KILLED proof of input_NdOCBnLBCd.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 10 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CompletionProof [UPPER BOUND(ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 2 ms] (20) CpxRNTS (21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (22) CdtProblem (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 1 ms] (32) CdtProblem (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) 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: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) f(0) -> s(0) f(s(x)) -> minus(s(x), g(f(x))) g(0) -> 0 g(s(x)) -> minus(s(x), f(g(x))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) f(0') -> s(0') f(s(x)) -> minus(s(x), g(f(x))) g(0') -> 0' g(s(x)) -> minus(s(x), f(g(x))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) f(0) -> s(0) f(s(x)) -> minus(s(x), g(f(x))) g(0) -> 0 g(s(x)) -> minus(s(x), f(g(x))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] f(0) -> s(0) [1] f(s(x)) -> minus(s(x), g(f(x))) [1] g(0) -> 0 [1] g(s(x)) -> minus(s(x), f(g(x))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] f(0) -> s(0) [1] f(s(x)) -> minus(s(x), g(f(x))) [1] g(0) -> 0 [1] g(s(x)) -> minus(s(x), f(g(x))) [1] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s f :: 0:s -> 0:s g :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: f_1 g_1 minus_2 Due to the following rules being added: minus(v0, v1) -> 0 [0] And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] f(0) -> s(0) [1] f(s(x)) -> minus(s(x), g(f(x))) [1] g(0) -> 0 [1] g(s(x)) -> minus(s(x), f(g(x))) [1] minus(v0, v1) -> 0 [0] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s f :: 0:s -> 0:s g :: 0:s -> 0:s 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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] f(0) -> s(0) [1] f(s(0)) -> minus(s(0), g(s(0))) [2] f(s(s(x'))) -> minus(s(s(x')), g(minus(s(x'), g(f(x'))))) [2] g(0) -> 0 [1] g(s(0)) -> minus(s(0), f(0)) [2] g(s(s(x''))) -> minus(s(s(x'')), f(minus(s(x''), f(g(x''))))) [2] minus(v0, v1) -> 0 [0] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s f :: 0:s -> 0:s g :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> minus(1 + 0, g(1 + 0)) :|: z = 1 + 0 f(z) -{ 2 }-> minus(1 + (1 + x'), g(minus(1 + x', g(f(x'))))) :|: x' >= 0, z = 1 + (1 + x') f(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 2 }-> minus(1 + 0, f(0)) :|: z = 1 + 0 g(z) -{ 2 }-> minus(1 + (1 + x''), f(minus(1 + x'', f(g(x''))))) :|: x'' >= 0, z = 1 + (1 + x'') g(z) -{ 1 }-> 0 :|: z = 0 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> minus(1 + 0, g(1 + 0)) :|: z = 1 + 0 f(z) -{ 2 }-> minus(1 + (1 + (z - 2)), g(minus(1 + (z - 2), g(f(z - 2))))) :|: z - 2 >= 0 f(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 2 }-> minus(1 + 0, f(0)) :|: z = 1 + 0 g(z) -{ 2 }-> minus(1 + (1 + (z - 2)), f(minus(1 + (z - 2), f(g(z - 2))))) :|: z - 2 >= 0 g(z) -{ 1 }-> 0 :|: z = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ---------------------------------------- (17) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: minus(v0, v1) -> null_minus [0] f(v0) -> null_f [0] g(v0) -> null_g [0] And the following fresh constants: null_minus, null_f, null_g ---------------------------------------- (18) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] f(0) -> s(0) [1] f(s(x)) -> minus(s(x), g(f(x))) [1] g(0) -> 0 [1] g(s(x)) -> minus(s(x), f(g(x))) [1] minus(v0, v1) -> null_minus [0] f(v0) -> null_f [0] g(v0) -> null_g [0] The TRS has the following type information: minus :: 0:s:null_minus:null_f:null_g -> 0:s:null_minus:null_f:null_g -> 0:s:null_minus:null_f:null_g 0 :: 0:s:null_minus:null_f:null_g s :: 0:s:null_minus:null_f:null_g -> 0:s:null_minus:null_f:null_g f :: 0:s:null_minus:null_f:null_g -> 0:s:null_minus:null_f:null_g g :: 0:s:null_minus:null_f:null_g -> 0:s:null_minus:null_f:null_g null_minus :: 0:s:null_minus:null_f:null_g null_f :: 0:s:null_minus:null_f:null_g null_g :: 0:s:null_minus:null_f:null_g Rewrite Strategy: INNERMOST ---------------------------------------- (19) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 null_minus => 0 null_f => 0 null_g => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> minus(1 + x, g(f(x))) :|: x >= 0, z = 1 + x f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 f(z) -{ 1 }-> 1 + 0 :|: z = 0 g(z) -{ 1 }-> minus(1 + x, f(g(x))) :|: x >= 0, z = 1 + x g(z) -{ 1 }-> 0 :|: z = 0 g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (21) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(z0, 0) -> c MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(0) -> c2 F(s(z0)) -> c3(MINUS(s(z0), g(f(z0))), G(f(z0)), F(z0)) G(0) -> c4 G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0)) S tuples: MINUS(z0, 0) -> c MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(0) -> c2 F(s(z0)) -> c3(MINUS(s(z0), g(f(z0))), G(f(z0)), F(z0)) G(0) -> c4 