MAYBE proof of input_1qpsj9z7GO.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 1 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 183 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 48 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 157 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 64 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 105 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 66 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 258 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 58 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 571 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 53 ms] (48) CpxRNTS (49) CompletionProof [UPPER BOUND(ID), 0 ms] (50) CpxTypedWeightedCompleteTrs (51) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (54) CdtProblem (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (60) CdtProblem (61) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 293 ms] (66) CdtProblem (67) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (88) CdtProblem (89) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (112) 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: fib(N) -> sel(N, fib1(s(0), s(0))) fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y))) add(0, X) -> X add(s(X), Y) -> s(add(X, Y)) sel(0, cons(X, XS)) -> X sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) fib1(X1, X2) -> n__fib1(X1, X2) activate(n__fib1(X1, X2)) -> fib1(X1, X2) 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: fib(N) -> sel(N, fib1(s(0'), s(0'))) fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y))) add(0', X) -> X add(s(X), Y) -> s(add(X, Y)) sel(0', cons(X, XS)) -> X sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) fib1(X1, X2) -> n__fib1(X1, X2) activate(n__fib1(X1, X2)) -> fib1(X1, X2) 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: fib(N) -> sel(N, fib1(s(0), s(0))) fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y))) add(0, X) -> X add(s(X), Y) -> s(add(X, Y)) sel(0, cons(X, XS)) -> X sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) fib1(X1, X2) -> n__fib1(X1, X2) activate(n__fib1(X1, X2)) -> fib1(X1, X2) 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: fib(N) -> sel(N, fib1(s(0), s(0))) [1] fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] sel(0, cons(X, XS)) -> X [1] sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) [1] fib1(X1, X2) -> n__fib1(X1, X2) [1] activate(n__fib1(X1, X2)) -> fib1(X1, X2) [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: fib(N) -> sel(N, fib1(s(0), s(0))) [1] fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] sel(0, cons(X, XS)) -> X [1] sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) [1] fib1(X1, X2) -> n__fib1(X1, X2) [1] activate(n__fib1(X1, X2)) -> fib1(X1, X2) [1] activate(X) -> X [1] The TRS has the following type information: fib :: 0:s -> 0:s sel :: 0:s -> n__fib1:cons -> 0:s fib1 :: 0:s -> 0:s -> n__fib1:cons s :: 0:s -> 0:s 0 :: 0:s cons :: 0:s -> n__fib1:cons -> n__fib1:cons n__fib1 :: 0:s -> 0:s -> n__fib1:cons add :: 0:s -> 0:s -> 0:s activate :: n__fib1:cons -> n__fib1: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: fib_1 sel_2 (c) The following functions are completely defined: fib1_2 activate_1 add_2 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: fib(N) -> sel(N, fib1(s(0), s(0))) [1] fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] sel(0, cons(X, XS)) -> X [1] sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) [1] fib1(X1, X2) -> n__fib1(X1, X2) [1] activate(n__fib1(X1, X2)) -> fib1(X1, X2) [1] activate(X) -> X [1] The TRS has the following type information: fib :: 0:s -> 0:s sel :: 0:s -> n__fib1:cons -> 0:s fib1 :: 0:s -> 0:s -> n__fib1:cons s :: 0:s -> 0:s 0 :: 0:s cons :: 0:s -> n__fib1:cons -> n__fib1:cons n__fib1 :: 0:s -> 0:s -> n__fib1:cons add :: 0:s -> 0:s -> 0:s activate :: n__fib1:cons -> n__fib1:cons const :: n__fib1: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: fib(N) -> sel(N, cons(s(0), n__fib1(s(0), add(s(0), s(0))))) [2] fib(N) -> sel(N, n__fib1(s(0), s(0))) [2] fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] sel(0, cons(X, XS)) -> X [1] sel(s(N), cons(X, n__fib1(X1', X2'))) -> sel(N, fib1(X1', X2')) [2] sel(s(N), cons(X, XS)) -> sel(N, XS) [2] fib1(X1, X2) -> n__fib1(X1, X2) [1] activate(n__fib1(X1, X2)) -> fib1(X1, X2) [1] activate(X) -> X [1] The TRS has the following type information: fib :: 0:s -> 0:s sel :: 0:s -> n__fib1:cons -> 0:s fib1 :: 0:s -> 0:s -> n__fib1:cons s :: 0:s -> 0:s 0 :: 0:s cons :: 0:s -> n__fib1:cons -> n__fib1:cons n__fib1 :: 0:s -> 0:s -> n__fib1:cons add :: 0:s -> 0:s -> 0:s activate :: n__fib1:cons -> n__fib1:cons const :: n__fib1: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 const => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 add(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0 fib(z) -{ 2 }-> sel(N, 1 + (1 + 0) + (1 + 0)) :|: z = N, N >= 0 fib(z) -{ 2 }-> sel(N, 1 + (1 + 0) + (1 + (1 + 0) + add(1 + 0, 1 + 0))) :|: z = N, N >= 0 fib1(z, z') -{ 1 }-> 1 + X + (1 + Y + add(X, Y)) :|: z' = Y, Y >= 0, X >= 0, z = X fib1(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(N, XS) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0 sel(z, z') -{ 2 }-> sel(N, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, z = 1 + N, X >= 0, z' = 1 + X + (1 + X1' + X2'), N >= 0 ---------------------------------------- (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: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + add(1 + 0, 1 + 0))) :|: z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 1 }-> 1 + z + (1 + z' + add(z, z')) :|: z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { add } { fib1 } { activate } { sel } { fib } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + add(1 + 0, 1 + 0))) :|: z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 1 }-> 1 + z + (1 + z' + add(z, z')) :|: z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 Function symbols to be analyzed: {add}, {fib1}, {activate}, {sel}, {fib} ---------------------------------------- (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: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + add(1 + 0, 1 + 0))) :|: z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 1 }-> 1 + z + (1 + z' + add(z, z')) :|: z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 Function symbols to be analyzed: {add}, {fib1}, {activate}, {sel}, {fib} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: add after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + add(1 + 0, 1 + 0))) :|: z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 1 }-> 1 + z + (1 + z' + add(z, z')) :|: z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 Function symbols to be analyzed: {add}, {fib1}, {activate}, {sel}, {fib} Previous analysis results are: add: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: add 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 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + add(1 + 0, 1 + 0))) :|: z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 1 }-> 1 + z + (1 + z' + add(z, z')) :|: z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 Function symbols to be analyzed: {fib1}, {activate}, {sel}, {fib} Previous analysis results are: add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0 fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 Function symbols to be analyzed: {fib1}, {activate}, {sel}, {fib} Previous analysis results are: add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: fib1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 2*z + 2*z' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0 fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 Function symbols to be analyzed: {fib1}, {activate}, {sel}, {fib} Previous analysis results are: add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] fib1: runtime: ?, size: O(n^1) [2 + 2*z + 2*z'] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: fib1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> z :|: z >= 0 activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0 fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 Function symbols to be analyzed: {activate}, {sel}, {fib} Previous analysis results are: add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z'] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0 fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 4 + X1' }-> sel(z - 1, s1) :|: s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 Function symbols to be analyzed: {activate}, {sel}, {fib} Previous analysis results are: add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z'] ---------------------------------------- (33) 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: 2*z ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0 fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 4 + X1' }-> sel(z - 1, s1) :|: