_1, c8_1, c11_2, c11_1, c12_1 ---------------------------------------- (41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(x0, A) -> c8(F'(x0, A)) by F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F(x0, B) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) S tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) F(x0, B) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: FOLDF_2, F'_2, F_2, F''_1 Compound Symbols: c7_2, c10_1, c8_1, c11_2, c11_1, c12_1 ---------------------------------------- (43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F'(triple(z0, z1, z2), B) -> c10(F(triple(z0, z1, z2), A)) by F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F(x0, B) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) S tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F(x0, B) -> c8(F'(x0, A)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: FOLDF_2, F_2, F'_2, F''_1 Compound Symbols: c7_2, c8_1, c11_2, c11_1, c12_1, c10_1 ---------------------------------------- (45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(x0, B) -> c8(F'(x0, A)) by F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) S tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F(x0, B) -> c8(F'(x0, B)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: FOLDF_2, F_2, F'_2, F''_1 Compound Symbols: c7_2, c8_1, c11_2, c11_1, c12_1, c10_1 ---------------------------------------- (47) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(x0, B) -> c8(F'(x0, B)) by F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) S tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F(x0, C) -> c8(F'(x0, A)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: FOLDF_2, F_2, F'_2, F''_1 Compound Symbols: c7_2, c8_1, c11_2, c11_1, c12_1, c10_1 ---------------------------------------- (49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(x0, C) -> c8(F'(x0, A)) by F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) S tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F(x0, C) -> c8(F'(x0, B)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: FOLDF_2, F_2, F'_2, F''_1 Compound Symbols: c7_2, c8_1, c11_2, c11_1, c12_1, c10_1 ---------------------------------------- (51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(x0, C) -> c8(F'(x0, B)) by F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) S tuples: FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: FOLDF_2, F'_2, F''_1, F_2 Compound Symbols: c7_2, c11_2, c11_1, c12_1, c8_1, c10_1 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FOLDF(z0, cons(z1, z2)) -> c7(F(foldf(z0, z2), z1), FOLDF(z0, z2)) by FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1), FOLDF(z0, nil)) FOLDF(z0, cons(x1, cons(z1, z2))) -> c7(F(f(foldf(z0, z2), z1), x1), FOLDF(z0, cons(z1, z2))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1), FOLDF(z0, nil)) FOLDF(z0, cons(x1, cons(z1, z2))) -> c7(F(f(foldf(z0, z2), z1), x1), FOLDF(z0, cons(z1, z2))) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1), FOLDF(z0, nil)) FOLDF(z0, cons(x1, cons(z1, z2))) -> c7(F(f(foldf(z0, z2), z1), x1), FOLDF(z0, cons(z1, z2))) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F''_1, F_2, FOLDF_2 Compound Symbols: c11_2, c11_1, c12_1, c8_1, c10_1, c7_2 ---------------------------------------- (55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, cons(z1, z2))) -> c7(F(f(foldf(z0, z2), z1), x1), FOLDF(z0, cons(z1, z2))) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, cons(z1, z2))) -> c7(F(f(foldf(z0, z2), z1), x1), FOLDF(z0, cons(z1, z2))) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F''_1, F_2, FOLDF_2 Compound Symbols: c11_2, c11_1, c12_1, c8_1, c10_1, c7_2, c7_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FOLDF(z0, cons(x1, cons(z1, z2))) -> c7(F(f(foldf(z0, z2), z1), x1), FOLDF(z0, cons(z1, z2))) by FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F''_1, F_2, FOLDF_2 Compound Symbols: c11_2, c11_1, c12_1, c8_1, c10_1, c7_1, c7_2 ---------------------------------------- (59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F'(triple(x0, cons(x1, x2), x3), A) -> c11(FOLDF(triple(cons(A, x0), nil, x3), cons(x1, x2))) by F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F''_1, F_2, FOLDF_2 Compound Symbols: c11_2, c12_1, c11_1, c8_1, c10_1, c7_1, c7_2 ---------------------------------------- (61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F''(triple(z0, z1, cons(y1, y2))) -> c12(FOLDF(triple(z0, z1, nil), cons(y1, y2))) by F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F_2, FOLDF_2, F''_1 Compound Symbols: c11_2, c11_1, c8_1, c10_1, c7_1, c7_2, c12_1 ---------------------------------------- (63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F'(triple(z0, nil, cons(y2, y3)), A) -> c11(F''(triple(cons(A, z0), nil, cons(y2, y3)))) by F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F_2, FOLDF_2, F''_1 Compound Symbols: c11_2, c8_1, c10_1, c7_1, c7_2, c11_1, c12_1 ---------------------------------------- (65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(triple(y0, nil, cons(y1, y2)), A) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) by F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F_2, FOLDF_2, F''_1 Compound Symbols: c11_2, c8_1, c10_1, c7_1, c7_2, c11_1, c12_1 ---------------------------------------- (67) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F'(triple(z0, nil, cons(y1, y2)), B) -> c10(F(triple(z0, nil, cons(y1, y2)), A)) by F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F_2, FOLDF_2, F''_1 Compound Symbols: c11_2, c8_1, c10_1, c7_1, c7_2, c11_1, c12_1 ---------------------------------------- (69) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) by