G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0)) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, F_1, G_1 Compound Symbols: c, c1_1, c2, c3_3, c4, c5_3 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: F(0) -> c2 G(0) -> c4 MINUS(z0, 0) -> c ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(z0)) -> c3(MINUS(s(z0), g(f(z0))), G(f(z0)), F(z0)) G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0)) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(z0)) -> c3(MINUS(s(z0), g(f(z0))), G(f(z0)), F(z0)) G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0)) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, F_1, G_1 Compound Symbols: c1_1, c3_3, c5_3 ---------------------------------------- (25) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c3(MINUS(s(z0), g(f(z0))), G(f(z0)), F(z0)) by F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0)), F(0)) F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0)) F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0)), F(0)) F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0)) F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0)), F(0)) F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, G_1, F_1 Compound Symbols: c1_1, c5_3, c3_3 ---------------------------------------- (27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0)) F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0)) F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, G_1, F_1 Compound Symbols: c1_1, c5_3, c3_3, c3_2 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0)) by G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0)), G(0)) G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0))) G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0)), G(0)) G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0))) G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0)), G(0)) G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, F_1, G_1 Compound Symbols: c1_1, c3_3, c3_2, c5_3 ---------------------------------------- (31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0))) G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0))) G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, F_1, G_1 Compound Symbols: c1_1, c3_3, c3_2, c5_3, c5_2 ---------------------------------------- (33) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) by F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0))) G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0))) G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, F_1, G_1 Compound Symbols: c1_1, c3_2, c5_3, c5_2, c3_3, c3_1 ---------------------------------------- (35) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0))) by F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) F(s(0)) -> c3(G(f(0))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) F(s(0)) -> c3(G(f(0))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) F(s(0)) -> c3(G(f(0))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, G_1, F_1 Compound Symbols: c1_1, c5_3, c5_2, c3_3, c3_1, c3_2 ---------------------------------------- (37) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) by G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) F(s(0)) -> c3(G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) F(s(0)) -> c3(G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, G_1, F_1 Compound Symbols: c1_1, c5_2, c3_3, c3_1, c3_2, c5_3, c5_1 ---------------------------------------- (39) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0))) by G(s(0)) -> c5(MINUS(s(0), s(0)), F(g(0))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) F(s(0)) -> c3(G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0)), F(g(0))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) F(s(0)) -> c3(G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0)), F(g(0))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, F_1, G_1 Compound Symbols: c1_1, c3_3, c3_1, c3_2, c5_3, c5_1, c5_2 ---------------------------------------- (41) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(s(0)) -> c5(MINUS(s(0), s(0)), F(g(0))) by G(s(0)) -> c5(MINUS(s(0), s(0)), F(0)) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) F(s(0)) -> c3(G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0)), F(0)) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) F(s(0)) -> c3(G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0)), F(0)) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, F_1, G_1 Compound Symbols: c1_1, c3_3, c3_1, c3_2, c5_3, c5_1, c5_2 ---------------------------------------- (43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) F(s(0)) -> c3(G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) F(s(0)) -> c3(G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, F_1, G_1 Compound Symbols: c1_1, c3_3, c3_1, c3_2, c5_3, c5_1 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0)) -> c3(G(f(0))) by