MAYBE proof of input_IT4Z00M2un.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CompletionProof [UPPER BOUND(ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (22) CdtProblem (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) 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), 0 ms] (34) CdtProblem (35) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem ---------------------------------------- (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(s(x)) -> s(f(f(p(s(x))))) f(0) -> 0 p(s(x)) -> x S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(s(x)) -> s(f(f(p(s(x))))) f(0') -> 0' p(s(x)) -> x S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(s(x)) -> s(f(f(p(s(x))))) f(0) -> 0 p(s(x)) -> x S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(s(x)) -> s(f(f(p(s(x))))) [1] f(0) -> 0 [1] p(s(x)) -> x [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(s(x)) -> s(f(f(p(s(x))))) [1] f(0) -> 0 [1] p(s(x)) -> x [1] The TRS has the following type information: f :: s:0 -> s:0 s :: s:0 -> s:0 p :: s:0 -> s:0 0 :: s:0 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 p_1 Due to the following rules being added: p(v0) -> 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: f(s(x)) -> s(f(f(p(s(x))))) [1] f(0) -> 0 [1] p(s(x)) -> x [1] p(v0) -> 0 [0] The TRS has the following type information: f :: s:0 -> s:0 s :: s:0 -> s:0 p :: s:0 -> s:0 0 :: s:0 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(s(x)) -> s(f(f(x))) [2] f(s(x)) -> s(f(f(0))) [1] f(0) -> 0 [1] p(s(x)) -> x [1] p(v0) -> 0 [0] The TRS has the following type information: f :: s:0 -> s:0 s :: s:0 -> s:0 p :: s:0 -> s:0 0 :: s:0 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) -{ 1 }-> 0 :|: z = 0 f(z) -{ 2 }-> 1 + f(f(x)) :|: x >= 0, z = 1 + x f(z) -{ 1 }-> 1 + f(f(0)) :|: x >= 0, z = 1 + x p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (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) -{ 1 }-> 0 :|: z = 0 f(z) -{ 1 }-> 1 + f(f(0)) :|: z - 1 >= 0 f(z) -{ 2 }-> 1 + f(f(z - 1)) :|: z - 1 >= 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 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: p(v0) -> null_p [0] f(v0) -> null_f [0] And the following fresh constants: null_p, 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(s(x)) -> s(f(f(p(s(x))))) [1] f(0) -> 0 [1] p(s(x)) -> x [1] p(v0) -> null_p [0] f(v0) -> null_f [0] The TRS has the following type information: f :: s:0:null_p:null_f -> s:0:null_p:null_f s :: s:0:null_p:null_f -> s:0:null_p:null_f p :: s:0:null_p:null_f -> s:0:null_p:null_f 0 :: s:0:null_p:null_f null_p :: s:0:null_p:null_f null_f :: s:0:null_p: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: 0 => 0 null_p => 0 null_f => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> 0 :|: z = 0 f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 f(z) -{ 1 }-> 1 + f(f(p(1 + x))) :|: x >= 0, z = 1 + x p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 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(s(z0)) -> s(f(f(p(s(z0))))) f(0) -> 0 p(s(z0)) -> z0 Tuples: F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))), P(s(z0))) F(0) -> c1 P(s(z0)) -> c2 S tuples: F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))), P(s(z0))) F(0) -> c1 P(s(z0)) -> c2 K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1, P_1 Compound Symbols: c_3, c1, c2 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: F(0) -> c1 P(s(z0)) -> c2 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: f(s(z0)) -> s(f(f(p(s(z0))))) f(0) -> 0 p(s(z0)) -> z0 Tuples: F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))), P(s(z0))) S tuples: F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0))), P(s(z0))) K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1 Compound Symbols: c_3 ---------------------------------------- (25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: f(s(z0)) -> s(f(f(p(s(z0))))) f(0) -> 0 p(s(z0)) -> z0 Tuples: F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0)))) S tuples: F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0)))) K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1 Compound Symbols: c_2 ---------------------------------------- (27) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c(F(f(p(s(z0)))), F(p(s(z0)))) by F(s(z0)) -> c(F(f(z0)), F(p(s(z0)))) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: f(s(z0)) -> s(f(f(p(s(z0))))) f(0) -> 0 p(s(z0)) -> z0 Tuples: F(s(z0)) -> c(F(f(z0)), F(p(s(z0)))) S tuples: F(s(z0)) -> c(F(f(z0)), F(p(s(z0)))) K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1 Compound Symbols: c_2 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c(F(f(z0)), F(p(s(z0)))) by F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0))))) F(s(0)) -> c(F(0), F(p(s(0)))) F(s(x0)) -> c(F(p(s(x0)))) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: