KILLED proof of input_T1EdOsr30l.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 315 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 50 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 3649 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 904 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (38) CdtProblem (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 109 ms] (42) CdtProblem (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 7 ms] (148) CdtProblem (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) 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: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) @(Nil, ys) -> ys binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs'))) binom(Cons(x, xs), Nil) -> Cons(Nil, Nil) binom(Nil, k) -> Cons(Nil, Nil) goal(x, y) -> binom(x, y) 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: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) @(Nil, ys) -> ys binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs'))) binom(Cons(x, xs), Nil) -> Cons(Nil, Nil) binom(Nil, k) -> Cons(Nil, Nil) goal(x, y) -> binom(x, y) 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: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) @(Nil, ys) -> ys binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs'))) binom(Cons(x, xs), Nil) -> Cons(Nil, Nil) binom(Nil, k) -> Cons(Nil, Nil) goal(x, y) -> binom(x, y) 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: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) [1] @(Nil, ys) -> ys [1] binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs'))) [1] binom(Cons(x, xs), Nil) -> Cons(Nil, Nil) [1] binom(Nil, k) -> Cons(Nil, Nil) [1] goal(x, y) -> binom(x, y) [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: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) [1] @(Nil, ys) -> ys [1] binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs'))) [1] binom(Cons(x, xs), Nil) -> Cons(Nil, Nil) [1] binom(Nil, k) -> Cons(Nil, Nil) [1] goal(x, y) -> binom(x, y) [1] The TRS has the following type information: @ :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil binom :: Cons:Nil -> Cons:Nil -> Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil 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: goal_2 (c) The following functions are completely defined: binom_2 @_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) [1] @(Nil, ys) -> ys [1] binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs'))) [1] binom(Cons(x, xs), Nil) -> Cons(Nil, Nil) [1] binom(Nil, k) -> Cons(Nil, Nil) [1] goal(x, y) -> binom(x, y) [1] The TRS has the following type information: @ :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil binom :: Cons:Nil -> Cons:Nil -> Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil 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: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) [1] @(Nil, ys) -> ys [1] binom(Cons(x, Cons(x'', xs'')), Cons(x', Cons(x''', xs'''))) -> @(@(binom(xs'', xs'''), binom(xs'', Cons(x''', xs'''))), @(binom(xs'', Cons(x''', xs''')), binom(xs'', Cons(x', Cons(x''', xs'''))))) [3] binom(Cons(x, Cons(x1, xs1)), Cons(x', Nil)) -> @(Cons(Nil, Nil), @(binom(xs1, Nil), binom(xs1, Cons(x', Nil)))) [3] binom(Cons(x, Nil), Cons(x', xs')) -> @(Cons(Nil, Nil), Cons(Nil, Nil)) [3] binom(Cons(x, xs), Nil) -> Cons(Nil, Nil) [1] binom(Nil, k) -> Cons(Nil, Nil) [1] goal(x, y) -> binom(x, y) [1] The TRS has the following type information: @ :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil binom :: Cons:Nil -> Cons:Nil -> Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil 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: Nil => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0 binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''') binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + x' + 0))) :|: x1 >= 0, z' = 1 + x' + 0, x >= 0, x' >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1) binom(z, z') -{ 3 }-> @(1 + 0 + 0, 1 + 0 + 0) :|: x >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs', z = 1 + x + 0 binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: k >= 0, z' = k, z = 0 goal(z, z') -{ 1 }-> binom(x, y) :|: x >= 0, y >= 0, z = x, z' = y ---------------------------------------- (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: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''') binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1) binom(z, z') -{ 3 }-> @(1 + 0 + 0, 1 + 0 + 0) :|: z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs' binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { @ } { binom } { goal } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''') binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1) binom(z, z') -{ 3 }-> @(1 + 0 + 0, 1 + 0 + 0) :|: z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs' binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0 Function symbols to be analyzed: {@}, {binom}, {goal} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''') binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1) binom(z, z') -{ 3 }-> @(1 + 0 + 0, 1 + 0 + 0) :|: z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs' binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0 Function symbols to be analyzed: {@}, {binom}, {goal} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: @ after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''') binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1) binom(z, z') -{ 3 }-> @(1 + 0 + 0, 1 + 0 + 0) :|: z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs' binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0 Function symbols to be analyzed: {@}, {binom}, {goal} Previous analysis results are: @: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: @ after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''') binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1) binom(z, z') -{ 3 }-> @(1 + 0 + 0, 1 + 0 + 0) :|: z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs' binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0 Function symbols to be analyzed: {binom}, {goal} Previous analysis results are: @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 binom(z, z') -{ 5 }-> s' :|: s' >= 0, s' <= 1 + 0 + 0 + (1 + 0 + 0), z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs' binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''') binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1) binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0 Function symbols to be analyzed: {binom}, {goal} Previous analysis results are: @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: binom after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 binom(z, z') -{ 5 }-> s' :|: s' >= 0, s' <= 1 + 0 + 0 + (1 + 0 + 0), z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs' binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''') binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1) binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0 Function symbols to be analyzed: {binom}, {goal} Previous analysis results are: @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] binom: runtime: ?, size: EXP ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: binom after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 @(z, z') -{ 2 + xs }-> 1 + x + s :|: s >= 0, s <= xs + z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0 binom(z, z') -{ 5 }-> s' :|: s' >= 0, s' <= 1 + 0 + 0 + (1 + 0 + 0), z - 1 >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs' binom(z, z') -{ 3 }-> @(@(binom(xs'', xs'''), binom(xs'', 1 + x''' + xs''')), @(binom(xs'', 1 + x''' + xs'''), binom(xs'', 1 + x' + (1 + x''' + xs''')))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x' >= 0, xs''' >= 0, x''' >= 0, x'' >= 0, z' = 1 + x' + (1 + x''' + xs''') binom(z, z') -{ 3 }-> @(1 + 0 + 0, @(binom(xs1, 0), binom(xs1, 1 + (z' - 1) + 0))) :|: x1 >= 0, x >= 0, z' - 1 >= 0, xs1 >= 0, z = 1 + x + (1 + x1 + xs1) binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 goal(z, z') -{ 1 }-> binom(z, z') :|: z >= 0, z' >= 0 Function symbols to be analyzed: {binom}, {goal} Previous analysis results are: @: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] binom: runtime: INF, size: EXP ---------------------------------------- (31) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (32) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) [1] @(Nil, ys) -> ys [1] binom(Cons(x, xs), Cons(x', xs')) -> @(binom(xs, xs'), binom(xs, Cons(x', xs'))) [1] binom(Cons(x, xs), Nil) -> Cons(Nil, Nil) [1] binom(Nil, k) -> Cons(Nil, Nil) [1] goal(x, y) -> binom(x, y) [1] The TRS has the following type information: @ :: Cons:Nil -> Cons:Nil -> Cons:Nil Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil Nil :: Cons:Nil binom :: Cons:Nil -> Cons:Nil -> Cons:Nil goal :: Cons:Nil -> Cons:Nil -> Cons:Nil Rewrite Strategy: INNERMOST ---------------------------------------- (33) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: Nil => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: @(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0 @(z, z') -{ 1 }-> 1 + x + @(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0 binom(z, z') -{ 1 }-> @(binom(xs, xs'), binom(xs, 1 + x' + xs')) :|: z = 1 + x + xs, xs >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs' binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0 binom(z, z') -{ 1 }-> 1 + 0 + 0 :|: k >= 0, z' = k, z = 0 goal(z, z') -{ 1 }-> binom(x, y) :|: x >= 0, y >= 0, z = x, z' = y Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) goal(z0, z1) -> binom(z0, z1) Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) @'(Nil, z0) -> c1 BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) BINOM(Cons(z0, z1), Nil) -> c4 BINOM(Nil, z0) -> c5 GOAL(z0, z1) -> c6(BINOM(z0, z1)) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) @'(Nil, z0) -> c1 BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) BINOM(Cons(z0, z1), Nil) -> c4 BINOM(Nil, z0) -> c5 GOAL(z0, z1) -> c6(BINOM(z0, z1)) K tuples:none Defined Rule Symbols: @_2, binom_2, goal_2 Defined Pair Symbols: @'_2, BINOM_2, GOAL_2 Compound Symbols: c_1, c1, c2_2, c3_2, c4, c5, c6_1 ---------------------------------------- (37) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: GOAL(z0, z1) -> c6(BINOM(z0, z1)) Removed 3 trailing nodes: BINOM(Nil, z0) -> c5 @'(Nil, z0) -> c1 BINOM(Cons(z0, z1), Nil) -> c4 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) goal(z0, z1) -> binom(z0, z1) Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) K tuples:none Defined Rule Symbols: @_2, binom_2, goal_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c2_2, c3_2 ---------------------------------------- (39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: goal(z0, z1) -> binom(z0, z1) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c2_2, c3_2 ---------------------------------------- (41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) We considered the (Usable) Rules:none And the Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) The