MAYBE proof of input_17QnAd4NBw.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 112 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), 164 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 47 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 1191 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 19 ms] (32) CpxRNTS (33) CompletionProof [UPPER BOUND(ID), 0 ms] (34) CpxTypedWeightedCompleteTrs (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 5 ms] (36) CpxRNTS (37) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (38) CdtProblem (39) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtInstantiationProof [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: c -> f(n__g(n__c)) f(n__g(X)) -> g(activate(X)) g(X) -> n__g(X) c -> n__c activate(n__g(X)) -> g(X) activate(n__c) -> c 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: c -> f(n__g(n__c)) f(n__g(X)) -> g(activate(X)) g(X) -> n__g(X) c -> n__c activate(n__g(X)) -> g(X) activate(n__c) -> c 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: c -> f(n__g(n__c)) f(n__g(X)) -> g(activate(X)) g(X) -> n__g(X) c -> n__c activate(n__g(X)) -> g(X) activate(n__c) -> c 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: c -> f(n__g(n__c)) [1] f(n__g(X)) -> g(activate(X)) [1] g(X) -> n__g(X) [1] c -> n__c [1] activate(n__g(X)) -> g(X) [1] activate(n__c) -> c [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: c -> f(n__g(n__c)) [1] f(n__g(X)) -> g(activate(X)) [1] g(X) -> n__g(X) [1] c -> n__c [1] activate(n__g(X)) -> g(X) [1] activate(n__c) -> c [1] activate(X) -> X [1] The TRS has the following type information: c :: n__c:n__g f :: n__c:n__g -> n__c:n__g n__g :: n__c:n__g -> n__c:n__g n__c :: n__c:n__g g :: n__c:n__g -> n__c:n__g activate :: n__c:n__g -> n__c:n__g 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: activate_1 c g_1 f_1 Due to the following rules being added: f(v0) -> n__c [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: c -> f(n__g(n__c)) [1] f(n__g(X)) -> g(activate(X)) [1] g(X) -> n__g(X) [1] c -> n__c [1] activate(n__g(X)) -> g(X) [1] activate(n__c) -> c [1] activate(X) -> X [1] f(v0) -> n__c [0] The TRS has the following type information: c :: n__c:n__g f :: n__c:n__g -> n__c:n__g n__g :: n__c:n__g -> n__c:n__g n__c :: n__c:n__g g :: n__c:n__g -> n__c:n__g activate :: n__c:n__g -> n__c:n__g 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: c -> f(n__g(n__c)) [1] f(n__g(n__g(X'))) -> g(g(X')) [2] f(n__g(n__c)) -> g(c) [2] f(n__g(X)) -> g(X) [2] g(X) -> n__g(X) [1] c -> n__c [1] activate(n__g(X)) -> g(X) [1] activate(n__c) -> c [1] activate(X) -> X [1] f(v0) -> n__c [0] The TRS has the following type information: c :: n__c:n__g f :: n__c:n__g -> n__c:n__g n__g :: n__c:n__g -> n__c:n__g n__c :: n__c:n__g g :: n__c:n__g -> n__c:n__g activate :: n__c:n__g -> n__c:n__g 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: n__c => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> g(X) :|: z = 1 + X, X >= 0 activate(z) -{ 1 }-> c :|: z = 0 c -{ 1 }-> f(1 + 0) :|: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> g(X) :|: z = 1 + X, X >= 0 f(z) -{ 2 }-> g(g(X')) :|: X' >= 0, z = 1 + (1 + X') f(z) -{ 2 }-> g(c) :|: z = 1 + 0 f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 g(z) -{ 1 }-> 1 + X :|: X >= 0, z = X ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: g(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) -{ 1 }-> c :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X' c -{ 1 }-> f(1 + 0) :|: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> g(c) :|: z = 1 + 0 f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 f(z) -{ 3 }-> 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X' f(z) -{ 4 }-> 1 + X'' :|: X' >= 0, z = 1 + (1 + X'), X >= 0, X' = X, X'' >= 0, 1 + X = X'' g(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) -{ 