s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 Function symbols to be analyzed: {activate}, {sel}, {fib} Previous analysis results are: add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z'] activate: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: activate after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0 fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 4 + X1' }-> sel(z - 1, s1) :|: s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 Function symbols to be analyzed: {sel}, {fib} Previous analysis results are: add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z'] activate: runtime: O(n^1) [2 + z], size: O(n^1) [2*z] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0 fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 4 + X1' }-> sel(z - 1, s1) :|: s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 Function symbols to be analyzed: {sel}, {fib} Previous analysis results are: add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z'] activate: runtime: O(n^1) [2 + z], size: O(n^1) [2*z] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: sel after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0 fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 4 + X1' }-> sel(z - 1, s1) :|: s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 Function symbols to be analyzed: {sel}, {fib} Previous analysis results are: add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z'] activate: runtime: O(n^1) [2 + z], size: O(n^1) [2*z] sel: runtime: ?, size: EXP ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: sel after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0 fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0 fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 sel(z, z') -{ 4 + X1' }-> sel(z - 1, s1) :|: s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 Function symbols to be analyzed: {fib} Previous analysis results are: add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z'] activate: runtime: O(n^1) [2 + z], size: O(n^1) [2*z] sel: runtime: EXP, size: EXP ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0 fib(z) -{ 4 + inf }-> s3 :|: s3 >= 0, s3 <= inf', s >= 0, s <= 1 + 0 + (1 + 0), z >= 0 fib(z) -{ 2 + inf'' }-> s4 :|: s4 >= 0, s4 <= inf1, z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 4 + X1' + inf2 }-> s5 :|: s5 >= 0, s5 <= inf3, s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 2 + inf4 }-> s6 :|: s6 >= 0, s6 <= inf5, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 Function symbols to be analyzed: {fib} Previous analysis results are: add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z'] activate: runtime: O(n^1) [2 + z], size: O(n^1) [2*z] sel: runtime: EXP, size: EXP ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: fib after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0 fib(z) -{ 4 + inf }-> s3 :|: s3 >= 0, s3 <= inf', s >= 0, s <= 1 + 0 + (1 + 0), z >= 0 fib(z) -{ 2 + inf'' }-> s4 :|: s4 >= 0, s4 <= inf1, z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 4 + X1' + inf2 }-> s5 :|: s5 >= 0, s5 <= inf3, s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 2 + inf4 }-> s6 :|: s6 >= 0, s6 <= inf5, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 Function symbols to be analyzed: {fib} Previous analysis results are: add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z'] activate: runtime: O(n^1) [2 + z], size: O(n^1) [2*z] sel: runtime: EXP, size: EXP fib: runtime: ?, size: INF ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: fib after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2 activate(z) -{ 1 }-> z :|: z >= 0 add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0 fib(z) -{ 4 + inf }-> s3 :|: s3 >= 0, s3 <= inf', s >= 0, s <= 1 + 0 + (1 + 0), z >= 0 fib(z) -{ 2 + inf'' }-> s4 :|: s4 >= 0, s4 <= inf1, z >= 0 fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 4 + X1' + inf2 }-> s5 :|: s5 >= 0, s5 <= inf3, s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0 sel(z, z') -{ 2 + inf4 }-> s6 :|: s6 >= 0, s6 <= inf5, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 Function symbols to be analyzed: {fib} Previous analysis results are: add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z'] activate: runtime: O(n^1) [2 + z], size: O(n^1) [2*z] sel: runtime: EXP, size: EXP fib: runtime: INF, size: INF ---------------------------------------- (49) 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: sel(v0, v1) -> null_sel [0] add(v0, v1) -> null_add [0] And the following fresh constants: null_sel, null_add, const ---------------------------------------- (50) 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: fib(N) -> sel(N, fib1(s(0), s(0))) [1] fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] sel(0, cons(X, XS)) -> X [1] sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) [1] fib1(X1, X2) -> n__fib1(X1, X2) [1] activate(n__fib1(X1, X2)) -> fib1(X1, X2) [1] activate(X) -> X [1] sel(v0, v1) -> null_sel [0] add(v0, v1) -> null_add [0] The TRS has the following type information: fib :: 0:s:null_sel:null_add -> 0:s:null_sel:null_add sel :: 0:s:null_sel:null_add -> n__fib1:cons -> 0:s:null_sel:null_add fib1 :: 0:s:null_sel:null_add -> 0:s:null_sel:null_add -> n__fib1:cons s :: 0:s:null_sel:null_add -> 0:s:null_sel:null_add 0 :: 0:s:null_sel:null_add cons :: 0:s:null_sel:null_add -> n__fib1:cons -> n__fib1:cons n__fib1 :: 0:s:null_sel:null_add -> 0:s:null_sel:null_add -> n__fib1:cons add :: 0:s:null_sel:null_add -> 0:s:null_sel:null_add -> 0:s:null_sel:null_add activate :: n__fib1:cons -> n__fib1:cons null_sel :: 0:s:null_sel:null_add null_add :: 0:s:null_sel:null_add const :: n__fib1:cons Rewrite Strategy: INNERMOST ---------------------------------------- (51) 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 null_sel => 0 null_add => 0 const => 0 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: activate(z) -{ 1 }-> X :|: X >= 0, z = X activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 add(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0 add(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 add(z, z') -{ 1 }-> 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0 fib(z) -{ 1 }-> sel(N, fib1(1 + 0, 1 + 0)) :|: z = N, N >= 0 fib1(z, z') -{ 1 }-> 1 + X + (1 + Y + add(X, Y)) :|: z' = Y, Y >= 0, X >= 0, z = X fib1(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 1 }-> sel(N, activate(XS)) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0 sel(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (53) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: fib(z0) -> sel(z0, fib1(s(0), s(0))) fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__fib1(z0, z1)) -> fib1(z0, z1) activate(z0) -> z0 Tuples: FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0))) FIB1(z0, z1) -> c1(ADD(z0, z1)) FIB1(z0, z1) -> c2 ADD(0, z0) -> c3 ADD(s(z0), z1) -> c4(ADD(z0, z1)) SEL(0, cons(z0, z1)) -> c5 SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) ACTIVATE(z0) -> c8 S tuples: FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0))) FIB1(z0, z1) -> c1(ADD(z0, z1)) FIB1(z0, z1) -> c2 ADD(0, z0) -> c3 ADD(s(z0), z1) -> c4(ADD(z0, z1)) SEL(0, cons(z0, z1)) -> c5 SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) ACTIVATE(z0) -> c8 K tuples:none Defined Rule Symbols: fib_1, fib1_2, add_2, sel_2, activate_1 Defined Pair Symbols: FIB_1, FIB1_2, ADD_2, SEL_2, ACTIVATE_1 Compound Symbols: c_2, c1_1, c2, c3, c4_1, c5, c6_2, c7_1, c8 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: FIB1(z0, z1) -> c2 ACTIVATE(z0) -> c8 SEL(0, cons(z0, z1)) -> c5 ADD(0, z0) -> c3 ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: fib(z0) -> sel(z0, fib1(s(0), s(0))) fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__fib1(z0, z1)) -> fib1(z0, z1) activate(z0) -> z0 Tuples: FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0))) FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) S tuples: FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0))) FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) K tuples:none Defined Rule Symbols: fib_1, fib1_2, add_2, sel_2, activate_1 Defined Pair Symbols: FIB_1, FIB1_2, ADD_2, SEL_2, ACTIVATE_1 Compound Symbols: c_2, c1_1, c4_1, c6_2, c7_1 ---------------------------------------- (57) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: fib(z0) -> sel(z0, fib1(s(0), s(0))) fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__fib1(z0, z1)) -> fib1(z0, z1) activate(z0) -> z0 Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) FIB(z0) -> c2(FIB1(s(0), s(0))) S tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) FIB(z0) -> c2(FIB1(s(0), s(0))) K tuples:none Defined Rule Symbols: fib_1, fib1_2, add_2, sel_2, activate_1 Defined Pair Symbols: FIB1_2, ADD_2, SEL_2, ACTIVATE_1, FIB_1 Compound Symbols: c1_1, c4_1, c6_2, c7_1, c2_1 ---------------------------------------- (59) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: FIB(z0) -> c2(FIB1(s(0), s(0))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: fib(z0) -> sel(z0, fib1(s(0), s(0))) fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__fib1(z0, z1)) -> fib1(z0, z1) activate(z0) -> z0 Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) S tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) K tuples:none Defined Rule Symbols: fib_1, fib1_2, add_2, sel_2, activate_1 Defined Pair Symbols: FIB1_2, ADD_2, SEL_2, ACTIVATE_1, FIB_1 Compound Symbols: c1_1, c4_1, c6_2, c7_1, c2_1 ---------------------------------------- (61) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: fib(z0) -> sel(z0, fib1(s(0), s(0))) fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) activate(n__fib1(z0, z1)) -> fib1(z0, z1) activate(z0) -> z0 Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) S tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) Defined Rule Symbols: fib_1, fib1_2, add_2, sel_2, activate_1 Defined Pair Symbols: FIB1_2, ADD_2, SEL_2, ACTIVATE_1, FIB_1 Compound Symbols: c1_1, c4_1, c6_2, c7_1, c2_1 ---------------------------------------- (63) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: fib(z0) -> sel(z0, fib1(s(0), s(0))) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__fib1(z0, z1)) -> fib1(z0, z1) activate(z0) -> z0 fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) S tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) Defined Rule Symbols: activate_1, fib1_2, add_2 Defined Pair Symbols: FIB1_2, ADD_2, SEL_2, ACTIVATE_1, FIB_1 Compound Symbols: c1_1, c4_1, c6_2, c7_1, c2_1 ---------------------------------------- (65) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) We considered the (Usable) Rules:none And the Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ACTIVATE(x_1)) = [1] POL(ADD(x_1, x_2)) = 0 POL(FIB(x_1)) = [2] + [2]x_1 + [2]x_1^2 POL(FIB1(x_1, x_2)) = 0 POL(SEL(x_1, x_2)) = [2]x_1 + [2]x_1^2 POL(activate(x_1)) = 0 POL(add(x_1, x_2)) = [1] + [2]x_1 + [2]x_2 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2 POL(c1(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c7(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 POL(fib1(x_1, x_2)) = 0 POL(n__fib1(x_1, x_2)) = 0 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__fib1(z0, z1)) -> fib1(z0, z1) activate(z0) -> z0 fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) S tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) Defined Rule Symbols: activate_1, fib1_2, add_2 Defined Pair Symbols: FIB1_2, ADD_2, SEL_2, ACTIVATE_1, FIB_1 Compound Symbols: c1_1, c4_1, c6_2, c7_1, c2_1 ---------------------------------------- (67) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FIB1(z0, z1) -> c1(ADD(z0, z1)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__fib1(z0, z1)) -> fib1(z0, z1) activate(z0) -> z0 fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) S tuples: ADD(s(z0), z1) -> c4(ADD(z0, z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB1(z0, z1) -> c1(ADD(z0, z1)) Defined Rule Symbols: activate_1, fib1_2, add_2 Defined Pair Symbols: FIB1_2, ADD_2, SEL_2, ACTIVATE_1, FIB_1 Compound Symbols: c1_1, c4_1, c6_2, c7_1, c2_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) by SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: activate(n__fib1(z0, z1)) -> fib1(z0, z1) activate(z0) -> z0 fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) S tuples: ADD(s(z0), z1) -> c4(ADD(z0, z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB1(z0, z1) -> c1(ADD(z0, z1)) Defined Rule Symbols: activate_1, fib1_2, add_2 Defined Pair Symbols: FIB1_2, ADD_2, ACTIVATE_1, FIB_1, SEL_2 Compound Symbols: c1_1, c4_1, c7_1, c2_1, c6_2 ---------------------------------------- (71) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: activate(n__fib1(z0, z1)) -> fib1(z0, z1) activate(z0) -> z0 ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) S tuples: ADD(s(z0), z1) -> c4(ADD(z0, z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB1(z0, z1) -> c1(ADD(z0, z1)) Defined Rule Symbols: fib1_2, add_2 Defined Pair Symbols: FIB1_2, ADD_2, ACTIVATE_1, FIB_1, SEL_2 Compound Symbols: c1_1, c4_1, c7_1, c2_1, c6_2 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) by FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), add(s(0), s(0)))))) FIB(x0) -> c2(SEL(x0, n__fib1(s(0), s(0)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), add(s(0), s(0)))))) FIB(x0) -> c2(SEL(x0, n__fib1(s(0), s(0)))) S tuples: ADD(s(z0), z1) -> c4(ADD(z0, z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB1(z0, z1) -> c1(ADD(z0, z1)) Defined Rule Symbols: fib1_2, add_2 Defined Pair Symbols: FIB1_2, ADD_2, ACTIVATE_1, SEL_2, FIB_1 Compound Symbols: c1_1, c4_1, c7_1, c6_2, c2_1 ---------------------------------------- (75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FIB(x0) -> c2(SEL(x0, n__fib1(s(0), s(0)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), add(s(0), s(0)))))) S tuples: ADD(s(z0), z1) -> c4(ADD(z0, z1)) K tuples: ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB1(z0, z1) -> c1(ADD(z0, z1)) Defined Rule Symbols: fib1_2, add_2 Defined Pair Symbols: FIB1_2, ADD_2, ACTIVATE_1, SEL_2, FIB_1 Compound Symbols: c1_1, c4_1, c7_1, c6_2, c2_1 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1))) by SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, n__fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), add(s(0), s(0)))))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, n__fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1))) S tuples: ADD(s(z0), z1) -> c4(ADD(z0, z1)) K tuples: ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB1(z0, z1) -> c1(ADD(z0, z1)) Defined Rule Symbols: fib1_2, add_2 Defined Pair Symbols: FIB1_2, ADD_2, ACTIVATE_1, SEL_2, FIB_1 Compound Symbols: c1_1, c4_1, c7_1, c6_2, c2_1 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), add(s(0), s(0)))))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1))) S tuples: ADD(s(z0), z1) -> c4(ADD(z0, z1)) K tuples: ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB1(z0, z1) -> c1(ADD(z0, z1)) Defined Rule Symbols: fib1_2, add_2 Defined Pair Symbols: FIB1_2, ADD_2, ACTIVATE_1, SEL_2, FIB_1 Compound Symbols: c1_1, c4_1, c7_1, c6_2, c2_1, c6_1 ---------------------------------------- (81) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1))) fib1(z0, z1) -> n__fib1(z0, z1) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), add(s(0), s(0)))))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1))) S tuples: ADD(s(z0), z1) -> c4(ADD(z0, z1)) K tuples: ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB1(z0, z1) -> c1(ADD(z0, z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: FIB1_2, ADD_2, ACTIVATE_1, SEL_2, FIB_1 Compound Symbols: c1_1, c4_1, c7_1, c6_2, c2_1, c6_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), add(s(0), s(0)))))) by FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), s(add(0, s(0))))))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1))) FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), s(add(0, s(0))))))) S tuples: ADD(s(z0), z1) -> c4(ADD(z0, z1)) K tuples: ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB1(z0, z1) -> c1(ADD(z0, z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: FIB1_2, ADD_2, ACTIVATE_1, SEL_2, FIB_1 Compound Symbols: c1_1, c4_1, c7_1, c6_2, c6_1, c2_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), s(add(0, s(0))))))) by FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), s(s(0)))))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1))) FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), s(s(0)))))) S tuples: ADD(s(z0), z1) -> c4(ADD(z0, z1)) K tuples: ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB1(z0, z1) -> c1(ADD(z0, z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: FIB1_2, ADD_2, ACTIVATE_1, SEL_2, FIB_1 Compound Symbols: c1_1, c4_1, c7_1, c6_2, c6_1, c2_1 ---------------------------------------- (87) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), s(s(0)))))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: FIB1(z0, z1) -> c1(ADD(z0, z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1))) S tuples: ADD(s(z0), z1) -> c4(ADD(z0, z1)) K tuples: ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB1(z0, z1) -> c1(ADD(z0, z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: FIB1_2, ADD_2, ACTIVATE_1, SEL_2 Compound Symbols: c1_1, c4_1, c7_1, c6_2, c6_1 ---------------------------------------- (89) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FIB1(z0, z1) -> c1(ADD(z0, z1)) by FIB1(s(y0), z1) -> c1(ADD(s(y0), z1)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: ADD(s(z0), z1) -> c4(ADD(z0, z1)) ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1))) FIB1(s(y0), z1) -> c1(ADD(s(y0), z1)) S tuples: ADD(s(z0), z1) -> c4(ADD(z0, z1)) K tuples: ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB1(s(y0), z1) -> c1(ADD(s(y0), z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: ADD_2, ACTIVATE_1, SEL_2, FIB1_2 Compound Symbols: c4_1, c7_1, c6_2, c6_1, c1_1 ---------------------------------------- (91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ADD(s(z0), z1) -> c4(ADD(z0, z1)) by ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1))) FIB1(s(y0), z1) -> c1(ADD(s(y0), z1)) ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) S tuples: ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) K tuples: ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) FIB1(s(y0), z1) -> c1(ADD(s(y0), z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: ACTIVATE_1, SEL_2, FIB1_2, ADD_2 Compound Symbols: c7_1, c6_2, c6_1, c1_1, c4_1 ---------------------------------------- (93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) by ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1))) FIB1(s(y0), z1) -> c1(ADD(s(y0), z1)) ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1)) S tuples: ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) K tuples: FIB1(s(y0), z1) -> c1(ADD(s(y0), z1)) ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: SEL_2, FIB1_2, ADD_2, ACTIVATE_1 Compound Symbols: c6_2, c6_1, c1_1, c4_1, c7_1 ---------------------------------------- (95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) by SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)), ACTIVATE(cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3))), ACTIVATE(cons(y1, n__fib1(y2, y3)))) SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(SEL(z0, n__fib1(s(y0), y1)), ACTIVATE(n__fib1(s(y0), y1))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1))) FIB1(s(y0), z1) -> c1(ADD(s(y0), z1)) ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1)) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)), ACTIVATE(cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3))), ACTIVATE(cons(y1, n__fib1(y2, y3)))) SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(SEL(z0, n__fib1(s(y0), y1)), ACTIVATE(n__fib1(s(y0), y1))) S tuples: ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) K tuples: FIB1(s(y0), z1) -> c1(ADD(s(y0), z1)) ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: SEL_2, FIB1_2, ADD_2, ACTIVATE_1 Compound Symbols: c6_2, c6_1, c1_1, c4_1, c7_1 ---------------------------------------- (97) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1))) FIB1(s(y0), z1) -> c1(ADD(s(y0), z1)) ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1)) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3)))) SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1))) S tuples: ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) K tuples: FIB1(s(y0), z1) -> c1(ADD(s(y0), z1)) ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: SEL_2, FIB1_2, ADD_2, ACTIVATE_1 Compound Symbols: c6_2, c6_1, c1_1, c4_1, c7_1 ---------------------------------------- (99) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1))) by SEL(s(z0), cons(z1, n__fib1(s(y0), z3))) -> c6(ACTIVATE(n__fib1(s(y0), z3))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) FIB1(s(y0), z1) -> c1(ADD(s(y0), z1)) ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1)) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3)))) SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1))) S tuples: ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) K tuples: FIB1(s(y0), z1) -> c1(ADD(s(y0), z1)) ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: SEL_2, FIB1_2, ADD_2, ACTIVATE_1 Compound Symbols: c6_2, c1_1, c4_1, c7_1, c6_1 ---------------------------------------- (101) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FIB1(s(y0), z1) -> c1(ADD(s(y0), z1)) by FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1)) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1)) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3)))) SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1))) FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1)) S tuples: ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) K tuples: ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1)) FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: SEL_2, ADD_2, ACTIVATE_1, FIB1_2 Compound Symbols: c6_2, c4_1, c7_1, c6_1, c1_1 ---------------------------------------- (103) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) by ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1)) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3)))) SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1))) FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1)) ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) S tuples: ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) K tuples: ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1)) FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: SEL_2, ACTIVATE_1, FIB1_2, ADD_2 Compound Symbols: c6_2, c7_1, c6_1, c1_1, c4_1 ---------------------------------------- (105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1)) by ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1)) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3)))) SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1))) FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1)) ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1)) S tuples: ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) K tuples: FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1)) ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: SEL_2, FIB1_2, ADD_2, ACTIVATE_1 Compound Symbols: c6_2, c6_1, c1_1, c4_1, c7_1 ---------------------------------------- (107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) by SEL(s(s(z0)), cons(z1, cons(z2, n__fib1(y2, y3)))) -> c6(SEL(s(z0), cons(z2, n__fib1(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__fib1(y3, y4))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__fib1(y3, y4))))) SEL(s(s(z0)), cons(z1, cons(z2, n__fib1(s(y2), y3)))) -> c6(SEL(s(z0), cons(z2, n__fib1(s(y2), y3)))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3)))) SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1))) FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1)) ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1)) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__fib1(y3, y4))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__fib1(y3, y4))))) SEL(s(s(z0)), cons(z1, cons(z2, n__fib1(s(y2), y3)))) -> c6(SEL(s(z0), cons(z2, n__fib1(s(y2), y3)))) S tuples: ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) K tuples: FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1)) ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: SEL_2, FIB1_2, ADD_2, ACTIVATE_1 Compound Symbols: c6_2, c6_1, c1_1, c4_1, c7_1 ---------------------------------------- (109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1))) by SEL(s(z0), cons(z1, n__fib1(s(s(y0)), z3))) -> c6(ACTIVATE(n__fib1(s(s(y0)), z3))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3)))) SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1))) FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1)) ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1)) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__fib1(y3, y4))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__fib1(y3, y4))))) SEL(s(s(z0)), cons(z1, cons(z2, n__fib1(s(y2), y3)))) -> c6(SEL(s(z0), cons(z2, n__fib1(s(y2), y3)))) SEL(s(z0), cons(z1, n__fib1(s(s(y0)), z3))) -> c6(ACTIVATE(n__fib1(s(s(y0)), z3))) S tuples: ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) K tuples: FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1)) ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: SEL_2, FIB1_2, ADD_2, ACTIVATE_1 Compound Symbols: c6_2, c6_1, c1_1, c4_1, c7_1 ---------------------------------------- (111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, n__fib1(s(y0), z3))) -> c6(ACTIVATE(n__fib1(s(y0), z3))) by SEL(s(z0), cons(z1, n__fib1(s(s(y0)), z3))) -> c6(ACTIVATE(n__fib1(s(s(y0)), z3))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: add(s(z0), z1) -> s(add(z0, z1)) add(0, z0) -> z0 Tuples: SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1))) SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3)))) FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1)) ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1)) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__fib1(y3, y4))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__fib1(y3, y4))))) SEL(s(s(z0)), cons(z1, cons(z2, n__fib1(s(y2), y3)))) -> c6(SEL(s(z0), cons(z2, n__fib1(s(y2), y3)))) SEL(s(z0), cons(z1, n__fib1(s(s(y0)), z3))) -> c6(ACTIVATE(n__fib1(s(s(y0)), z3))) S tuples: ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) K tuples: FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1)) ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1)) Defined Rule Symbols: add_2 Defined Pair Symbols: SEL_2, FIB1_2, ADD_2, ACTIVATE_1 Compound Symbols: c6_2, c6_1, c1_1, c4_1, c7_1