F(s(0)) -> c3(G(s(0))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) F(s(0)) -> c3(G(s(0))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) F(s(0)) -> c3(G(s(0))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, F_1, G_1 Compound Symbols: c1_1, c3_3, c3_1, c3_2, c5_3, c5_1 ---------------------------------------- (47) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) by F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) F(s(0)) -> c3(G(s(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) F(s(0)) -> c3(G(s(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, F_1, G_1 Compound Symbols: c1_1, c3_3, c3_1, c3_2, c5_3, c5_1 ---------------------------------------- (49) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) by F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) F(s(0)) -> c3(G(s(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(s(x0))) -> c3(G(f(s(x0)))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) F(s(0)) -> c3(G(s(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, F_1, G_1 Compound Symbols: c1_1, c3_1, c3_2, c5_3, c5_1, c3_3 ---------------------------------------- (51) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(x0))) -> c3(G(f(s(x0)))) by F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) F(s(0)) -> c3(G(s(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) F(s(0)) -> c3(G(s(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, F_1, G_1 Compound Symbols: c1_1, c3_2, c5_3, c5_1, c3_1, c3_3 ---------------------------------------- (53) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) by F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(s(0))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) F(s(0)) -> c3(G(s(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(s(0))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) F(s(0)) -> c3(G(s(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(s(0))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, G_1, F_1 Compound Symbols: c1_1, c5_3, c5_1, c3_1, c3_3, c3_2 ---------------------------------------- (55) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) F(s(0)) -> c3(G(s(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) F(s(0)) -> c(MINUS(s(0), minus(s(0), f(g(0))))) F(s(0)) -> c(G(s(0))) S tuples: MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) F(s(0)) -> c3(G(s(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) F(s(0)) -> c(MINUS(s(0), minus(s(0), f(g(0))))) F(s(0)) -> c(G(s(0))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: MINUS_2, G_1, F_1 Compound Symbols: c1_1, c5_3, c5_1, c3_1, c3_3, c_1 ---------------------------------------- (57) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) by MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) F(s(0)) -> c3(G(s(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) F(s(0)) -> c(MINUS(s(0), minus(s(0), f(g(0))))) F(s(0)) -> c(G(s(0))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) S tuples: G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) G(s(0)) -> c5(MINUS(s(0), s(0))) F(s(0)) -> c3(G(s(0))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) F(s(0)) -> c(MINUS(s(0), minus(s(0), f(g(0))))) F(s(0)) -> c(G(s(0))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: G_1, F_1, MINUS_2 Compound Symbols: c5_3, c5_1, c3_1, c3_3, c_1, c1_1 ---------------------------------------- (59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: F(s(0)) -> c(MINUS(s(0), minus(s(0), f(g(0))))) G(s(0)) -> c5(MINUS(s(0), s(0))) F(s(0)) -> c3(G(s(0))) F(s(0)) -> c(G(s(0))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) S tuples: G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: G_1, F_1, MINUS_2 Compound Symbols: c5_3, c5_1, c3_3, c3_1, c1_1 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0)))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0))))) S tuples: G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) G(s(s(x0))) -> c5(F(g(s(x0)))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0)))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0))))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: G_1, F_1, MINUS_2 Compound Symbols: c5_3, c5_1, c3_3, c3_1, c1_1, c5_2, c3_2 ---------------------------------------- (63) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) by G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: G(s(s(x0))) -> c5(F(g(s(x0)))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0)))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) S tuples: G(s(s(x0))) -> c5(F(g(s(x0)))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0)))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: G_1, F_1, MINUS_2 Compound Symbols: c5_1, c3_3, c3_1, c1_1, c5_2, c3_2, c5_3 ---------------------------------------- (65) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(s(s(x0))) -> c5(F(g(s(x0)))) by G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0)))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) S tuples: F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0)))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: F_1, MINUS_2, G_1 Compound Symbols: c3_3, c3_1, c1_1, c5_2, c3_2, c5_3, c5_1 ---------------------------------------- (67) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) by F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0)))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0)))) S tuples: F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0)))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0)))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: F_1, MINUS_2, G_1 Compound Symbols: c3_1, c1_1, c5_2, c3_2, c5_3, c5_1, c3_3 ---------------------------------------- (69) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0)))) by G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(minus(s(0), f(g(0))))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0)))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(minus(s(0), f(g(0))))) S tuples: F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0)))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(minus(s(0), f(g(0))))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: F_1, MINUS_2, G_1 Compound Symbols: c3_1, c1_1, c3_2, c5_3, c5_1, c3_3, c5_2 ---------------------------------------- (71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) by MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0)))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(minus(s(0), f(g(0))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1)))) S tuples: F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0)))) G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(minus(s(0), f(g(0))))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1)))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: F_1, G_1, MINUS_2 Compound Symbols: c3_1, c3_2, c5_3, c5_1, c3_3, c5_2, c1_1 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1)))) F(s(s(0))) -> c3(G(minus(s(0), g(f(0))))) G(s(s(0))) -> c5(F(minus(s(0), f(g(0))))) S tuples: F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1)))) F(s(s(0))) -> c3(G(minus(s(0), g(f(0))))) G(s(s(0))) -> c5(F(minus(s(0), f(g(0))))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: F_1, G_1, MINUS_2 Compound Symbols: c3_1, c5_3, c5_1, c3_3, c1_1 ---------------------------------------- (75) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) by G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(minus(s(z0), f(g(z0)))))), G(s(s(z0)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1)))) F(s(s(0))) -> c3(G(minus(s(0), g(f(0))))) G(s(s(0))) -> c5(F(minus(s(0), f(g(0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(minus(s(z0), f(g(z0)))))), G(s(s(z0)))) S tuples: F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1)))) F(s(s(0))) -> c3(G(minus(s(0), g(f(0))))) G(s(s(0))) -> c5(F(minus(s(0), f(g(0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(minus(s(z0), f(g(z0)))))), G(s(s(z0)))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: F_1, G_1, MINUS_2 Compound Symbols: c3_1, c5_1, c3_3, c1_1, c5_3 ---------------------------------------- (77) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(0))) -> c3(G(minus(s(0), g(f(0))))) by F(s(s(0))) -> c3(G(minus(s(0), g(s(0))))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1)))) G(s(s(0))) -> c5(F(minus(s(0), f(g(0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(minus(s(z0), f(g(z0)))))), G(s(s(z0)))) F(s(s(0))) -> c3(G(minus(s(0), g(s(0))))) S tuples: F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1)))) G(s(s(0))) -> c5(F(minus(s(0), f(g(0))))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(minus(s(z0), f(g(z0)))))), G(s(s(z0)))) F(s(s(0))) -> c3(G(minus(s(0), g(s(0))))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: F_1, G_1, MINUS_2 Compound Symbols: c3_1, c5_1, c3_3, c1_1, c5_3 ---------------------------------------- (79) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace G(s(s(0))) -> c5(F(minus(s(0), f(g(0))))) by G(s(s(0))) -> c5(F(minus(s(0), f(0)))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) f(0) -> s(0) f(s(z0)) -> minus(s(z0), g(f(z0))) g(0) -> 0 g(s(z0)) -> minus(s(z0), f(g(z0))) Tuples: F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1)))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(minus(s(z0), f(g(z0)))))), G(s(s(z0)))) F(s(s(0))) -> c3(G(minus(s(0), g(s(0))))) G(s(s(0))) -> c5(F(minus(s(0), f(0)))) S tuples: F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0))))) G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0))))) F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1)))) G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(minus(s(z0), f(g(z0)))))), G(s(s(z0)))) F(s(s(0))) -> c3(G(minus(s(0), g(s(0))))) G(s(s(0))) -> c5(F(minus(s(0), f(0)))) K tuples:none Defined Rule Symbols: minus_2, f_1, g_1 Defined Pair Symbols: F_1, G_1, MINUS_2 Compound Symbols: c3_1, c5_1, c3_3, c1_1, c5_3