KILLED proof of input_JZxXcbKr8w.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CompletionProof [UPPER BOUND(ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (22) CdtProblem (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (34) CdtProblem (35) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) 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: f(g(X)) -> g(f(f(X))) f(h(X)) -> h(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: f(g(X)) -> g(f(f(X))) f(h(X)) -> h(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: f(g(X)) -> g(f(f(X))) f(h(X)) -> h(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: f(g(X)) -> g(f(f(X))) [1] f(h(X)) -> h(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: f(g(X)) -> g(f(f(X))) [1] f(h(X)) -> h(g(X)) [1] The TRS has the following type information: f :: g:h -> g:h g :: g:h -> g:h h :: g:h -> g:h 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 Due to the following rules being added: f(v0) -> const [0] And the following fresh constants: const ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(g(X)) -> g(f(f(X))) [1] f(h(X)) -> h(g(X)) [1] f(v0) -> const [0] The TRS has the following type information: f :: g:h:const -> g:h:const g :: g:h:const -> g:h:const h :: g:h:const -> g:h:const const :: g:h:const Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(g(g(X'))) -> g(f(g(f(f(X'))))) [2] f(g(h(X''))) -> g(f(h(g(X'')))) [2] f(g(X)) -> g(f(const)) [1] f(h(X)) -> h(g(X)) [1] f(v0) -> const [0] The TRS has the following type information: f :: g:h:const -> g:h:const g :: g:h:const -> g:h:const h :: g:h:const -> g:h:const const :: g:h:const Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: const => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 f(z) -{ 1 }-> 1 + f(0) :|: z = 1 + X, X >= 0 f(z) -{ 2 }-> 1 + f(1 + f(f(X'))) :|: X' >= 0, z = 1 + (1 + X') f(z) -{ 2 }-> 1 + f(1 + (1 + X'')) :|: z = 1 + (1 + X''), X'' >= 0 f(z) -{ 1 }-> 1 + (1 + X) :|: z = 1 + X, X >= 0 ---------------------------------------- (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) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 1 }-> 1 + f(0) :|: z - 1 >= 0 f(z) -{ 2 }-> 1 + f(1 + f(f(z - 2))) :|: z - 2 >= 0 f(z) -{ 2 }-> 1 + f(1 + (1 + (z - 2))) :|: z - 2 >= 0 f(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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: f(v0) -> null_f [0] And the following fresh constants: null_f ---------------------------------------- (18) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(g(X)) -> g(f(f(X))) [1] f(h(X)) -> h(g(X)) [1] f(v0) -> null_f [0] The TRS has the following type information: f :: g:h:null_f -> g:h:null_f g :: g:h:null_f -> g:h:null_f h :: g:h:null_f -> g:h:null_f null_f :: g:h:null_f Rewrite Strategy: INNERMOST ---------------------------------------- (19) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: null_f => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 f(z) -{ 1 }-> 1 + f(f(X)) :|: z = 1 + X, X >= 0 f(z) -{ 1 }-> 1 + (1 + X) :|: z = 1 + X, X >= 0 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: f(g(z0)) -> g(f(f(z0))) f(h(z0)) -> h(g(z0)) Tuples: F(g(z0)) -> c(F(f(z0)), F(z0)) F(h(z0)) -> c1 S tuples: F(g(z0)) -> c(F(f(z0)), F(z0)) F(h(z0)) -> c1 K tuples:none Defined Rule Symbols: f_1 Defined Pair Symbols: F_1 Compound Symbols: c_2, c1 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(h(z0)) -> c1 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0)) -> g(f(f(z0))) f(h(z0)) -> h(g(z0)) Tuples: F(g(z0)) -> c(F(f(z0)), F(z0)) S tuples: F(g(z0)) -> c(F(f(z0)), F(z0)) K tuples:none Defined Rule Symbols: f_1 Defined