order we found is given by the following interpretation: Polynomial interpretation : POL(@(x_1, x_2)) = [1] + x_2 POL(@'(x_1, x_2)) = 0 POL(BINOM(x_1, x_2)) = x_1 + x_2 POL(Cons(x_1, x_2)) = [1] + x_1 + x_2 POL(Nil) = 0 POL(binom(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2)) = x_1 + x_2 ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c2_2, c3_2 ---------------------------------------- (43) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) by BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c2(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), z3)) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil)), BINOM(Nil, x3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Nil)) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))), BINOM(Nil, z0)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c2(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), z3)) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil)), BINOM(Nil, x3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Nil)) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))), BINOM(Nil, z0)) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c3_2, c2_2 ---------------------------------------- (45) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c2(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil)))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c3_2, c2_2, c2_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) by BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)), BINOM(Nil, Cons(x2, x3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))), BINOM(Nil, Cons(x2, z0))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c2(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil)))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil)), BINOM(Nil, Cons(x2, x3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0))), BINOM(Nil, Cons(x2, z0))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c2_2, c2_1, c3_2 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c2(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil)))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c2_2, c2_1, c3_2, c3_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c2(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), z3)) by BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c2(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))), BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Nil)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c2(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))), BINOM(Cons(x1, Nil), z0)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Nil)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil)))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c2(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))), BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Nil)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c2(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))), BINOM(Cons(x1, Nil), z0)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Nil)) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c2_2, c2_1, c3_2, c3_1 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil)))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c2(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))), BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c2(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))), BINOM(Cons(x1, Nil), z0)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c2_2, c2_1, c3_2, c3_1 ---------------------------------------- (55) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil)))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c2_2, c2_1, c3_2, c3_1, c1_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) by BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, Cons(x4, x5))) -> c2(@'(@(binom(z1, x5), binom(z1, Cons(x4, x5))), @(binom(z1, Cons(x4, x5)), binom(z1, Cons(z2, Cons(x4, x5))))), BINOM(Cons(z0, z1), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c2(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))), BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c2(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))), BINOM(Cons(x1, Nil), Cons(x4, z0))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil)))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c2(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))), BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c2(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))), BINOM(Cons(x1, Nil), Cons(x4, z0))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c2_1, c3_2, c3_1, c2_2, c1_1 ---------------------------------------- (59) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil)))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c2_1, c3_2, c3_1, c2_2, c1_1, c4_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c2(@'(binom(Nil, x3), Cons(Nil, Nil))) by BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil)))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c2_1, c3_2, c3_1, c2_2, c1_1, c4_1 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c2(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil)))) by BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(z2, Nil))))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c2_1, c3_2, c3_1, c2_2, c1_1, c4_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c2(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) by