KILLED proof of input_VfeNxaapBw.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 413 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 61 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 293 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 26 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 5766 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 2036 ms] (36) CpxRNTS (37) CompletionProof [UPPER BOUND(ID), 0 ms] (38) CpxTypedWeightedCompleteTrs (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (48) CdtProblem (49) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 86 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (74) 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, 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, XS) 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, 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, XS) 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, 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, XS) 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, 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, XS) [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, 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, XS) [1] The TRS has the following type information: fib :: 0:s -> 0:s sel :: 0:s -> cons -> 0:s fib1 :: 0:s -> 0:s -> cons s :: 0:s -> 0:s 0 :: 0:s cons :: 0:s -> cons -> cons add :: 0:s -> 0:s -> 0:s 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 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, 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, XS) [1] The TRS has the following type information: fib :: 0:s -> 0:s sel :: 0:s -> cons -> 0:s fib1 :: 0:s -> 0:s -> cons s :: 0:s -> 0:s 0 :: 0:s cons :: 0:s -> cons -> cons add :: 0:s -> 0:s -> 0:s const :: 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), fib1(s(0), add(s(0), s(0))))) [2] fib1(0, Y) -> cons(0, fib1(Y, Y)) [2] fib1(s(X'), Y) -> cons(s(X'), fib1(Y, s(add(X', Y)))) [2] 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, XS) [1] The TRS has the following type information: fib :: 0:s -> 0:s sel :: 0:s -> cons -> 0:s fib1 :: 0:s -> 0:s -> cons s :: 0:s -> 0:s 0 :: 0:s cons :: 0:s -> cons -> cons add :: 0:s -> 0:s -> 0:s const :: 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: 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) + fib1(1 + 0, add(1 + 0, 1 + 0))) :|: z = N, N >= 0 fib1(z, z') -{ 2 }-> 1 + 0 + fib1(Y, Y) :|: z' = Y, Y >= 0, z = 0 fib1(z, z') -{ 2 }-> 1 + (1 + X') + fib1(Y, 1 + add(X', Y)) :|: z' = Y, Y >= 0, X' >= 0, z = 1 + X' sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 1 }-> sel(N, XS) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, 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: 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) + fib1(1 + 0, add(1 + 0, 1 + 0))) :|: z >= 0 fib1(z, z') -{ 2 }-> 1 + 0 + fib1(z', z') :|: z' >= 0, z = 0 fib1(z, z') -{ 2 }-> 1 + (1 + (z - 1)) + fib1(z', 1 + add(z - 1, z')) :|: z' >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 1 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { sel } { add } { fib1 } { fib } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: 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) + fib1(1 + 0, add(1 + 0, 1 + 0))) :|: z >= 0 fib1(z, z') -{ 2 }-> 1 + 0 + fib1(z', z') :|: z' >= 0, z = 0 fib1(z, z') -{ 2 }-> 1 + (1 + (z - 1)) + fib1(z', 1 + add(z - 1, z')) :|: z' >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 1 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 Function symbols to be analyzed: {sel}, {add}, {fib1}, {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: 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) + fib1(1 + 0, add(1 + 0, 1 + 0))) :|: z >= 0 fib1(z, z') -{ 2 }-> 1 + 0 + fib1(z', z') :|: z' >= 0, z = 0 fib1(z, z') -{ 2 }-> 1 + (1 + (z - 1)) + fib1(z', 1 + add(z - 1, z')) :|: z' >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 1 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 Function symbols to be analyzed: {sel}, {add}, {fib1}, {fib} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: sel after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: 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) + fib1(1 + 0, add(1 + 0, 1 + 0))) :|: z >= 0 fib1(z, z') -{ 2 }-> 1 + 0 + fib1(z', z') :|: z' >= 0, z = 0 fib1(z, z') -{ 2 }-> 1 + (1 + (z - 1)) + fib1(z', 1 + add(z - 1, z')) :|: z' >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 1 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 Function symbols to be analyzed: {sel}, {add}, {fib1}, {fib} Previous analysis results are: sel: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: sel 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: 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) + fib1(1 + 0, add(1 + 0, 1 + 0))) :|: z >= 0 fib1(z, z') -{ 2 }-> 1 + 0 + fib1(z', z') :|: z' >= 0, z = 0 fib1(z, z') -{ 2 }-> 1 + (1 + (z - 1)) + fib1(z', 1 + add(z - 1, z')) :|: z' >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 1 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 Function symbols to be analyzed: {add}, {fib1}, {fib} Previous analysis results are: sel: runtime: O(n^1) [1 + z], size: O(n^1) [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: 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) + fib1(1 + 0, add(1 + 0, 1 + 0))) :|: z >= 0 fib1(z, z') -{ 2 }-> 1 + 0 + fib1(z', z') :|: z' >= 0, z = 0 fib1(z, z') -{ 2 }-> 1 + (1 + (z - 1)) + fib1(z', 1 + add(z - 1, z')) :|: z' >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 1 + z }-> s :|: s >= 0, s <= XS, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 Function symbols to be analyzed: {add}, {fib1}, {fib} Previous analysis results are: sel: runtime: O(n^1) [1 + z], size: O(n^1) [z'] ---------------------------------------- (27) 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' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: 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) + fib1(1 + 0, add(1 + 0, 1 + 0))) :|: z >= 0 fib1(z, z') -{ 2 }-> 1 + 0 + fib1(z', z') :|: z' >= 0, z = 0 fib1(z, z') -{ 2 }-> 1 + (1 + (z - 1)) + fib1(z', 1 + add(z - 1, z')) :|: z' >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 1 + z }-> s :|: s >= 0, s <= XS, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 Function symbols to be analyzed: {add}, {fib1}, {fib} Previous analysis results are: sel: runtime: O(n^1) [1 + z], size: O(n^1) [z'] add: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (29) 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 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: 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) + fib1(1 + 0, add(1 + 0, 1 + 0))) :|: z >= 0 fib1(z, z') -{ 2 }-> 1 + 0 + fib1(z', z') :|: z' >= 0, z = 0 fib1(z, z') -{ 2 }-> 1 + (1 + (z - 1)) + fib1(z', 1 + add(z - 1, z')) :|: z' >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 1 + z }-> s :|: s >= 0, s <= XS, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 Function symbols to be analyzed: {fib1}, {fib} Previous analysis results are: sel: runtime: O(n^1) [1 + z], size: O(n^1) [z'] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + 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: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z' >= 0, z - 1 >= 0 fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + fib1(1 + 0, s')) :|: s' >= 0, s' <= 1 + 0 + (1 + 0), z >= 0 fib1(z, z') -{ 2 }-> 1 + 0 + fib1(z', z') :|: z' >= 0, z = 0 fib1(z, z') -{ 2 + z }-> 1 + (1 + (z - 1)) + fib1(z', 1 + s'') :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 1 + z }-> s :|: s >= 0, s <= XS, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 Function symbols to be analyzed: {fib1}, {fib} Previous analysis results are: sel: runtime: O(n^1) [1 + z], size: O(n^1) [z'] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: fib1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z' >= 0, z - 1 >= 0 fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + fib1(1 + 0, s')) :|: s' >= 0, s' <= 1 + 0 + (1 + 0), z >= 0 fib1(z, z') -{ 2 }-> 1 + 0 + fib1(z', z') :|: z' >= 0, z = 0 fib1(z, z') -{ 2 + z }-> 1 + (1 + (z - 1)) + fib1(z', 1 + s'') :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 1 + z }-> s :|: s >= 0, s <= XS, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 Function symbols to be analyzed: {fib1}, {fib} Previous analysis results are: sel: runtime: O(n^1) [1 + z], size: O(n^1) [z'] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] fib1: runtime: ?, size: O(1) [0] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: fib1 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z' >= 0, z - 1 >= 0 fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + fib1(1 + 0, s')) :|: s' >= 0, s' <= 1 + 0 + (1 + 0), z >= 0 fib1(z, z') -{ 2 }-> 1 + 0 + fib1(z', z') :|: z' >= 0, z = 0 fib1(z, z') -{ 2 + z }-> 1 + (1 + (z - 1)) + fib1(z', 1 + s'') :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0 sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 1 + z }-> s :|: s >= 0, s <= XS, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0 Function symbols to be analyzed: {fib1}, {fib} Previous analysis results are: sel: runtime: O(n^1) [1 + z], size: O(n^1) [z'] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] fib1: runtime: INF, size: O(1) [0] ---------------------------------------- (37) 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 ---------------------------------------- (38) 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, 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, XS) [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 -> cons -> 0:s:null_sel:null_add fib1 :: 0:s:null_sel:null_add -> 0:s:null_sel:null_add -> 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 -> cons -> cons add :: 0:s:null_sel:null_add -> 0:s:null_sel:null_add -> 0:s:null_sel:null_add null_sel :: 0:s:null_sel:null_add null_add :: 0:s:null_sel:null_add const :: cons Rewrite Strategy: INNERMOST ---------------------------------------- (39) 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 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: 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 + fib1(Y, add(X, Y)) :|: z' = Y, Y >= 0, X >= 0, z = X sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0 sel(z, z') -{ 1 }-> sel(N, 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. ---------------------------------------- (41) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: fib(z0) -> sel(z0, fib1(s(0), s(0))) fib1(z0, z1) -> cons(z0, fib1(z1, add(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, z2) Tuples: FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0))) FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(0, z0) -> c2 ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(0, cons(z0, z1)) -> c4 SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) S tuples: FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0))) FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(0, z0) -> c2 ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(0, cons(z0, z1)) -> c4 SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) K tuples:none Defined Rule Symbols: fib_1, fib1_2, add_2, sel_2 Defined Pair Symbols: FIB_1, FIB1_2, ADD_2, SEL_2 Compound Symbols: c_2, c1_2, c2, c3_1, c4, c5_1 ---------------------------------------- (43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: ADD(0, z0) -> c2 SEL(0, cons(z0, z1)) -> c4 ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: fib(z0) -> sel(z0, fib1(s(0), s(0))) fib1(z0, z1) -> cons(z0, fib1(z1, add(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, z2) Tuples: FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0))) FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) S tuples: FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0))) FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) K tuples:none Defined Rule Symbols: fib_1, fib1_2, add_2, sel_2 Defined Pair Symbols: FIB_1, FIB1_2, ADD_2, SEL_2 Compound Symbols: c_2, c1_2, c3_1, c5_1 ---------------------------------------- (45) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: fib(z0) -> sel(z0, fib1(s(0), s(0))) fib1(z0, z1) -> cons(z0, fib1(z1, add(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, z2) Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) FIB(z0) -> c2(FIB1(s(0), s(0))) S tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) 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 Defined Pair Symbols: FIB1_2, ADD_2, SEL_2, FIB_1 Compound Symbols: c1_2, c3_1, c5_1, c2_1 ---------------------------------------- (47) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: FIB(z0) -> c2(FIB1(s(0), s(0))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: fib(z0) -> sel(z0, fib1(s(0), s(0))) fib1(z0, z1) -> cons(z0, fib1(z1, add(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, z2) Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) S tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) K tuples:none Defined Rule Symbols: fib_1, fib1_2, add_2, sel_2 Defined Pair Symbols: FIB1_2, ADD_2, SEL_2, FIB_1 Compound Symbols: c1_2, c3_1, c5_1, c2_1 ---------------------------------------- (49) 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)))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: fib(z0) -> sel(z0, fib1(s(0), s(0))) fib1(z0, z1) -> cons(z0, fib1(z1, add(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, z2) Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) S tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) Defined Rule Symbols: fib_1, fib1_2, add_2, sel_2 Defined Pair Symbols: FIB1_2, ADD_2, SEL_2, FIB_1 Compound Symbols: c1_2, c3_1, c5_1, c2_1 ---------------------------------------- (51) 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, z2) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1))) Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) S tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) Defined Rule Symbols: add_2, fib1_2 Defined Pair Symbols: FIB1_2, ADD_2, SEL_2, FIB_1 Compound Symbols: c1_2, c3_1, c5_1, c2_1 ---------------------------------------- (53) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) We considered the (Usable) Rules:none And the Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [3] POL(ADD(x_1, x_2)) = 0 POL(FIB(x_1)) = x_1 POL(FIB1(x_1, x_2)) = 0 POL(SEL(x_1, x_2)) = x_1 POL(add(x_1, x_2)) = [1] + [3]x_1 + [2]x_2 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(cons(x_1, x_2)) = [3] + x_1 POL(fib1(x_1, x_2)) = 0 POL(s(x_1)) = [3] + x_1 ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1))) Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) S tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) Defined Rule Symbols: add_2, fib1_2 Defined Pair Symbols: FIB1_2, ADD_2, SEL_2, FIB_1 Compound Symbols: c1_2, c3_1, c5_1, c2_1 ---------------------------------------- (55) 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), fib1(s(0), add(s(0), s(0)))))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1))) Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) FIB(x0) -> c2(SEL(x0, cons(s(0), fib1(s(0), add(s(0), s(0)))))) S tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) Defined Rule Symbols: add_2, fib1_2 Defined Pair Symbols: FIB1_2, ADD_2, SEL_2, FIB_1 Compound Symbols: c1_2, c3_1, c5_1, c2_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIB(x0) -> c2(SEL(x0, cons(s(0), fib1(s(0), add(s(0), s(0)))))) by FIB(x0) -> c2(SEL(x0, cons(s(0), cons(s(0), fib1(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))) FIB(x0) -> c2(SEL(x0, cons(s(0), fib1(s(0), s(add(0, s(0))))))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1))) Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) FIB(x0) -> c2(SEL(x0, cons(s(0), cons(s(0), fib1(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))) FIB(x0) -> c2(SEL(x0, cons(s(0), fib1(s(0), s(add(0, s(0))))))) S tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) Defined Rule Symbols: add_2, fib1_2 Defined Pair Symbols: FIB1_2, ADD_2, SEL_2, FIB_1 Compound Symbols: c1_2, c3_1, c5_1, c2_1 ---------------------------------------- (59) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(x0) -> c2(SEL(x0, cons(s(0), cons(s(0), fib1(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))) by FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), fib1(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1))) Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) FIB(x0) -> c2(SEL(x0, cons(s(0), fib1(s(0), s(add(0, s(0))))))) FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), fib1(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))))) S tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) Defined Rule Symbols: add_2, fib1_2 Defined Pair Symbols: FIB1_2, ADD_2, SEL_2, FIB_1 Compound Symbols: c1_2, c3_1, c5_1, c2_1 ---------------------------------------- (61) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(x0) -> c2(SEL(x0, cons(s(0), fib1(s(0), s(add(0, s(0))))))) by FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), fib1(s(add(0, s(0))), add(s(0), s(add(0, s(0))))))))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1))) Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), fib1(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))))) FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), fib1(s(add(0, s(0))), add(s(0), s(add(0, s(0))))))))) S tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) Defined Rule Symbols: add_2, fib1_2 Defined Pair Symbols: FIB1_2, ADD_2, SEL_2, FIB_1 Compound Symbols: c1_2, c3_1, c5_1, c2_1 ---------------------------------------- (63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ADD(s(z0), z1) -> c3(ADD(z0, z1)) by ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1))) Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), fib1(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))))) FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), fib1(s(add(0, s(0))), add(s(0), s(add(0, s(0))))))))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) S tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) Defined Rule Symbols: add_2, fib1_2 Defined Pair Symbols: FIB1_2, SEL_2, FIB_1, ADD_2 Compound Symbols: c1_2, c5_1, c2_1, c3_1 ---------------------------------------- (65) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), fib1(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))))) by FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), cons(add(s(0), add(s(0), s(0))), fib1(add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))), add(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))))))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1))) Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), fib1(s(add(0, s(0))), add(s(0), s(add(0, s(0))))))))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), cons(add(s(0), add(s(0), s(0))), fib1(add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))), add(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))))))) S tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) Defined Rule Symbols: add_2, fib1_2 Defined Pair Symbols: FIB1_2, SEL_2, FIB_1, ADD_2 Compound Symbols: c1_2, c5_1, c2_1, c3_1 ---------------------------------------- (67) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) by SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1))) Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), fib1(s(add(0, s(0))), add(s(0), s(add(0, s(0))))))))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), cons(add(s(0), add(s(0), s(0))), fib1(add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))), add(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))))))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2))) S tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2))) Defined Rule Symbols: add_2, fib1_2 Defined Pair Symbols: FIB1_2, FIB_1, ADD_2, SEL_2 Compound Symbols: c1_2, c2_1, c3_1, c5_1 ---------------------------------------- (69) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), fib1(s(add(0, s(0))), add(s(0), s(add(0, s(0))))))))) by FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(s(add(0, s(0))), fib1(add(s(0), s(add(0, s(0)))), add(s(add(0, s(0))), add(s(0), s(add(0, s(0))))))))))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1))) Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), cons(add(s(0), add(s(0), s(0))), fib1(add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))), add(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))))))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2))) FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(s(add(0, s(0))), fib1(add(s(0), s(add(0, s(0)))), add(s(add(0, s(0))), add(s(0), s(add(0, s(0))))))))))) S tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2))) Defined Rule Symbols: add_2, fib1_2 Defined Pair Symbols: FIB1_2, ADD_2, FIB_1, SEL_2 Compound Symbols: c1_2, c3_1, c2_1, c5_1 ---------------------------------------- (71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) by ADD(s(s(s(y0))), z1) -> c3(ADD(s(s(y0)), z1)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1))) Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), cons(add(s(0), add(s(0), s(0))), fib1(add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))), add(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))))))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2))) FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(s(add(0, s(0))), fib1(add(s(0), s(add(0, s(0)))), add(s(add(0, s(0))), add(s(0), s(add(0, s(0))))))))))) ADD(s(s(s(y0))), z1) -> c3(ADD(s(s(y0)), z1)) S tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(s(s(y0))), z1) -> c3(ADD(s(s(y0)), z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2))) Defined Rule Symbols: add_2, fib1_2 Defined Pair Symbols: FIB1_2, FIB_1, SEL_2, ADD_2 Compound Symbols: c1_2, c2_1, c5_1, c3_1 ---------------------------------------- (73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2))) by SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1))) Tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), cons(add(s(0), add(s(0), s(0))), fib1(add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))), add(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))))))) FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(s(add(0, s(0))), fib1(add(s(0), s(add(0, s(0)))), add(s(add(0, s(0))), add(s(0), s(add(0, s(0))))))))))) ADD(s(s(s(y0))), z1) -> c3(ADD(s(s(y0)), z1)) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) S tuples: FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1)) ADD(s(s(s(y0))), z1) -> c3(ADD(s(s(y0)), z1)) K tuples: FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) Defined Rule Symbols: add_2, fib1_2 Defined Pair Symbols: FIB1_2, FIB_1, ADD_2, SEL_2 Compound Symbols: c1_2, c2_1, c3_1, c5_1