Pair Symbols: F_1 Compound Symbols: c_2 ---------------------------------------- (25) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(g(z0)) -> c(F(f(z0)), F(z0)) by F(g(g(z0))) -> c(F(g(f(f(z0)))), F(g(z0))) F(g(h(z0))) -> c(F(h(g(z0))), F(h(z0))) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0)) -> g(f(f(z0))) f(h(z0)) -> h(g(z0)) Tuples: F(g(g(z0))) -> c(F(g(f(f(z0)))), F(g(z0))) F(g(h(z0))) -> c(F(h(g(z0))), F(h(z0))) S tuples: F(g(g(z0))) -> c(F(g(f(f(z0)))), F(g(z0))) F(g(h(z0))) -> c(F(h(g(z0))), F(h(z0))) K tuples:none Defined Rule Symbols: f_1 Defined Pair Symbols: F_1 Compound Symbols: c_2 ---------------------------------------- (27) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(g(h(z0))) -> c(F(h(g(z0))), F(h(z0))) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0)) -> g(f(f(z0))) f(h(z0)) -> h(g(z0)) Tuples: F(g(g(z0))) -> c(F(g(f(f(z0)))), F(g(z0))) S tuples: F(g(g(z0))) -> c(F(g(f(f(z0)))), F(g(z0))) K tuples:none Defined Rule Symbols: f_1 Defined Pair Symbols: F_1 Compound Symbols: c_2 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(g(g(z0))) -> c(F(g(f(f(z0)))), F(g(z0))) by F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0)))) F(g(g(h(z0)))) -> c(F(g(f(h(g(z0))))), F(g(h(z0)))) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0)) -> g(f(f(z0))) f(h(z0)) -> h(g(z0)) Tuples: F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0)))) F(g(g(h(z0)))) -> c(F(g(f(h(g(z0))))), F(g(h(z0)))) S tuples: F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0)))) F(g(g(h(z0)))) -> c(F(g(f(h(g(z0))))), F(g(h(z0)))) K tuples:none Defined Rule Symbols: f_1 Defined Pair Symbols: F_1 Compound Symbols: c_2 ---------------------------------------- (31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0)) -> g(f(f(z0))) f(h(z0)) -> h(g(z0)) Tuples: F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0)))) F(g(g(h(z0)))) -> c(F(g(f(h(g(z0)))))) S tuples: F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0)))) F(g(g(h(z0)))) -> c(F(g(f(h(g(z0)))))) K tuples:none Defined Rule Symbols: f_1 Defined Pair Symbols: F_1 Compound Symbols: c_2, c_1 ---------------------------------------- (33) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(g(g(h(z0)))) -> c(F(g(f(h(g(z0)))))) by F(g(g(h(x0)))) -> c(F(g(h(g(g(x0)))))) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0)) -> g(f(f(z0))) f(h(z0)) -> h(g(z0)) Tuples: F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0)))) F(g(g(h(x0)))) -> c(F(g(h(g(g(x0)))))) S tuples: F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0)))) F(g(g(h(x0)))) -> c(F(g(h(g(g(x0)))))) K tuples:none Defined Rule Symbols: f_1 Defined Pair Symbols: F_1 Compound Symbols: c_2, c_1 ---------------------------------------- (35) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(g(g(h(x0)))) -> c(F(g(h(g(g(x0)))))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0)) -> g(f(f(z0))) f(h(z0)) -> h(g(z0)) Tuples: F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0)))) S tuples: F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0)))) K tuples:none Defined Rule Symbols: f_1 Defined Pair Symbols: F_1 Compound Symbols: c_2 ---------------------------------------- (37) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0)))) by F(g(g(g(z0)))) -> c(F(g(g(f(f(f(f(z0))))))), F(g(g(z0)))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: f(g(z0)) -> g(f(f(z0))) f(h(z0)) -> h(g(z0)) Tuples: F(g(g(g(z0)))) -> c(F(g(g(f(f(f(f(z0))))))), F(g(g(z0)))) S tuples: F(g(g(g(z0)))) -> c(F(g(g(f(f(f(f(z0))))))), F(g(g(z0)))) K tuples:none Defined Rule Symbols: f_1 Defined Pair Symbols: F_1 Compound Symbols: c_2