f(s(z0)) -> s(f(f(p(s(z0))))) f(0) -> 0 p(s(z0)) -> z0 Tuples: F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0))))) F(s(0)) -> c(F(0), F(p(s(0)))) F(s(x0)) -> c(F(p(s(x0)))) S tuples: F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0))))) F(s(0)) -> c(F(0), F(p(s(0)))) F(s(x0)) -> c(F(p(s(x0)))) K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1 Compound Symbols: c_2, c_1 ---------------------------------------- (31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: f(s(z0)) -> s(f(f(p(s(z0))))) f(0) -> 0 p(s(z0)) -> z0 Tuples: F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0))))) F(s(x0)) -> c(F(p(s(x0)))) F(s(0)) -> c(F(p(s(0)))) S tuples: F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0))))) F(s(x0)) -> c(F(p(s(x0)))) F(s(0)) -> c(F(p(s(0)))) K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1 Compound Symbols: c_2, c_1 ---------------------------------------- (33) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(0)) -> c(F(p(s(0)))) by F(s(0)) -> c(F(0)) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: f(s(z0)) -> s(f(f(p(s(z0))))) f(0) -> 0 p(s(z0)) -> z0 Tuples: F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0))))) F(s(x0)) -> c(F(p(s(x0)))) F(s(0)) -> c(F(0)) S tuples: F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0))))) F(s(x0)) -> c(F(p(s(x0)))) F(s(0)) -> c(F(0)) K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1 Compound Symbols: c_2, c_1 ---------------------------------------- (35) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: F(s(0)) -> c(F(0)) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: f(s(z0)) -> s(f(f(p(s(z0))))) f(0) -> 0 p(s(z0)) -> z0 Tuples: F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0))))) F(s(x0)) -> c(F(p(s(x0)))) S tuples: F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0))))) F(s(x0)) -> c(F(p(s(x0)))) K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1 Compound Symbols: c_2, c_1 ---------------------------------------- (37) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(p(s(s(z0))))) by F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(s(z0))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: f(s(z0)) -> s(f(f(p(s(z0))))) f(0) -> 0 p(s(z0)) -> z0 Tuples: F(s(x0)) -> c(F(p(s(x0)))) F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(s(z0))) S tuples: F(s(x0)) -> c(F(p(s(x0)))) F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(s(z0))) K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1 Compound Symbols: c_1, c_2 ---------------------------------------- (39) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c(F(p(s(x0)))) by F(s(z0)) -> c(F(z0)) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: f(s(z0)) -> s(f(f(p(s(z0))))) f(0) -> 0 p(s(z0)) -> z0 Tuples: F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(s(z0))) F(s(z0)) -> c(F(z0)) S tuples: F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(s(z0))) F(s(z0)) -> c(F(z0)) K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1 Compound Symbols: c_2, c_1 ---------------------------------------- (41) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0))) -> c(F(s(f(f(p(s(z0)))))), F(s(z0))) by F(s(s(z0))) -> c(F(s(f(f(z0)))), F(s(z0))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: f(s(z0)) -> s(f(f(p(s(z0))))) f(0) -> 0 p(s(z0)) -> z0 Tuples: F(s(z0)) -> c(F(z0)) F(s(s(z0))) -> c(F(s(f(f(z0)))), F(s(z0))) S tuples: F(s(z0)) -> c(F(z0)) F(s(s(z0))) -> c(F(s(f(f(z0)))), F(s(z0))) K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1 Compound Symbols: c_1, c_2 ---------------------------------------- (43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(z0)) -> c(F(z0)) by F(s(s(y0))) -> c(F(s(y0))) F(s(s(s(y0)))) -> c(F(s(s(y0)))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: f(s(z0)) -> s(f(f(p(s(z0))))) f(0) -> 0 p(s(z0)) -> z0 Tuples: F(s(s(z0))) -> c(F(s(f(f(z0)))), F(s(z0))) F(s(s(y0))) -> c(F(s(y0))) F(s(s(s(y0)))) -> c(F(s(s(y0)))) S tuples: F(s(s(z0))) -> c(F(s(f(f(z0)))), F(s(z0))) F(s(s(y0))) -> c(F(s(y0))) F(s(s(s(y0)))) -> c(F(s(s(y0)))) K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1 Compound Symbols: c_2, c_1 ---------------------------------------- (45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(s(s(y0))) -> c(F(s(y0))) by F(s(s(s(y0)))) -> c(F(s(s(y0)))) F(s(s(s(s(y0))))) -> c(F(s(s(s(y0))))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: f(s(z0)) -> s(f(f(p(s(z0))))) f(0) -> 0 p(s(z0)) -> z0 Tuples: F(s(s(z0))) -> c(F(s(f(f(z0)))), F(s(z0))) F(s(s(s(y0)))) -> c(F(s(s(y0)))) F(s(s(s(s(y0))))) -> c(F(s(s(s(y0))))) S tuples: F(s(s(z0))) -> c(F(s(f(f(z0)))), F(s(z0))) F(s(s(s(y0)))) -> c(F(s(s(y0)))) F(s(s(s(s(y0))))) -> c(F(s(s(s(y0))))) K tuples:none Defined Rule Symbols: f_1, p_1 Defined Pair Symbols: F_1 Compound Symbols: c_2, c_1