F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F_2, FOLDF_2, F''_1 Compound Symbols: c11_2, c8_1, c10_1, c7_1, c7_2, c11_1, c12_1 ---------------------------------------- (71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(triple(y0, nil, cons(y1, y2)), B) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) by F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F_2, FOLDF_2, F''_1 Compound Symbols: c11_2, c8_1, c10_1, c7_1, c7_2, c11_1, c12_1 ---------------------------------------- (73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), A)) by F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F_2, FOLDF_2, F''_1 Compound Symbols: c11_2, c8_1, c10_1, c7_1, c7_2, c11_1, c12_1 ---------------------------------------- (75) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(triple(y0, nil, cons(y1, y2)), C) -> c8(F'(triple(y0, nil, cons(y1, y2)), B)) by F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F_2, FOLDF_2, F''_1 Compound Symbols: c11_2, c8_1, c10_1, c7_1, c7_2, c11_1, c12_1 ---------------------------------------- (77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FOLDF(z0, cons(x1, nil)) -> c7(F(z0, x1)) by FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) K tuples:none Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F_2, FOLDF_2, F''_1 Compound Symbols: c11_2, c8_1, c10_1, c7_2, c11_1, c12_1, c7_1 ---------------------------------------- (79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) We considered the (Usable) Rules: f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f(z0, z1) -> f'(z0, g(z1)) foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) And the Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) The order we found is given by the following interpretation: Polynomial interpretation : POL(A) = 0 POL(B) = 0 POL(C) = 0 POL(F(x_1, x_2)) = 0 POL(F'(x_1, x_2)) = x_2 POL(F''(x_1)) = x_1 POL(FOLDF(x_1, x_2)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(f(x_1, x_2)) = x_1 POL(f'(x_1, x_2)) = x_1 POL(f''(x_1)) = x_1 POL(foldf(x_1, x_2)) = x_1 POL(g(x_1)) = [1] + x_1 POL(nil) = 0 POL(triple(x_1, x_2, x_3)) = x_2 ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) K tuples: FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F_2, FOLDF_2, F''_1 Compound Symbols: c11_2, c8_1, c10_1, c7_2, c11_1, c12_1, c7_1 ---------------------------------------- (81) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F'(triple(z0, cons(z1, nil), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, nil))) by F'(triple(z0, cons(A, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil))) F'(triple(z0, cons(B, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil))) F'(triple(z0, cons(C, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) F'(triple(z0, cons(A, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil))) F'(triple(z0, cons(B, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil))) F'(triple(z0, cons(C, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil))) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) F'(triple(z0, cons(A, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil))) F'(triple(z0, cons(B, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil))) F'(triple(z0, cons(C, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil))) K tuples: FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F_2, FOLDF_2, F''_1 Compound Symbols: c11_2, c8_1, c10_1, c7_2, c11_1, c12_1, c7_1 ---------------------------------------- (83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F''(triple(z0, z1, cons(z2, nil))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, nil))) by F''(triple(z0, cons(y1, y2), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(A, nil))) F''(triple(z0, cons(y1, nil), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(A, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(A, nil))) F''(triple(z0, cons(y1, y2), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(B, nil))) F''(triple(z0, cons(y1, nil), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(B, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(B, nil))) F''(triple(z0, cons(y1, y2), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(C, nil))) F''(triple(z0, cons(y1, nil), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(C, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(C, nil))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) F'(triple(z0, cons(A, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil))) F'(triple(z0, cons(B, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil))) F'(triple(z0, cons(C, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil))) F''(triple(z0, cons(y1, y2), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(A, nil))) F''(triple(z0, cons(y1, nil), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(A, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(A, nil))) F''(triple(z0, cons(y1, y2), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(B, nil))) F''(triple(z0, cons(y1, nil), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(B, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(B, nil))) F''(triple(z0, cons(y1, y2), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(C, nil))) F''(triple(z0, cons(y1, nil), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(C, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(C, nil))) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) F'(triple(z0, cons(A, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil))) F'(triple(z0, cons(B, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil))) F'(triple(z0, cons(C, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil))) F''(triple(z0, cons(y1, y2), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(A, nil))) F''(triple(z0, cons(y1, nil), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(A, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(A, nil))) F''(triple(z0, cons(y1, y2), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(B, nil))) F''(triple(z0, cons(y1, nil), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(B, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(B, nil))) F''(triple(z0, cons(y1, y2), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(C, nil))) F''(triple(z0, cons(y1, nil), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(C, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(C, nil))) K