1 }-> c :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' c -{ 1 }-> f(1 + 0) :|: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> g(c) :|: z = 1 + 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X'' g(z) -{ 1 }-> 1 + z :|: z >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { g } { c, f } { activate } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> c :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' c -{ 1 }-> f(1 + 0) :|: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> g(c) :|: z = 1 + 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X'' g(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {g}, {c,f}, {activate} ---------------------------------------- (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) -{ 1 }-> c :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' c -{ 1 }-> f(1 + 0) :|: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> g(c) :|: z = 1 + 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X'' g(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {g}, {c,f}, {activate} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> c :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' c -{ 1 }-> f(1 + 0) :|: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> g(c) :|: z = 1 + 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X'' g(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {g}, {c,f}, {activate} Previous analysis results are: g: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> c :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' c -{ 1 }-> f(1 + 0) :|: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> g(c) :|: z = 1 + 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X'' g(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {c,f}, {activate} Previous analysis results are: g: runtime: O(1) [1], size: O(n^1) [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) -{ 1 }-> c :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' c -{ 1 }-> f(1 + 0) :|: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> g(c) :|: z = 1 + 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X'' g(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {c,f}, {activate} Previous analysis results are: g: runtime: O(1) [1], size: O(n^1) [1 + z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: c after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> c :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' c -{ 1 }-> f(1 + 0) :|: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> g(c) :|: z = 1 + 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X'' g(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {c,f}, {activate} Previous analysis results are: g: runtime: O(1) [1], size: O(n^1) [1 + z] c: runtime: ?, size: INF f: runtime: ?, size: INF ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: c after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> c :|: z = 0 activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' c -{ 1 }-> f(1 + 0) :|: c -{ 1 }-> 0 :|: f(z) -{ 2 }-> g(c) :|: z = 1 + 0 f(z) -{ 0 }-> 0 :|: z >= 0 f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X' f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X'' g(z) -{ 1 }-> 1 + z :|: z >= 0 Function symbols to be analyzed: {c,f}, {activate} Previous analysis results are: g: runtime: O(1) [1], size: O(n^1) [1 + z] c: runtime: INF, size: INF f: runtime: ?, size: INF ---------------------------------------- (33) 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 ---------------------------------------- (34) 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: c -> f(n__g(n__c)) [1] f(n__g(X)) -> g(activate(X)) [1] g(X) -> n__g(X) [1] c -> n__c [1] activate(n__g(X)) -> g(X) [1] activate(n__c) -> c [1] activate(X) -> X [1] f(v0) -> null_f [0] The TRS has the following type information: c :: n__c:n__g:null_f f :: n__c:n__g:null_f -> n__c:n__g:null_f n__g :: n__c:n__g:null_f -> n__c:n__g:null_f n__c :: n__c:n__g:null_f g :: n__c:n__g:null_f -> n__c:n__g:null_f activate :: n__c:n__g:null_f -> n__c:n__g:null_f null_f :: n__c:n__g:null_f Rewrite Strategy: INNERMOST ---------------------------------------- (35) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: n__c => 0 null_f => 0 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> g(X) :|: z = 1 + X, X >= 0 activate(z) -{ 1 }-> c :|: z = 0 c -{ 1 }-> f(1 + 0) :|: c -{ 1 }-> 0 :|: f(z) -{ 1 }-> g(activate(X)) :|: z = 1 + X, X >= 0 f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 g(z) -{ 1 }-> 1 + X :|: X >= 0, z = X Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (37) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: c -> f(n__g(n__c)) c -> n__c f(n__g(z0)) -> g(activate(z0)) g(z0) -> n__g(z0) activate(n__g(z0)) -> g(z0) activate(n__c) -> c activate(z0) -> z0 Tuples: C -> c1(F(n__g(n__c))) C -> c2 F(n__g(z0)) -> c3(G(activate(z0)), ACTIVATE(z0)) G(z0) -> c4 ACTIVATE(n__g(z0)) -> c5(G(z0)) ACTIVATE(n__c) -> c6(C) ACTIVATE(z0) -> c7 S tuples: C -> c1(F(n__g(n__c))) C -> c2 F(n__g(z0)) -> c3(G(activate(z0)), ACTIVATE(z0)) G(z0) -> c4 ACTIVATE(n__g(z0)) -> c5(G(z0)) ACTIVATE(n__c) -> c6(C) ACTIVATE(z0) -> c7 K tuples:none Defined Rule Symbols: c, f_1, g_1, activate_1 Defined Pair Symbols: C, F_1, G_1, ACTIVATE_1 Compound Symbols: c1_1, c2, c3_2, c4, c5_1, c6_1, c7 ---------------------------------------- (39) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: C -> c2 ACTIVATE(z0) -> c7 G(z0) -> c4 ACTIVATE(n__g(z0)) -> c5(G(z0)) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: c -> f(n__g(n__c)) c -> n__c f(n__g(z0)) -> g(activate(z0)) g(z0) -> n__g(z0) activate(n__g(z0)) -> g(z0) activate(n__c) -> c activate(z0) -> z0 Tuples: C -> c1(F(n__g(n__c))) F(n__g(z0)) -> c3(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__c) -> c6(C) S tuples: C -> c1(F(n__g(n__c))) F(n__g(z0)) -> c3(G(activate(z0)), ACTIVATE(z0)) ACTIVATE(n__c) -> c6(C) K tuples:none Defined Rule Symbols: c, f_1, g_1, activate_1 Defined Pair Symbols: C, F_1, ACTIVATE_1 Compound Symbols: c1_1, c3_2, c6_1 ---------------------------------------- (41) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: c -> f(n__g(n__c)) c -> n__c f(n__g(z0)) -> g(activate(z0)) g(z0) -> n__g(z0) activate(n__g(z0)) -> g(z0) activate(n__c) -> c activate(z0) -> z0 Tuples: C -> c1(F(n__g(n__c))) ACTIVATE(n__c) -> c6(C) F(n__g(z0)) -> c3(ACTIVATE(z0)) S tuples: C -> c1(F(n__g(n__c))) ACTIVATE(n__c) -> c6(C) F(n__g(z0)) -> c3(ACTIVATE(z0)) K tuples:none Defined Rule Symbols: c, f_1, g_1, activate_1 Defined Pair Symbols: C, ACTIVATE_1, F_1 Compound Symbols: c1_1, c6_1, c3_1 ---------------------------------------- (43) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: c -> f(n__g(n__c)) c -> n__c f(n__g(z0)) -> g(activate(z0)) g(z0) -> n__g(z0) activate(n__g(z0)) -> g(z0) activate(n__c) -> c activate(z0) -> z0 ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: C -> c1(F(n__g(n__c))) ACTIVATE(n__c) -> c6(C) F(n__g(z0)) -> c3(ACTIVATE(z0)) S tuples: C -> c1(F(n__g(n__c))) ACTIVATE(n__c) -> c6(C) F(n__g(z0)) -> c3(ACTIVATE(z0)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: C, ACTIVATE_1, F_1 Compound Symbols: c1_1, c6_1, c3_1 ---------------------------------------- (45) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace F(n__g(z0)) -> c3(ACTIVATE(z0)) by F(n__g(n__c)) -> c3(ACTIVATE(n__c)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: C -> c1(F(n__g(n__c))) ACTIVATE(n__c) -> c6(C) F(n__g(n__c)) -> c3(ACTIVATE(n__c)) S tuples: C -> c1(F(n__g(n__c))) ACTIVATE(n__c) -> c6(C) F(n__g(n__c)) -> c3(ACTIVATE(n__c)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: C, ACTIVATE_1, F_1 Compound Symbols: c1_1, c6_1, c3_1