BINOM(Cons(x0, Nil), Cons(x1, x2)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c3_2, c3_1, c2_2, c2_1, c1_1, c4_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, z3)) -> c3(@'(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(z2, z3))) by BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c3(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))), BINOM(Cons(x1, Nil), Cons(x3, x4))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c3(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))), BINOM(Cons(x1, Nil), Cons(x3, z0))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c3(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil))), BINOM(Cons(x1, Nil), Cons(x3, x4))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c3(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0)))), BINOM(Cons(x1, Nil), Cons(x3, z0))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c3_2, c3_1, c2_2, c2_1, c1_1, c4_1 ---------------------------------------- (69) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c3_2, c3_1, c2_2, c2_1, c1_1, c4_1, c5_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x2, Cons(z2, z3)))), BINOM(Cons(z0, z1), Cons(x2, Cons(z2, z3)))) by BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, Cons(x4, x5))) -> c3(@'(@(binom(z1, x5), binom(z1, Cons(x4, x5))), @(binom(z1, Cons(x4, x5)), binom(z1, Cons(z2, Cons(x4, x5))))), BINOM(Cons(z0, z1), Cons(z2, Cons(x4, x5)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c3(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))), BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c3(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))), BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c3(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))), BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c3(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))), BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, z0)))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c3_2, c3_1, c2_2, c2_1, c1_1, c4_1, c5_1 ---------------------------------------- (73) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c3_2, c3_1, c2_2, c2_1, c1_1, c4_1, c5_1, c6_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BINOM(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c3(@'(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x2, Nil))), BINOM(Cons(z0, z1), Cons(x2, Nil))) by BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(z2, Nil)))), BINOM(Cons(z0, z1), Cons(z2, Nil))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c3_1, c2_2, c2_1, c1_1, c4_1, c3_2, c5_1, c6_1 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BINOM(Cons(x0, Nil), Cons(x2, x3)) -> c3(@'(binom(Nil, x3), Cons(Nil, Nil))) by BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c3_1, c2_2, c2_1, c1_1, c4_1, c3_2, c5_1, c6_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace BINOM(Cons(x0, Nil), Cons(x2, z0)) -> c3(@'(Cons(Nil, Nil), binom(Nil, Cons(x2, z0)))) by BINOM(Cons(x0, Nil), Cons(x1, x2)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c2_2, c2_1, c1_1, c4_1, c3_2, c5_1, c6_1, c3_1 ---------------------------------------- (81) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c2_2, c2_1, c1_1, c4_1, c3_2, c5_1, c6_1, c3_1 ---------------------------------------- (83) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil))))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) S tuples: @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: @'_2, BINOM_2 Compound Symbols: c_1, c2_2, c2_1, c1_1, c4_1, c3_2, c5_1, c6_1, c3_1 ---------------------------------------- (85) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace @'(Cons(z0, z1), z2) -> c(@'(z1, z2)) by @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(BINOM(Cons(x1, Nil), x4)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples: BINOM(Cons(z0, z1), Cons(z2, z3)) -> c2(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, z3)) BINOM(Cons(z0, z1), Cons(z2, z3)) -> c3(@'(binom(z1, z3), binom(z1, Cons(z2, z3))), BINOM(z1, Cons(z2, z3))) Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c2_1, c1_1, c4_1, c3_2, c5_1, c6_1, c3_1, c_1 ---------------------------------------- (87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: BINOM(Cons(x0, Nil), Cons(x1, z0)) -> c2(@'(Cons(Nil, Nil), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(BINOM(Cons(x1, Nil), Cons(x3, Cons(x4, x5)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(z2, Nil)) -> c2(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(z2, Nil))))) BINOM(Cons(x0, Nil), Cons(x1, x2)) -> c3(@'(Cons(Nil, Nil), Cons(Nil, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(BINOM(Cons(x1, Nil), z0)) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(BINOM(Cons(x1, Nil), Cons(x4, x5))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(BINOM(Cons(x1, Nil), Cons(x3, x4))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Nil)) -> c3(@'(Cons(Nil, Nil), @(binom(z1, Nil), binom(z1, Cons(x3, Nil)))), BINOM(Cons(z0, z1), Cons(x3, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c1_1, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1 ---------------------------------------- (89) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c1_1, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1 ---------------------------------------- (91) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c1(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c1_1, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1 ---------------------------------------- (93) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c1(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1 ---------------------------------------- (95) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1 ---------------------------------------- (97) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c2(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Cons(z2, z3)))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1 ---------------------------------------- (99) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x4, Nil))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1 ---------------------------------------- (101) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c4(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c4_1, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1, c4_1 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Cons(z2, z3)), @(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x3, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1, c4_1 ---------------------------------------- (107) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Nil)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), Nil), @(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x3, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Nil))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1, c4_1 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, x4)) -> c5(@'(binom(Cons(x1, Nil), x4), @(binom(Nil, x4), Cons(Nil, Nil)))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c5_1, c6_1, c3_1, c2_1, c_1, c1_1, c4_1 ---------------------------------------- (111) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, z0)) -> c5(@'(binom(Cons(x1, Nil), z0), @(Cons(Nil, Nil), binom(Nil, Cons(x3, z0))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c6_1, c3_1, c2_1, c_1, c1_1, c4_1, c5_1 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(z2, z3)))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c6_1, c3_1, c2_1, c_1, c1_1, c4_1, c5_1 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Cons(z2, z3)))) -> c3(@'(@(@(binom(z1, z3), binom(z1, Cons(z2, z3))), binom(Cons(z0, z1), Cons(x4, Cons(z2, z3)))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Cons(z2, z3))))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c6_1, c3_1, c2_1, c_1, c1_1, c4_1, c5_1 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(x3, Cons(x4, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z0, z1), Cons(x4, Nil))), binom(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))), BINOM(Cons(x1, Cons(z0, z1)), Cons(x3, Cons(x4, Nil)))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c6_1, c3_1, c2_1, c_1, c1_1, c4_1, c5_1 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, x5))) -> c6(@'(@(binom(Nil, x5), Cons(Nil, Nil)), binom(Cons(x1, Nil), Cons(x3, Cons(x4, x5))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c6_1, c3_1, c2_1, c_1, c1_1, c4_1, c5_1 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(x0, Cons(x1, Nil)), Cons(x3, Cons(x4, z0))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(x4, z0))), binom(Cons(x1, Nil), Cons(x3, Cons(x4, z0))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c2_1, c_1, c1_1, c4_1, c5_1, c6_1 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c2_1, c_1, c1_1, c4_1, c5_1, c6_1 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c_1, c1_1, c4_1, c5_1, c6_1, c2_1 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c_1, c1_1, c4_1, c5_1, c6_1, c2_1 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3))))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3))))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c_1, c4_1, c5_1, c6_1, c2_1, c1_1 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) by BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3))))) BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c_1, c4_1, c5_1, c6_1, c2_1, c1_1 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3))))) BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c_1, c4_1, c5_1, c6_1, c2_1, c1_1 ---------------------------------------- (135) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3))))) BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c_1, c4_1, c5_1, c6_1, c2_1, c1_1 ---------------------------------------- (137) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3))))) BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c_1, c4_1, c5_1, c6_1, c2_1, c1_1 ---------------------------------------- (139) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil)))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3))))) BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil)))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c_1, c4_1, c5_1, c6_1, c2_1, c1_1 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil)))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3))))) BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil)))) S tuples: @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c_1, c5_1, c6_1, c2_1, c1_1, c4_1 ---------------------------------------- (143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace @'(Cons(z0, Cons(y0, y1)), z2) -> c(@'(Cons(y0, y1), z2)) by @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3)) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3))))) BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil)))) @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3)) S tuples: @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c5_1, c6_1, c2_1, c1_1, c4_1, c_1 