tuples: FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F_2, FOLDF_2, F''_1 Compound Symbols: c11_2, c8_1, c10_1, c7_2, c11_1, c12_1, c7_1 ---------------------------------------- (85) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 13 trailing nodes: F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), A)) F'(triple(z0, cons(C, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(C, nil))) FOLDF(triple(y0, nil, cons(y1, nil)), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), C)) F'(triple(z0, cons(A, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(A, nil))) F(triple(z0, nil, cons(z1, nil)), B) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), B)) F'(triple(z0, nil, cons(z1, nil)), B) -> c10(F(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, nil)), C) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) F(triple(z0, nil, cons(z1, nil)), A) -> c8(F'(triple(z0, nil, cons(z1, nil)), A)) FOLDF(triple(y0, nil, cons(y1, nil)), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, nil)), B)) F'(triple(z0, cons(B, nil), cons(y1, nil)), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, nil)), cons(B, nil))) F'(triple(z0, nil, cons(z1, nil)), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, nil)))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil))) F''(triple(z0, cons(y1, y2), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(A, nil))) F''(triple(z0, cons(y1, nil), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(A, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(A, nil))) F''(triple(z0, cons(y1, y2), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(B, nil))) F''(triple(z0, cons(y1, nil), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(B, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(B, nil))) F''(triple(z0, cons(y1, y2), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(C, nil))) F''(triple(z0, cons(y1, nil), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(C, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(C, nil))) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil))) F''(triple(z0, cons(y1, y2), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(A, nil))) F''(triple(z0, cons(y1, nil), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(A, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(A, nil))) F''(triple(z0, cons(y1, y2), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(B, nil))) F''(triple(z0, cons(y1, nil), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(B, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(B, nil))) F''(triple(z0, cons(y1, y2), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(C, nil))) F''(triple(z0, cons(y1, nil), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(C, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(C, nil))) K tuples: FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F_2, FOLDF_2, F''_1 Compound Symbols: c11_2, c8_1, c10_1, c7_2, c11_1, c12_1, c7_1 ---------------------------------------- (87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F''(triple(z0, cons(y1, y2), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(A, nil))) F''(triple(z0, cons(y1, nil), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(A, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(A, nil))) We considered the (Usable) Rules: f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f(z0, z1) -> f'(z0, g(z1)) foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) And the Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil))) F''(triple(z0, cons(y1, y2), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(A, nil))) F''(triple(z0, cons(y1, nil), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(A, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(A, nil))) F''(triple(z0, cons(y1, y2), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(B, nil))) F''(triple(z0, cons(y1, nil), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(B, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(B, nil))) F''(triple(z0, cons(y1, y2), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(C, nil))) F''(triple(z0, cons(y1, nil), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(C, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(C, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(A) = [1] POL(B) = 0 POL(C) = 0 POL(F(x_1, x_2)) = x_1 POL(F'(x_1, x_2)) = x_1 POL(F''(x_1)) = x_1 POL(FOLDF(x_1, x_2)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 POL(f(x_1, x_2)) = 0 POL(f'(x_1, x_2)) = 0 POL(f''(x_1)) = 0 POL(foldf(x_1, x_2)) = x_1 POL(g(x_1)) = [1] + x_1 POL(nil) = 0 POL(triple(x_1, x_2, x_3)) = x_3 ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil))) F''(triple(z0, cons(y1, y2), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(A, nil))) F''(triple(z0, cons(y1, nil), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(A, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(A, nil))) F''(triple(z0, cons(y1, y2), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(B, nil))) F''(triple(z0, cons(y1, nil), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(B, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(B, nil))) F''(triple(z0, cons(y1, y2), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(C, nil))) F''(triple(z0, cons(y1, nil), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(C, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(C, nil))) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil))) F''(triple(z0, cons(y1, y2), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(B, nil))) F''(triple(z0, cons(y1, nil), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(B, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(B, nil))) F''(triple(z0, cons(y1, y2), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(C, nil))) F''(triple(z0, cons(y1, nil), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(C, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(C, nil))) K tuples: FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) F''(triple(z0, cons(y1, y2), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(A, nil))) F''(triple(z0, cons(y1, nil), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(A, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(A, nil))) Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F_2, FOLDF_2, F''_1 Compound Symbols: c11_2, c8_1, c10_1, c7_2, c11_1, c12_1, c7_1 ---------------------------------------- (89) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F''(triple(z0, cons(y1, y2), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(B, nil))) F''(triple(z0, cons(y1, nil), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(B, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(B, nil))) F''(triple(z0, cons(y1, y2), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(C, nil))) F''(triple(z0, cons(y1, nil), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(C, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(C, nil))) We considered the (Usable) Rules: f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f(z0, z1) -> f'(z0, g(z1)) foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) And the Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil))) F''(triple(z0, cons(y1, y2), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(A, nil))) F''(triple(z0, cons(y1, nil), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(A, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(A, nil))) F''(triple(z0, cons(y1, y2), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(B, nil))) F''(triple(z0, cons(y1, nil), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(B, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(B, nil))) F''(triple(z0, cons(y1, y2), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(C, nil))) F''(triple(z0, cons(y1, nil), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(C, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(C, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(A) = 0 POL(B) = 0 POL(C) = 0 POL(F(x_1, x_2)) = 0 POL(F'(x_1, x_2)) = x_2 POL(F''(x_1)) = x_1 POL(FOLDF(x_1, x_2)) = 0 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c12(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] POL(f(x_1, x_2)) = x_1 POL(f'(x_1, x_2)) = x_1 POL(f''(x_1)) = x_1 POL(foldf(x_1, x_2)) = x_1 POL(g(x_1)) = [1] + x_1 POL(nil) = 0 POL(triple(x_1, x_2, x_3)) = x_2 ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(z0, nil) -> z0 foldf(z0, cons(z1, z2)) -> f(foldf(z0, z2), z1) f(z0, z1) -> f'(z0, g(z1)) f'(triple(z0, z1, z2), C) -> triple(z0, z1, cons(C, z2)) f'(triple(z0, z1, z2), B) -> f(triple(z0, z1, z2), A) f'(triple(z0, z1, z2), A) -> f''(foldf(triple(cons(A, z0), nil, z2), z1)) f''(triple(z0, z1, z2)) -> foldf(triple(z0, z1, nil), z2) Tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil))) F''(triple(z0, cons(y1, y2), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(A, nil))) F''(triple(z0, cons(y1, nil), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(A, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(A, nil))) F''(triple(z0, cons(y1, y2), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(B, nil))) F''(triple(z0, cons(y1, nil), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(B, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(B, nil))) F''(triple(z0, cons(y1, y2), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(C, nil))) F''(triple(z0, cons(y1, nil), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(C, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(C, nil))) S tuples: F'(triple(x0, cons(z1, x2), x3), A) -> c11(F''(f'(foldf(triple(cons(A, x0), nil, x3), x2), g(z1))), FOLDF(triple(cons(A, x0), nil, x3), cons(z1, x2))) F'(triple(x0, cons(x1, nil), x3), A) -> c11(F''(f(triple(cons(A, x0), nil, x3), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, nil))) F'(triple(x0, cons(x1, cons(z1, z2)), x3), A) -> c11(F''(f(f(foldf(triple(cons(A, x0), nil, x3), z2), z1), x1)), FOLDF(triple(cons(A, x0), nil, x3), cons(x1, cons(z1, z2)))) F(triple(y0, cons(y1, y2), y3), A) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), A) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), A) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F'(triple(z0, cons(y1, y2), z2), B) -> c10(F(triple(z0, cons(y1, y2), z2), A)) F'(triple(z0, cons(y1, nil), z2), B) -> c10(F(triple(z0, cons(y1, nil), z2), A)) F'(triple(z0, cons(y1, cons(y2, y3)), z2), B) -> c10(F(triple(z0, cons(y1, cons(y2, y3)), z2), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), B) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), B) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