MAYBE proof of input_IHsaVKHQg0.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 715 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 177 ms] (24) CpxRNTS (25) CompletionProof [UPPER BOUND(ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (30) CdtProblem (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms] (36) CdtProblem (37) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (44) 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: first(0, X) -> nil first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) from(X) -> cons(X, from(s(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: first(0', X) -> nil first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) from(X) -> cons(X, from(s(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: first(0, X) -> nil first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) from(X) -> cons(X, from(s(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: first(0, X) -> nil [1] first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1] from(X) -> cons(X, from(s(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: first(0, X) -> nil [1] first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1] from(X) -> cons(X, from(s(X))) [1] The TRS has the following type information: first :: 0:s -> nil:cons -> nil:cons 0 :: 0:s nil :: nil:cons s :: 0:s -> 0:s cons :: 0:s -> nil:cons -> nil:cons from :: 0:s -> nil:cons 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: first_2 from_1 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: first(0, X) -> nil [1] first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1] from(X) -> cons(X, from(s(X))) [1] The TRS has the following type information: first :: 0:s -> nil:cons -> nil:cons 0 :: 0:s nil :: nil:cons s :: 0:s -> 0:s cons :: 0:s -> nil:cons -> nil:cons from :: 0:s -> nil:cons 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: first(0, X) -> nil [1] first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1] from(X) -> cons(X, from(s(X))) [1] The TRS has the following type information: first :: 0:s -> nil:cons -> nil:cons 0 :: 0:s nil :: nil:cons s :: 0:s -> 0:s cons :: 0:s -> nil:cons -> nil:cons from :: 0:s -> nil:cons 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 nil => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: first(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X ---------------------------------------- (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: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { from } { first } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 Function symbols to be analyzed: {from}, {first} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 Function symbols to be analyzed: {from}, {first} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: from after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 Function symbols to be analyzed: {from}, {first} Previous analysis results are: from: runtime: ?, size: O(1) [0] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: from after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 first(z, z') -{ 1 }-> 1 + Y + first(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 Function symbols to be analyzed: {from}, {first} Previous analysis results are: from: runtime: INF, size: O(1) [0] ---------------------------------------- (25) 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: first(v0, v1) -> null_first [0] And the following fresh constants: null_first ---------------------------------------- (26) 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: first(0, X) -> nil [1] first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) [1] from(X) -> cons(X, from(s(X))) [1] first(v0, v1) -> null_first [0] The TRS has the following type information: first :: 0:s -> nil:cons:null_first -> nil:cons:null_first 0 :: 0:s nil :: nil:cons:null_first s :: 0:s -> 0:s cons :: 0:s -> nil:cons:null_first -> nil:cons:null_first from :: 0:s -> nil:cons:null_first null_first :: nil:cons:null_first Rewrite Strategy: INNERMOST ---------------------------------------- (27) 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 nil => 0 null_first => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: first(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0 first(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 first(z, z') -{ 1 }-> 1 + Y + first(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (29) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: first(0, z0) -> nil first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2)) from(z0) -> cons(z0, from(s(z0))) Tuples: FIRST(0, z0) -> c FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) S tuples: FIRST(0, z0) -> c FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) K tuples:none Defined Rule Symbols: first_2, from_1 Defined Pair Symbols: FIRST_2, FROM_1 Compound Symbols: c, c1_1, c2_1 ---------------------------------------- (31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FIRST(0, z0) -> c ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: first(0, z0) -> nil first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2)) from(z0) -> cons(z0, from(s(z0))) Tuples: FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) S tuples: FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) K tuples:none Defined Rule Symbols: first_2, from_1 Defined Pair Symbols: FIRST_2, FROM_1 Compound Symbols: c1_1, c2_1 ---------------------------------------- (33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: first(0, z0) -> nil first(s(z0), cons(z1, z2)) -> cons(z1, first(z0, z2)) from(z0) -> cons(z0, from(s(z0))) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) S tuples: FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: FIRST_2, FROM_1 Compound Symbols: c1_1, c2_1 ---------------------------------------- (35) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2)) We considered the (Usable) Rules:none And the Tuples: FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(FIRST(x_1, x_2)) = x_1 + x_2 POL(FROM(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(s(x_1)) = x_1 ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) S tuples: FROM(z0) -> c2(FROM(s(z0))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2)) Defined Rule Symbols:none Defined Pair Symbols: FIRST_2, FROM_1 Compound Symbols: c1_1, c2_1 ---------------------------------------- (37) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FROM(z0) -> c2(FROM(s(z0))) by FROM(s(x0)) -> c2(FROM(s(s(x0)))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2)) FROM(s(x0)) -> c2(FROM(s(s(x0)))) S tuples: FROM(s(x0)) -> c2(FROM(s(s(x0)))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2)) Defined Rule Symbols:none Defined Pair Symbols: FIRST_2, FROM_1 Compound Symbols: c1_1, c2_1 ---------------------------------------- (39) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FROM(s(x0)) -> c2(FROM(s(s(x0)))) by FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2)) FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) S tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) K tuples: FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2)) Defined Rule Symbols:none Defined Pair Symbols: FIRST_2, FROM_1 Compound Symbols: c1_1, c2_1 ---------------------------------------- (41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FIRST(s(z0), cons(z1, z2)) -> c1(FIRST(z0, z2)) by FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FIRST(s(y0), cons(y1, y2))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FIRST(s(y0), cons(y1, y2))) S tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) K tuples: FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FIRST(s(y0), cons(y1, y2))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, FIRST_2 Compound Symbols: c2_1, c1_1 ---------------------------------------- (43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FIRST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FIRST(s(y0), cons(y1, y2))) by FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c1(FIRST(s(s(y0)), cons(z2, cons(y2, y3)))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c1(FIRST(s(s(y0)), cons(z2, cons(y2, y3)))) S tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) K tuples: FIRST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c1(FIRST(s(s(y0)), cons(z2, cons(y2, y3)))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, FIRST_2 Compound Symbols: c2_1, c1_1