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Cons(z5, z6)), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3))))) BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil)))) @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3)) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) S tuples: @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c5_1, c6_1, c2_1, c1_1, c4_1, c_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, @(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) by BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3))))) BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil)))) @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3)) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) S tuples: @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c5_1, c6_1, c2_1, c1_1, c4_1, c_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), @(Cons(Nil, Nil), Cons(Nil, Nil)))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3))))) BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil)))) @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3)) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) S tuples: @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c5_1, c6_1, c2_1, c1_1, c4_1, c_1 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, binom(Nil, Cons(z2, z3)))))) by BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3))))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: binom(Cons(z0, z1), Cons(z2, z3)) -> @(binom(z1, z3), binom(z1, Cons(z2, z3))) binom(Cons(z0, z1), Nil) -> Cons(Nil, Nil) binom(Nil, z0) -> Cons(Nil, Nil) @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2)) @(Nil, z0) -> z0 Tuples: BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c2(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), z3)) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c2(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(z2, z3))) BINOM(Cons(x0, Cons(x1, Cons(z0, z1))), Cons(z2, z3)) -> c3(@'(binom(Cons(x1, Cons(z0, z1)), z3), @(binom(Cons(z0, z1), z3), @(binom(z1, z3), binom(z1, Cons(z2, z3))))), BINOM(Cons(x1, Cons(z0, z1)), Cons(z2, z3))) BINOM(Cons(x0, Cons(z0, z1)), Cons(x3, Cons(z2, z3))) -> c3(@'(@(binom(z1, z3), binom(z1, Cons(z2, z3))), @(binom(z1, Cons(z2, z3)), binom(z1, Cons(x3, Cons(z2, z3))))), BINOM(Cons(z0, z1), Cons(x3, Cons(z2, z3)))) BINOM(Cons(x0, Cons(x1, x2)), Cons(x3, Nil)) -> c3(BINOM(Cons(x1, x2), Cons(x3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), binom(Cons(z2, z3), Cons(z4, Cons(z5, z6))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c3(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z4, Cons(z5, Cons(z6, z7)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Cons(z6, z7))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c3(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), binom(Cons(z2, z3), Cons(z4, Cons(z5, Nil))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c6(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), binom(Nil, Cons(z2, Cons(z3, z4)))))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c2(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c1(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3))))) BINOM(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, y2))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, y2)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(y1, Nil))), Cons(z3, Nil)) -> c3(BINOM(Cons(z1, Cons(y1, Nil)), Cons(z3, Nil))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c2(@'(@(binom(Cons(z2, z3), z6), @(binom(z3, z6), binom(z3, Cons(z5, z6)))), @(binom(Cons(z2, z3), Cons(z5, z6)), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, z6))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Cons(z6, z7)))) -> c2(@'(@(@(binom(z3, z7), binom(z3, Cons(z6, z7))), binom(Cons(z2, z3), Cons(z5, Cons(z6, z7)))), @(binom(Cons(z2, z3), Cons(z5, Cons(z6, z7))), @(binom(z3, Cons(z5, Cons(z6, z7))), binom(z3, Cons(z4, Cons(z5, Cons(z6, z7))))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Cons(z6, z7)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, Nil))) -> c2(@'(@(Cons(Nil, Nil), binom(Cons(z2, z3), Cons(z5, Nil))), @(binom(Cons(z2, z3), Cons(z5, Nil)), @(binom(z3, Cons(z5, Nil)), binom(z3, Cons(z4, Cons(z5, Nil)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z5, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(binom(Nil, z4), Cons(Nil, Nil)), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil)))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, Cons(z3, z4))) -> c4(@'(@(Cons(Nil, Nil), binom(Nil, Cons(z3, z4))), @(binom(Nil, Cons(z3, z4)), Cons(Nil, Nil)))) @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3)) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Cons(z5, z6))) -> c3(@'(@(binom(Cons(z2, z3), z6), binom(Cons(z2, z3), Cons(z5, z6))), @(@(binom(z3, z6), binom(z3, Cons(z5, z6))), @(binom(z3, Cons(z5, z6)), binom(z3, Cons(z4, Cons(z5, z6)))))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Cons(z5, z6)))) BINOM(Cons(z0, Cons(z1, Cons(z2, z3))), Cons(z4, Nil)) -> c3(@'(binom(Cons(z1, Cons(z2, z3)), Nil), Cons(Nil, binom(Cons(z2, z3), Cons(z4, Nil)))), BINOM(Cons(z1, Cons(z2, z3)), Cons(z4, Nil))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, @(Nil, Cons(Nil, Nil))))) BINOM(Cons(z0, Cons(z1, Nil)), Cons(z2, z3)) -> c5(@'(binom(Cons(z1, Nil), z3), Cons(Nil, binom(Nil, Cons(z2, z3))))) S tuples: @'(Cons(z0, Cons(z1, Cons(y1, y2))), z3) -> c(@'(Cons(z1, Cons(y1, y2)), z3)) K tuples:none Defined Rule Symbols: binom_2, @_2 Defined Pair Symbols: BINOM_2, @'_2 Compound Symbols: c2_2, c3_2, c3_1, c6_1, c2_1, c1_1, c4_1, c_1, c5_1