MAYBE proof of input_otvooEpeMC.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 187 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), 90 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 24 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 20 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 33 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 168 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 65 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 449 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 137 ms] (44) CpxRNTS (45) CompletionProof [UPPER BOUND(ID), 0 ms] (46) CpxTypedWeightedCompleteTrs (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 16 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (82) 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: g(X) -> h(activate(X)) c -> d h(n__d) -> g(n__c) d -> n__d c -> n__c activate(n__d) -> d 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: g(X) -> h(activate(X)) c -> d h(n__d) -> g(n__c) d -> n__d c -> n__c activate(n__d) -> d 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: g(X) -> h(activate(X)) c -> d h(n__d) -> g(n__c) d -> n__d c -> n__c activate(n__d) -> d 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: g(X) -> h(activate(X)) [1] c -> d [1] h(n__d) -> g(n__c) [1] d -> n__d [1] c -> n__c [1] activate(n__d) -> d [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: g(X) -> h(activate(X)) [1] c -> d [1] h(n__d) -> g(n__c) [1] d -> n__d [1] c -> n__c [1] activate(n__d) -> d [1] activate(n__c) -> c [1] activate(X) -> X [1] The TRS has the following type information: g :: n__d:n__c -> g:h h :: n__d:n__c -> g:h activate :: n__d:n__c -> n__d:n__c c :: n__d:n__c d :: n__d:n__c n__d :: n__d:n__c n__c :: n__d:n__c 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: g_1 h_1 (c) The following functions are completely defined: activate_1 d c Due to the following rules being added: none And the following fresh constants: const ---------------------------------------- (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: g(X) -> h(activate(X)) [1] c -> d [1] h(n__d) -> g(n__c) [1] d -> n__d [1] c -> n__c [1] activate(n__d) -> d [1] activate(n__c) -> c [1] activate(X) -> X [1] The TRS has the following type information: g :: n__d:n__c -> g:h h :: n__d:n__c -> g:h activate :: n__d:n__c -> n__d:n__c c :: n__d:n__c d :: n__d:n__c n__d :: n__d:n__c n__c :: n__d:n__c const :: g: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: g(n__d) -> h(d) [2] g(n__c) -> h(c) [2] g(X) -> h(X) [2] c -> d [1] h(n__d) -> g(n__c) [1] d -> n__d [1] c -> n__c [1] activate(n__d) -> d [1] activate(n__c) -> c [1] activate(X) -> X [1] The TRS has the following type information: g :: n__d:n__c -> g:h h :: n__d:n__c -> g:h activate :: n__d:n__c -> n__d:n__c c :: n__d:n__c d :: n__d:n__c n__d :: n__d:n__c n__c :: n__d:n__c const :: g: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: n__d => 1 n__c => 0 const => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> d :|: z = 1 activate(z) -{ 1 }-> c :|: z = 0 c -{ 1 }-> d :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 2 }-> h(X) :|: X >= 0, z = X g(z) -{ 2 }-> h(d) :|: z = 1 g(z) -{ 2 }-> h(c) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: d -{ 1 }-> 1 :|: c -{ 1 }-> d :|: c -{ 1 }-> 0 :|: ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 2 }-> d :|: z = 0 activate(z) -{ 2 }-> 1 :|: z = 1 activate(z) -{ 2 }-> 0 :|: z = 0 c -{ 2 }-> 1 :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 2 }-> h(X) :|: X >= 0, z = X g(z) -{ 3 }-> h(d) :|: z = 0 g(z) -{ 3 }-> h(1) :|: z = 1 g(z) -{ 3 }-> h(0) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 ---------------------------------------- (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 }-> d :|: z = 0 activate(z) -{ 2 }-> 1 :|: z = 1 activate(z) -{ 2 }-> 0 :|: z = 0 c -{ 2 }-> 1 :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 2 }-> h(z) :|: z >= 0 g(z) -{ 3 }-> h(d) :|: z = 0 g(z) -{ 3 }-> h(1) :|: z = 1 g(z) -{ 3 }-> h(0) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { d } { c } { activate } { h, g } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> d :|: z = 0 activate(z) -{ 2 }-> 1 :|: z = 1 activate(z) -{ 2 }-> 0 :|: z = 0 c -{ 2 }-> 1 :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 2 }-> h(z) :|: z >= 0 g(z) -{ 3 }-> h(d) :|: z = 0 g(z) -{ 3 }-> h(1) :|: z = 1 g(z) -{ 3 }-> h(0) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 Function symbols to be analyzed: {d}, {c}, {activate}, {h,g} ---------------------------------------- (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 }-> d :|: z = 0 activate(z) -{ 2 }-> 1 :|: z = 1 activate(z) -{ 2 }-> 0 :|: z = 0 c -{ 2 }-> 1 :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 2 }-> h(z) :|: z >= 0 g(z) -{ 3 }-> h(d) :|: z = 0 g(z) -{ 3 }-> h(1) :|: z = 1 g(z) -{ 3 }-> h(0) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 Function symbols to be analyzed: {d}, {c}, {activate}, {h,g} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: d after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> d :|: z = 0 activate(z) -{ 2 }-> 1 :|: z = 1 activate(z) -{ 2 }-> 0 :|: z = 0 c -{ 2 }-> 1 :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 2 }-> h(z) :|: z >= 0 g(z) -{ 3 }-> h(d) :|: z = 0 g(z) -{ 3 }-> h(1) :|: z = 1 g(z) -{ 3 }-> h(0) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 Function symbols to be analyzed: {d}, {c}, {activate}, {h,g} Previous analysis results are: d: runtime: ?, size: O(1) [1] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: d 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) -{ 2 }-> d :|: z = 0 activate(z) -{ 2 }-> 1 :|: z = 1 activate(z) -{ 2 }-> 0 :|: z = 0 c -{ 2 }-> 1 :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 2 }-> h(z) :|: z >= 0 g(z) -{ 3 }-> h(d) :|: z = 0 g(z) -{ 3 }-> h(1) :|: z = 1 g(z) -{ 3 }-> h(0) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 Function symbols to be analyzed: {c}, {activate}, {h,g} Previous analysis results are: d: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (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) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 :|: z = 1 activate(z) -{ 2 }-> 0 :|: z = 0 c -{ 2 }-> 1 :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0 g(z) -{ 2 }-> h(z) :|: z >= 0 g(z) -{ 3 }-> h(1) :|: z = 1 g(z) -{ 3 }-> h(0) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 Function symbols to be analyzed: {c}, {activate}, {h,g} Previous analysis results are: d: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: c after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 :|: z = 1 activate(z) -{ 2 }-> 0 :|: z = 0 c -{ 2 }-> 1 :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0 g(z) -{ 2 }-> h(z) :|: z >= 0 g(z) -{ 3 }-> h(1) :|: z = 1 g(z) -{ 3 }-> h(0) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 Function symbols to be analyzed: {c}, {activate}, {h,g} Previous analysis results are: d: runtime: O(1) [1], size: O(1) [1] c: runtime: ?, size: O(1) [1] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: c after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 :|: z = 1 activate(z) -{ 2 }-> 0 :|: z = 0 c -{ 2 }-> 1 :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0 g(z) -{ 2 }-> h(z) :|: z >= 0 g(z) -{ 3 }-> h(1) :|: z = 1 g(z) -{ 3 }-> h(0) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 Function symbols to be analyzed: {activate}, {h,g} Previous analysis results are: d: runtime: O(1) [1], size: O(1) [1] c: runtime: O(1) [2], size: O(1) [1] ---------------------------------------- (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) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 :|: z = 1 activate(z) -{ 2 }-> 0 :|: z = 0 c -{ 2 }-> 1 :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0 g(z) -{ 2 }-> h(z) :|: z >= 0 g(z) -{ 3 }-> h(1) :|: z = 1 g(z) -{ 3 }-> h(0) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 Function symbols to be analyzed: {activate}, {h,g} Previous analysis results are: d: runtime: O(1) [1], size: O(1) [1] c: runtime: O(1) [2], size: O(1) [1] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: activate after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 :|: z = 1 activate(z) -{ 2 }-> 0 :|: z = 0 c -{ 2 }-> 1 :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0 g(z) -{ 2 }-> h(z) :|: z >= 0 g(z) -{ 3 }-> h(1) :|: z = 1 g(z) -{ 3 }-> h(0) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 Function symbols to be analyzed: {activate}, {h,g} Previous analysis results are: d: runtime: O(1) [1], size: O(1) [1] c: runtime: O(1) [2], size: O(1) [1] activate: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: activate after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 :|: z = 1 activate(z) -{ 2 }-> 0 :|: z = 0 c -{ 2 }-> 1 :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0 g(z) -{ 2 }-> h(z) :|: z >= 0 g(z) -{ 3 }-> h(1) :|: z = 1 g(z) -{ 3 }-> h(0) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 Function symbols to be analyzed: {h,g} Previous analysis results are: d: runtime: O(1) [1], size: O(1) [1] c: runtime: O(1) [2], size: O(1) [1] activate: runtime: O(1) [3], size: O(n^1) [1 + z] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 :|: z = 1 activate(z) -{ 2 }-> 0 :|: z = 0 c -{ 2 }-> 1 :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0 g(z) -{ 2 }-> h(z) :|: z >= 0 g(z) -{ 3 }-> h(1) :|: z = 1 g(z) -{ 3 }-> h(0) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 Function symbols to be analyzed: {h,g} Previous analysis results are: d: runtime: O(1) [1], size: O(1) [1] c: runtime: O(1) [2], size: O(1) [1] activate: runtime: O(1) [3], size: O(n^1) [1 + z] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: h after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed SIZE bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 :|: z = 1 activate(z) -{ 2 }-> 0 :|: z = 0 c -{ 2 }-> 1 :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0 g(z) -{ 2 }-> h(z) :|: z >= 0 g(z) -{ 3 }-> h(1) :|: z = 1 g(z) -{ 3 }-> h(0) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 Function symbols to be analyzed: {h,g} Previous analysis results are: d: runtime: O(1) [1], size: O(1) [1] c: runtime: O(1) [2], size: O(1) [1] activate: runtime: O(1) [3], size: O(n^1) [1 + z] h: runtime: ?, size: O(1) [0] g: runtime: ?, size: O(1) [0] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: h after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0 activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 2 }-> 1 :|: z = 1 activate(z) -{ 2 }-> 0 :|: z = 0 c -{ 2 }-> 1 :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0 g(z) -{ 2 }-> h(z) :|: z >= 0 g(z) -{ 3 }-> h(1) :|: z = 1 g(z) -{ 3 }-> h(0) :|: z = 0 h(z) -{ 1 }-> g(0) :|: z = 1 Function symbols to be analyzed: {h,g} Previous analysis results are: d: runtime: O(1) [1], size: O(1) [1] c: runtime: O(1) [2], size: O(1) [1] activate: runtime: O(1) [3], size: O(n^1) [1 + z] h: runtime: INF, size: O(1) [0] g: runtime: ?, size: O(1) [0] ---------------------------------------- (45) 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: h(v0) -> null_h [0] And the following fresh constants: null_h ---------------------------------------- (46) 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: g(X) -> h(activate(X)) [1] c -> d [1] h(n__d) -> g(n__c) [1] d -> n__d [1] c -> n__c [1] activate(n__d) -> d [1] activate(n__c) -> c [1] activate(X) -> X [1] h(v0) -> null_h [0] The TRS has the following type information: g :: n__d:n__c -> null_h h :: n__d:n__c -> null_h activate :: n__d:n__c -> n__d:n__c c :: n__d:n__c d :: n__d:n__c n__d :: n__d:n__c n__c :: n__d:n__c null_h :: null_h Rewrite Strategy: INNERMOST ---------------------------------------- (47) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: n__d => 1 n__c => 0 null_h => 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> d :|: z = 1 activate(z) -{ 1 }-> c :|: z = 0 c -{ 1 }-> d :|: c -{ 1 }-> 0 :|: d -{ 1 }-> 1 :|: g(z) -{ 1 }-> h(activate(X)) :|: X >= 0, z = X h(z) -{ 1 }-> g(0) :|: z = 1 h(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (49) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: g(z0) -> h(activate(z0)) c -> d c -> n__c h(n__d) -> g(n__c) d -> n__d activate(n__d) -> d activate(n__c) -> c activate(z0) -> z0 Tuples: G(z0) -> c1(H(activate(z0)), ACTIVATE(z0)) C -> c2(D) C -> c3 H(n__d) -> c4(G(n__c)) D -> c5 