MAYBE proof of input_heOVRPZ99w.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) InliningProof [UPPER BOUND(ID), 27 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 117 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 158 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 43 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 262 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 55 ms] (38) CpxRNTS (39) CompletionProof [UPPER BOUND(ID), 0 ms] (40) CpxTypedWeightedCompleteTrs (41) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (44) CdtProblem (45) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (48) 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(X) -> g(n__h(f(X))) h(X) -> n__h(X) activate(n__h(X)) -> h(X) activate(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(X) -> g(n__h(f(X))) h(X) -> n__h(X) activate(n__h(X)) -> h(X) activate(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(X) -> g(n__h(f(X))) h(X) -> n__h(X) activate(n__h(X)) -> h(X) activate(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(X) -> g(n__h(f(X))) [1] h(X) -> n__h(X) [1] activate(n__h(X)) -> h(X) [1] activate(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(X) -> g(n__h(f(X))) [1] h(X) -> n__h(X) [1] activate(n__h(X)) -> h(X) [1] activate(X) -> X [1] The TRS has the following type information: f :: a -> g g :: n__h -> g n__h :: g -> n__h h :: g -> n__h activate :: n__h -> n__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: f_1 h_1 activate_1 (c) The following functions are completely defined: none Due to the following rules being added: none And the following fresh constants: const, const1, const2 ---------------------------------------- (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(X) -> g(n__h(f(X))) [1] h(X) -> n__h(X) [1] activate(n__h(X)) -> h(X) [1] activate(X) -> X [1] The TRS has the following type information: f :: a -> g g :: n__h -> g n__h :: g -> n__h h :: g -> n__h activate :: n__h -> n__h const :: g const1 :: a const2 :: n__h 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(X) -> g(n__h(f(X))) [1] h(X) -> n__h(X) [1] activate(n__h(X)) -> h(X) [1] activate(X) -> X [1] The TRS has the following type information: f :: a -> g g :: n__h -> g n__h :: g -> n__h h :: g -> n__h activate :: n__h -> n__h const :: g const1 :: a const2 :: n__h 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 const1 => 0 const2 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> h(X) :|: z = 1 + X, X >= 0 f(z) -{ 1 }-> 1 + (1 + f(X)) :|: X >= 0, z = X h(z) -{ 1 }-> 1 + X :|: X >= 0, z = X ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: h(z) -{ 1 }-> 1 + X :|: X >= 0, z = X ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 2 }-> 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X' f(z) -{ 1 }-> 1 + (1 + f(X)) :|: X >= 0, z = X h(z) -{ 1 }-> 1 + X :|: X >= 0, z = X ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 h(z) -{ 1 }-> 1 + z :|: z >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { activate } { h } { f } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 h(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {activate}, {h}, {f} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 h(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {activate}, {h}, {f} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: activate after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 h(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {activate}, {h}, {f} Previous analysis results are: activate: runtime: ?, size: O(n^1) [z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: activate after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 h(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {h}, {f} Previous analysis results are: activate: runtime: O(1) [3], size: O(n^1) [z] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 h(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {h}, {f} Previous analysis results are: activate: runtime: O(1) [3], size: O(n^1) [z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: h after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 h(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {h}, {f} Previous analysis results are: activate: runtime: O(1) [3], size: O(n^1) [z] h: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: h after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 h(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {f} Previous analysis results are: activate: runtime: O(1) [3], size: O(n^1) [z] h: runtime: O(1) [1], size: O(n^1) [1 + z] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 h(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {f} Previous analysis results are: activate: runtime: O(1) [3], size: O(n^1) [z] h: runtime: O(1) [1], size: O(n^1) [1 + z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 h(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {f} Previous analysis results are: activate: runtime: O(1) [3], size: O(n^1) [z] h: runtime: O(1) [1], size: O(n^1) [1 + z] f: runtime: ?, size: O(1) [0] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0 h(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {f} Previous analysis results are: activate: runtime: O(1) [3], size: O(n^1) [z] h: runtime: O(1) [1], size: O(n^1) [1 + z] f: runtime: INF, size: O(1) [0] ---------------------------------------- (39) 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: none And the following fresh constants: const, const1, const2 ---------------------------------------- (40) 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(X) -> g(n__h(f(X))) [1] h(X) -> n__h(X) [1] activate(n__h(X)) -> h(X) [1] activate(X) -> X [1] The TRS has the following type information: f :: a -> g g :: n__h -> g n__h :: g -> n__h h :: g -> n__h activate :: n__h -> n__h const :: g const1 :: a const2 :: n__h Rewrite Strategy: INNERMOST ---------------------------------------- (41) 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 const1 => 0 const2 => 0 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> h(X) :|: z = 1 + X, X >= 0 f(z) -{ 1 }-> 1 + (1 + f(X)) :|: X >= 0, z = X h(z) -{ 1 }-> 1 + X :|: X >= 0, z = X Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (43) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: f(z0) -> g(n__h(f(z0))) h(z0) -> n__h(z0) activate(n__h(z0)) -> h(z0) activate(z0) -> z0 Tuples: F(z0) -> c(F(z0)) H(z0) -> c1 ACTIVATE(n__h(z0)) -> c2(H(z0)) ACTIVATE(z0) -> c3 S tuples: F(z0) -> c(F(z0)) H(z0) -> c1 ACTIVATE(n__h(z0)) -> c2(H(z0)) ACTIVATE(z0) -> c3 K tuples:none Defined Rule Symbols: f_1, h_1, activate_1 Defined Pair Symbols: F_1, H_1, ACTIVATE_1 Compound Symbols: c_1, c1, c2_1, c3 ---------------------------------------- (45) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: H(z0) -> c1 ACTIVATE(n__h(z0)) -> c2(H(z0)) ACTIVATE(z0) -> c3 ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: f(z0) -> g(n__h(f(z0))) h(z0) -> n__h(z0) activate(n__h(z0)) -> h(z0) activate(z0) -> z0 Tuples: F(z0) -> c(F(z0)) S tuples: F(z0) -> c(F(z0)) K tuples:none Defined Rule Symbols: f_1, h_1, activate_1 Defined Pair Symbols: F_1 Compound Symbols: c_1 ---------------------------------------- (47) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(z0) -> g(n__h(f(z0))) h(z0) -> n__h(z0) activate(n__h(z0)) -> h(z0) activate(z0) -> z0 ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: F(z0) -> c(F(z0)) S tuples: F(z0) -> c(F(z0)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: F_1 Compound Symbols: c_1