), B) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), A)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), A)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), A)) F(triple(y0, cons(y1, y2), y3), C) -> c8(F'(triple(y0, cons(y1, y2), y3), B)) F(triple(y0, cons(y1, nil), y2), C) -> c8(F'(triple(y0, cons(y1, nil), y2), B)) F(triple(y0, cons(y1, cons(y2, y3)), y4), C) -> c8(F'(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(x0, cons(x1, cons(z1, x3))) -> c7(F(f'(foldf(x0, x3), g(z1)), x1), FOLDF(x0, cons(z1, x3))) FOLDF(z0, cons(x1, cons(x2, nil))) -> c7(F(f(z0, x2), x1), FOLDF(z0, cons(x2, nil))) FOLDF(z0, cons(x1, cons(x2, cons(z1, z2)))) -> c7(F(f(f(foldf(z0, z2), z1), x2), x1), FOLDF(z0, cons(x2, cons(z1, z2)))) F'(triple(z0, cons(z1, cons(y2, y3)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, y3)))) F'(triple(z0, cons(z1, cons(y2, nil)), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, nil)))) F'(triple(z0, cons(z1, cons(y2, cons(y3, y4))), z3), A) -> c11(FOLDF(triple(cons(A, z0), nil, z3), cons(z1, cons(y2, cons(y3, y4))))) F''(triple(z0, z1, cons(z2, cons(y2, y3)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, y3)))) F''(triple(z0, z1, cons(z2, cons(y2, nil)))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, nil)))) F''(triple(z0, z1, cons(z2, cons(y2, cons(y3, y4))))) -> c12(FOLDF(triple(z0, z1, nil), cons(z2, cons(y2, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, y4))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, y4))))) F'(triple(z0, nil, cons(z1, cons(y3, nil))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, nil))))) F'(triple(z0, nil, cons(z1, cons(y3, cons(y4, y5)))), A) -> c11(F''(triple(cons(A, z0), nil, cons(z1, cons(y3, cons(y4, y5)))))) F(triple(z0, nil, cons(z1, cons(y2, y3))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F'(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F'(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c10(F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), A)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), A)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), A)) F(triple(z0, nil, cons(z1, cons(y2, y3))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, y3))), B)) F(triple(z0, nil, cons(z1, cons(y2, nil))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, nil))), B)) F(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), C) -> c8(F'(triple(z0, nil, cons(z1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), A)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), B)) FOLDF(triple(y0, nil, cons(y1, cons(y2, y3))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, y3))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, nil))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, nil))), C)) FOLDF(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil)) -> c7(F(triple(y0, nil, cons(y1, cons(y2, cons(y3, y4)))), C)) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(A, nil))) F'(triple(z0, cons(A, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(A, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(B, nil))) F'(triple(z0, cons(B, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(B, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, y3))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, y3))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, nil))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, nil))), cons(C, nil))) F'(triple(z0, cons(C, nil), cons(y1, cons(y2, cons(y3, y4)))), A) -> c11(FOLDF(triple(cons(A, z0), nil, cons(y1, cons(y2, cons(y3, y4)))), cons(C, nil))) K tuples: FOLDF(triple(y0, cons(y1, y2), y3), cons(A, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), A)) FOLDF(triple(y0, cons(y1, nil), y2), cons(A, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), A)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(A, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), A)) FOLDF(triple(y0, cons(y1, y2), y3), cons(B, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), B)) FOLDF(triple(y0, cons(y1, nil), y2), cons(B, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), B)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(B, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), B)) FOLDF(triple(y0, cons(y1, y2), y3), cons(C, nil)) -> c7(F(triple(y0, cons(y1, y2), y3), C)) FOLDF(triple(y0, cons(y1, nil), y2), cons(C, nil)) -> c7(F(triple(y0, cons(y1, nil), y2), C)) FOLDF(triple(y0, cons(y1, cons(y2, y3)), y4), cons(C, nil)) -> c7(F(triple(y0, cons(y1, cons(y2, y3)), y4), C)) F''(triple(z0, cons(y1, y2), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(A, nil))) F''(triple(z0, cons(y1, nil), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(A, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(A, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(A, nil))) F''(triple(z0, cons(y1, y2), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(B, nil))) F''(triple(z0, cons(y1, nil), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(B, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(B, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(B, nil))) F''(triple(z0, cons(y1, y2), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, y2), nil), cons(C, nil))) F''(triple(z0, cons(y1, nil), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, nil), nil), cons(C, nil))) F''(triple(z0, cons(y1, cons(y2, y3)), cons(C, nil))) -> c12(FOLDF(triple(z0, cons(y1, cons(y2, y3)), nil), cons(C, nil))) Defined Rule Symbols: g_1, foldf_2, f_2, f'_2, f''_1 Defined Pair Symbols: F'_2, F_2, FOLDF_2, F''_1 Compound Symbols: c11_2, c8_1, c10_1, c7_2, c11_1, c12_1, c7_1