ACTIVATE(n__d) -> c6(D) ACTIVATE(n__c) -> c7(C) ACTIVATE(z0) -> c8 S tuples: G(z0) -> c1(H(activate(z0)), ACTIVATE(z0)) C -> c2(D) C -> c3 H(n__d) -> c4(G(n__c)) D -> c5 ACTIVATE(n__d) -> c6(D) ACTIVATE(n__c) -> c7(C) ACTIVATE(z0) -> c8 K tuples:none Defined Rule Symbols: g_1, c, h_1, d, activate_1 Defined Pair Symbols: G_1, C, H_1, D, ACTIVATE_1 Compound Symbols: c1_2, c2_1, c3, c4_1, c5, c6_1, c7_1, c8 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: ACTIVATE(n__d) -> c6(D) D -> c5 ACTIVATE(z0) -> c8 ACTIVATE(n__c) -> c7(C) C -> c2(D) C -> c3 ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: g(z0) -> h(activate(z0)) c -> d c -> n__c h(n__d) -> g(n__c) d -> n__d activate(n__d) -> d activate(n__c) -> c activate(z0) -> z0 Tuples: G(z0) -> c1(H(activate(z0)), ACTIVATE(z0)) H(n__d) -> c4(G(n__c)) S tuples: G(z0) -> c1(H(activate(z0)), ACTIVATE(z0)) H(n__d) -> c4(G(n__c)) K tuples:none Defined Rule Symbols: g_1, c, h_1, d, activate_1 Defined Pair Symbols: G_1, H_1 Compound Symbols: c1_2, c4_1 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: g(z0) -> h(activate(z0)) c -> d c -> n__c h(n__d) -> g(n__c) d -> n__d activate(n__d) -> d activate(n__c) -> c activate(z0) -> z0 Tuples: H(n__d) -> c4(G(n__c)) G(z0) -> c1(H(activate(z0))) S tuples: H(n__d) -> c4(G(n__c)) G(z0) -> c1(H(activate(z0))) K tuples:none Defined Rule Symbols: g_1, c, h_1, d, activate_1 Defined Pair Symbols: H_1, G_1 Compound Symbols: c4_1, c1_1 ---------------------------------------- (55) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: g(z0) -> h(activate(z0)) h(n__d) -> g(n__c) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__d) -> d activate(n__c) -> c activate(z0) -> z0 d -> n__d c -> d c -> n__c Tuples: H(n__d) -> c4(G(n__c)) G(z0) -> c1(H(activate(z0))) S tuples: H(n__d) -> c4(G(n__c)) G(z0) -> c1(H(activate(z0))) K tuples:none Defined Rule Symbols: activate_1, d, c Defined Pair Symbols: H_1, G_1 Compound Symbols: c4_1, c1_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(z0) -> c1(H(activate(z0))) by G(n__d) -> c1(H(d)) G(n__c) -> c1(H(c)) G(z0) -> c1(H(z0)) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__d) -> d activate(n__c) -> c activate(z0) -> z0 d -> n__d c -> d c -> n__c Tuples: H(n__d) -> c4(G(n__c)) G(n__d) -> c1(H(d)) G(n__c) -> c1(H(c)) G(z0) -> c1(H(z0)) S tuples: H(n__d) -> c4(G(n__c)) G(n__d) -> c1(H(d)) G(n__c) -> c1(H(c)) G(z0) -> c1(H(z0)) K tuples:none Defined Rule Symbols: activate_1, d, c Defined Pair Symbols: H_1, G_1 Compound Symbols: c4_1, c1_1 ---------------------------------------- (59) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: G(n__d) -> c1(H(d)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__d) -> d activate(n__c) -> c activate(z0) -> z0 d -> n__d c -> d c -> n__c Tuples: H(n__d) -> c4(G(n__c)) G(n__d) -> c1(H(d)) G(n__c) -> c1(H(c)) G(z0) -> c1(H(z0)) S tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(c)) G(z0) -> c1(H(z0)) K tuples: G(n__d) -> c1(H(d)) Defined Rule Symbols: activate_1, d, c Defined Pair Symbols: H_1, G_1 Compound Symbols: c4_1, c1_1 ---------------------------------------- (61) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: activate(n__d) -> d activate(n__c) -> c activate(z0) -> z0 ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: d -> n__d c -> d c -> n__c Tuples: H(n__d) -> c4(G(n__c)) G(n__d) -> c1(H(d)) G(n__c) -> c1(H(c)) G(z0) -> c1(H(z0)) S tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(c)) G(z0) -> c1(H(z0)) K tuples: G(n__d) -> c1(H(d)) Defined Rule Symbols: d, c Defined Pair Symbols: H_1, G_1 Compound Symbols: c4_1, c1_1 ---------------------------------------- (63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. G(z0) -> c1(H(z0)) We considered the (Usable) Rules: c -> d d -> n__d c -> n__c And the Tuples: H(n__d) -> c4(G(n__c)) G(n__d) -> c1(H(d)) G(n__c) -> c1(H(c)) G(z0) -> c1(H(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(G(x_1)) = [2] + [2]x_1 POL(H(x_1)) = x_1 POL(c) = [2] POL(c1(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(d) = [2] POL(n__c) = 0 POL(n__d) = [2] ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: d -> n__d c -> d c -> n__c Tuples: H(n__d) -> c4(G(n__c)) G(n__d) -> c1(H(d)) G(n__c) -> c1(H(c)) G(z0) -> c1(H(z0)) S tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(c)) K tuples: G(n__d) -> c1(H(d)) G(z0) -> c1(H(z0)) Defined Rule Symbols: d, c Defined Pair Symbols: H_1, G_1 Compound Symbols: c4_1, c1_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(n__d) -> c1(H(d)) by G(n__d) -> c1(H(n__d)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: d -> n__d c -> d c -> n__c Tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(c)) G(z0) -> c1(H(z0)) G(n__d) -> c1(H(n__d)) S tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(c)) K tuples: G(n__d) -> c1(H(d)) G(z0) -> c1(H(z0)) Defined Rule Symbols: d, c Defined Pair Symbols: H_1, G_1 Compound Symbols: c4_1, c1_1 ---------------------------------------- (67) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: G(n__d) -> c1(H(n__d)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: d -> n__d c -> d c -> n__c Tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(c)) G(z0) -> c1(H(z0)) S tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(c)) K tuples: G(z0) -> c1(H(z0)) Defined Rule Symbols: d, c Defined Pair Symbols: H_1, G_1 Compound Symbols: c4_1, c1_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(n__c) -> c1(H(c)) by G(n__c) -> c1(H(d)) G(n__c) -> c1(H(n__c)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: d -> n__d c -> d c -> n__c Tuples: H(n__d) -> c4(G(n__c)) G(z0) -> c1(H(z0)) G(n__c) -> c1(H(d)) G(n__c) -> c1(H(n__c)) S tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(d)) G(n__c) -> c1(H(n__c)) K tuples: G(z0) -> c1(H(z0)) Defined Rule Symbols: d, c Defined Pair Symbols: H_1, G_1 Compound Symbols: c4_1, c1_1 ---------------------------------------- (71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: G(n__c) -> c1(H(n__c)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: d -> n__d c -> d c -> n__c Tuples: H(n__d) -> c4(G(n__c)) G(z0) -> c1(H(z0)) G(n__c) -> c1(H(d)) S tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(d)) K tuples: G(z0) -> c1(H(z0)) Defined Rule Symbols: d, c Defined Pair Symbols: H_1, G_1 Compound Symbols: c4_1, c1_1 ---------------------------------------- (73) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: c -> d c -> n__c ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: d -> n__d Tuples: H(n__d) -> c4(G(n__c)) G(z0) -> c1(H(z0)) G(n__c) -> c1(H(d)) S tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(d)) K tuples: G(z0) -> c1(H(z0)) Defined Rule Symbols: d Defined Pair Symbols: H_1, G_1 Compound Symbols: c4_1, c1_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace G(n__c) -> c1(H(d)) by G(n__c) -> c1(H(n__d)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: d -> n__d Tuples: H(n__d) -> c4(G(n__c)) G(z0) -> c1(H(z0)) G(n__c) -> c1(H(n__d)) S tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(n__d)) K tuples: G(z0) -> c1(H(z0)) Defined Rule Symbols: d Defined Pair Symbols: H_1, G_1 Compound Symbols: c4_1, c1_1 ---------------------------------------- (77) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: d -> n__d ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: H(n__d) -> c4(G(n__c)) G(z0) -> c1(H(z0)) G(n__c) -> c1(H(n__d)) S tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(n__d)) K tuples: G(z0) -> c1(H(z0)) Defined Rule Symbols:none Defined Pair Symbols: H_1, G_1 Compound Symbols: c4_1, c1_1 ---------------------------------------- (79) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace G(z0) -> c1(H(z0)) by G(n__c) -> c1(H(n__c)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(n__d)) G(n__c) -> c1(H(n__c)) S tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(n__d)) K tuples: G(n__c) -> c1(H(n__c)) Defined Rule Symbols:none Defined Pair Symbols: H_1, G_1 Compound Symbols: c4_1, c1_1 ---------------------------------------- (81) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: G(n__c) -> c1(H(n__c)) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(n__d)) S tuples: H(n__d) -> c4(G(n__c)) G(n__c) -> c1(H(n__d)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: H_1, G_1 Compound Symbols: c4_1, c1_1