MAYBE proof of input_P4ptHu8xZf.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) CompletionProof [UPPER BOUND(ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxTypedWeightedCompleteTrs (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) SimplificationProof [BOTH BOUNDS(ID, ID), 1 ms] (20) CpxRNTS (21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (22) 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(h(x), y) -> h(f(y, f(x, h(a)))) 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(h(x), y) -> h(f(y, f(x, h(a)))) 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(h(x), y) -> h(f(y, f(x, h(a)))) 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(h(x), y) -> h(f(y, f(x, h(a)))) [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(h(x), y) -> h(f(y, f(x, h(a)))) [1] The TRS has the following type information: f :: h:a -> h:a -> h:a h :: h:a -> h:a a :: h:a Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: f(v0, v1) -> null_f [0] And the following fresh constants: null_f ---------------------------------------- (10) 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(h(x), y) -> h(f(y, f(x, h(a)))) [1] f(v0, v1) -> null_f [0] The TRS has the following type information: f :: h:a:null_f -> h:a:null_f -> h:a:null_f h :: h:a:null_f -> h:a:null_f a :: h:a:null_f null_f :: h:a:null_f Rewrite Strategy: INNERMOST ---------------------------------------- (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: a => 0 null_f => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 f(z, z') -{ 1 }-> 1 + f(y, f(x, 1 + 0)) :|: x >= 0, y >= 0, z = 1 + x, z' = y Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (13) 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_2 Due to the following rules being added: f(v0, v1) -> a [0] And the following fresh constants: none ---------------------------------------- (14) 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(h(x), y) -> h(f(y, f(x, h(a)))) [1] f(v0, v1) -> a [0] The TRS has the following type information: f :: h:a -> h:a -> h:a h :: h:a -> h:a a :: h:a Rewrite Strategy: INNERMOST ---------------------------------------- (15) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (16) 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(h(h(x')), y) -> h(f(y, h(f(h(a), f(x', h(a)))))) [2] f(h(x), y) -> h(f(y, a)) [1] f(v0, v1) -> a [0] The TRS has the following type information: f :: h:a -> h:a -> h:a h :: h:a -> h:a a :: h:a Rewrite Strategy: INNERMOST ---------------------------------------- (17) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: a => 0 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 f(z, z') -{ 1 }-> 1 + f(y, 0) :|: x >= 0, y >= 0, z = 1 + x, z' = y f(z, z') -{ 2 }-> 1 + f(y, 1 + f(1 + 0, f(x', 1 + 0))) :|: x' >= 0, y >= 0, z = 1 + (1 + x'), z' = y ---------------------------------------- (19) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 f(z, z') -{ 1 }-> 1 + f(z', 0) :|: z - 1 >= 0, z' >= 0 f(z, z') -{ 2 }-> 1 + f(z', 1 + f(1 + 0, f(z - 2, 1 + 0))) :|: z - 2 >= 0, z' >= 0 ---------------------------------------- (21) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: f(h(z0), z1) -> h(f(z1, f(z0, h(a)))) Tuples: F(h(z0), z1) -> c(F(z1, f(z0, h(a))), F(z0, h(a))) S tuples: F(h(z0), z1) -> c(F(z1, f(z0, h(a))), F(z0, h(a))) K tuples:none Defined Rule Symbols: f_2 Defined Pair Symbols: F_2 Compound Symbols: c_2