KILLED proof of input_bLguWnZxwP.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 4 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 249 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 97 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 311 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 83 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 4977 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 1581 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 [ComplexityIfPolyImplication, 0 ms] (44) CdtProblem (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 22 ms] (56) CdtProblem (57) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (60) CdtProblem (61) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (64) CdtProblem (65) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtInstantiationProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtLeafRemovalProof [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 ---------------------------------------- (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: plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) fib(x) -> fibiter(x, 0, 0, s(0)) fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y) if(false, b, c, x, y) -> x if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y)) 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: plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) lt(0', s(y)) -> true lt(x, 0') -> false lt(s(x), s(y)) -> lt(x, y) fib(x) -> fibiter(x, 0', 0', s(0')) fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y) if(false, b, c, x, y) -> x if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y)) 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: plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) fib(x) -> fibiter(x, 0, 0, s(0)) fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y) if(false, b, c, x, y) -> x if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y)) 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: plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] lt(0, s(y)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] fib(x) -> fibiter(x, 0, 0, s(0)) [1] fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y) [1] if(false, b, c, x, y) -> x [1] if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y)) [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: plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] lt(0, s(y)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] fib(x) -> fibiter(x, 0, 0, s(0)) [1] fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y) [1] if(false, b, c, x, y) -> x [1] if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y)) [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s lt :: 0:s -> 0:s -> true:false true :: true:false false :: true:false fib :: 0:s -> 0:s fibiter :: 0:s -> 0:s -> 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 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 fibiter_4 if_5 (c) The following functions are completely defined: plus_2 lt_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] lt(0, s(y)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] fib(x) -> fibiter(x, 0, 0, s(0)) [1] fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y) [1] if(false, b, c, x, y) -> x [1] if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y)) [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s lt :: 0:s -> 0:s -> true:false true :: true:false false :: true:false fib :: 0:s -> 0:s fibiter :: 0:s -> 0:s -> 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s 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: plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] lt(0, s(y)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] fib(x) -> fibiter(x, 0, 0, s(0)) [1] fibiter(s(y'), 0, x, y) -> if(true, s(y'), 0, x, y) [2] fibiter(0, c, x, y) -> if(false, 0, c, x, y) [2] fibiter(s(y''), s(x'), x, y) -> if(lt(x', y''), s(y''), s(x'), x, y) [2] if(false, b, c, x, y) -> x [1] if(true, b, c, 0, y) -> fibiter(b, s(c), y, y) [2] if(true, b, c, s(x''), y) -> fibiter(b, s(c), y, s(plus(x'', y))) [2] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s lt :: 0:s -> 0:s -> true:false true :: true:false false :: true:false fib :: 0:s -> 0:s fibiter :: 0:s -> 0:s -> 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s 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 true => 1 false => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> fibiter(x, 0, 0, 1 + 0) :|: x >= 0, z = x fibiter(z, z', z'', z1) -{ 2 }-> if(lt(x', y''), 1 + y'', 1 + x', x, y) :|: z = 1 + y'', z1 = y, z' = 1 + x', x' >= 0, x >= 0, y >= 0, z'' = x, y'' >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + y', 0, x, y) :|: z1 = y, x >= 0, y >= 0, z = 1 + y', z'' = x, y' >= 0, z' = 0 fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, c, x, y) :|: z1 = y, c >= 0, x >= 0, y >= 0, z'' = x, z' = c, z = 0 if(z, z', z'', z1, z2) -{ 1 }-> x :|: b >= 0, z2 = y, c >= 0, x >= 0, y >= 0, z' = b, z = 0, z'' = c, z1 = x if(z, z', z'', z1, z2) -{ 2 }-> fibiter(b, 1 + c, y, y) :|: b >= 0, z1 = 0, z2 = y, c >= 0, z = 1, y >= 0, z' = b, z'' = c if(z, z', z'', z1, z2) -{ 2 }-> fibiter(b, 1 + c, y, 1 + plus(x'', y)) :|: b >= 0, z2 = y, z1 = 1 + x'', c >= 0, z = 1, y >= 0, z' = b, x'' >= 0, z'' = c lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x lt(z, z') -{ 1 }-> 1 :|: z' = 1 + y, y >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y ---------------------------------------- (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: fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(lt(z' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { lt } { plus } { if, fibiter } { fib } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(lt(z' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {lt}, {plus}, {if,fibiter}, {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: fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(lt(z' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {lt}, {plus}, {if,fibiter}, {fib} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(lt(z' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {lt}, {plus}, {if,fibiter}, {fib} Previous analysis results are: lt: runtime: ?, size: O(1) [1] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: lt after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(lt(z' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 0 lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {plus}, {if,fibiter}, {fib} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (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: fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0 fibiter(z, z', z'', z1) -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {plus}, {if,fibiter}, {fib} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus 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: fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0 fibiter(z, z', z'', z1) -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {plus}, {if,fibiter}, {fib} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus 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: fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0 fibiter(z, z', z'', z1) -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {if,fibiter}, {fib} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: 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: fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0 fibiter(z, z', z'', z1) -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0 if(z, z', z'', z1, z2) -{ 2 + z1 }-> fibiter(z', 1 + z'', z2, 1 + s1) :|: s1 >= 0, s1 <= z1 - 1 + z2, z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {if,fibiter}, {fib} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: fibiter after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0 fibiter(z, z', z'', z1) -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0 if(z, z', z'', z1, z2) -{ 2 + z1 }-> fibiter(z', 1 + z'', z2, 1 + s1) :|: s1 >= 0, s1 <= z1 - 1 + z2, z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {if,fibiter}, {fib} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] if: runtime: ?, size: INF fibiter: runtime: ?, size: INF ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0 fibiter(z, z', z'', z1) -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0 fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0 fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0 if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0 if(z, z', z'', z1, z2) -{ 2 + z1 }-> fibiter(z', 1 + z'', z2, 1 + s1) :|: s1 >= 0, s1 <= z1 - 1 + z2, z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 0 lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0 lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {if,fibiter}, {fib} Previous analysis results are: lt: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] if: runtime: INF, size: INF fibiter: runtime: ?, size: INF ---------------------------------------- (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: none And the following fresh constants: none ---------------------------------------- (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: plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] lt(0, s(y)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] fib(x) -> fibiter(x, 0, 0, s(0)) [1] fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y) [1] if(false, b, c, x, y) -> x [1] if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y)) [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s lt :: 0:s -> 0:s -> true:false true :: true:false false :: true:false fib :: 0:s -> 0:s fibiter :: 0:s -> 0:s -> 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s 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 true => 1 false => 0 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: fib(z) -{ 1 }-> fibiter(x, 0, 0, 1 + 0) :|: x >= 0, z = x fibiter(z, z', z'', z1) -{ 1 }-> if(lt(c, b), b, c, x, y) :|: b >= 0, z1 = y, c >= 0, x >= 0, y >= 0, z'' = x, z = b, z' = c if(z, z', z'', z1, z2) -{ 1 }-> x :|: b >= 0, z2 = y, c >= 0, x >= 0, y >= 0, z' = b, z = 0, z'' = c, z1 = x if(z, z', z'', z1, z2) -{ 1 }-> fibiter(b, 1 + c, y, plus(x, y)) :|: b >= 0, z2 = y, c >= 0, z = 1, x >= 0, y >= 0, z' = b, z'' = c, z1 = x lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x lt(z, z') -{ 1 }-> 1 :|: z' = 1 + y, y >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 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: plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) fib(z0) -> fibiter(z0, 0, 0, s(0)) fibiter(z0, z1, z2, z3) -> if(lt(z1, z0), z0, z1, z2, z3) if(false, z0, z1, z2, z3) -> z2 if(true, z0, z1, z2, z3) -> fibiter(z0, s(z1), z3, plus(z2, z3)) Tuples: PLUS(0, z0) -> c PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(0, s(z0)) -> c2 LT(z0, 0) -> c3 LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIB(z0) -> c5(FIBITER(z0, 0, 0, s(0))) FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0)) IF(false, z0, z1, z2, z3) -> c7 IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) S tuples: PLUS(0, z0) -> c PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(0, s(z0)) -> c2 LT(z0, 0) -> c3 LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIB(z0) -> c5(FIBITER(z0, 0, 0, s(0))) FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0)) IF(false, z0, z1, z2, z3) -> c7 IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) K tuples:none Defined Rule Symbols: plus_2, lt_2, fib_1, fibiter_4, if_5 Defined Pair Symbols: PLUS_2, LT_2, FIB_1, FIBITER_4, IF_5 Compound Symbols: c, c1_1, c2, c3, c4_1, c5_1, c6_2, c7, c8_2 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: FIB(z0) -> c5(FIBITER(z0, 0, 0, s(0))) Removed 4 trailing nodes: IF(false, z0, z1, z2, z3) -> c7 LT(0, s(z0)) -> c2 LT(z0, 0) -> c3 PLUS(0, z0) -> c ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) fib(z0) -> fibiter(z0, 0, 0, s(0)) fibiter(z0, z1, z2, z3) -> if(lt(z1, z0), z0, z1, z2, z3) if(false, z0, z1, z2, z3) -> z2 if(true, z0, z1, z2, z3) -> fibiter(z0, s(z1), z3, plus(z2, z3)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0)) IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0)) IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) K tuples:none Defined Rule Symbols: plus_2, lt_2, fib_1, fibiter_4, if_5 Defined Pair Symbols: PLUS_2, LT_2, FIBITER_4, IF_5 Compound Symbols: c1_1, c4_1, c6_2, c8_2 ---------------------------------------- (45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: fib(z0) -> fibiter(z0, 0, 0, s(0)) fibiter(z0, z1, z2, z3) -> if(lt(z1, z0), z0, z1, z2, z3) if(false, z0, z1, z2, z3) -> z2 if(true, z0, z1, z2, z3) -> fibiter(z0, s(z1), z3, plus(z2, z3)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0)) IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0)) IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) K tuples:none Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, LT_2, FIBITER_4, IF_5 Compound Symbols: c1_1, c4_1, c6_2, c8_2 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0)) by FIBITER(s(z0), 0, x2, x3) -> c6(IF(true, s(z0), 0, x2, x3), LT(0, s(z0))) FIBITER(0, z0, x2, x3) -> c6(IF(false, 0, z0, x2, x3), LT(z0, 0)) FIBITER(s(z1), s(z0), x2, x3) -> c6(IF(lt(z0, z1), s(z1), s(z0), x2, x3), LT(s(z0), s(z1))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) FIBITER(s(z0), 0, x2, x3) -> c6(IF(true, s(z0), 0, x2, x3), LT(0, s(z0))) FIBITER(0, z0, x2, x3) -> c6(IF(false, 0, z0, x2, x3), LT(z0, 0)) FIBITER(s(z1), s(z0), x2, x3) -> c6(IF(lt(z0, z1), s(z1), s(z0), x2, x3), LT(s(z0), s(z1))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) FIBITER(s(z0), 0, x2, x3) -> c6(IF(true, s(z0), 0, x2, x3), LT(0, s(z0))) FIBITER(0, z0, x2, x3) -> c6(IF(false, 0, z0, x2, x3), LT(z0, 0)) FIBITER(s(z1), s(z0), x2, x3) -> c6(IF(lt(z0, z1), s(z1), s(z0), x2, x3), LT(s(z0), s(z1))) K tuples:none Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, LT_2, IF_5, FIBITER_4 Compound Symbols: c1_1, c4_1, c8_2, c6_2 ---------------------------------------- (49) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: FIBITER(s(z0), 0, x2, x3) -> c6(IF(true, s(z0), 0, x2, x3), LT(0, s(z0))) Removed 1 trailing nodes: FIBITER(0, z0, x2, x3) -> c6(IF(false, 0, z0, x2, x3), LT(z0, 0)) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) FIBITER(s(z1), s(z0), x2, x3) -> c6(IF(lt(z0, z1), s(z1), s(z0), x2, x3), LT(s(z0), s(z1))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) FIBITER(s(z1), s(z0), x2, x3) -> c6(IF(lt(z0, z1), s(z1), s(z0), x2, x3), LT(s(z0), s(z1))) K tuples:none Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, LT_2, IF_5, FIBITER_4 Compound Symbols: c1_1, c4_1, c8_2, c6_2 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FIBITER(s(z1), s(z0), x2, x3) -> c6(IF(lt(z0, z1), s(z1), s(z0), x2, x3), LT(s(z0), s(z1))) by FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0)))) FIBITER(s(0), s(z0), x2, x3) -> c6(IF(false, s(0), s(z0), x2, x3), LT(s(z0), s(0))) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0)))) FIBITER(s(0), s(z0), x2, x3) -> c6(IF(false, s(0), s(z0), x2, x3), LT(s(z0), s(0))) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0)))) FIBITER(s(0), s(z0), x2, x3) -> c6(IF(false, s(0), s(z0), x2, x3), LT(s(z0), s(0))) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) K tuples:none Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, LT_2, IF_5, FIBITER_4 Compound Symbols: c1_1, c4_1, c8_2, c6_2, c6_1 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0)))) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0)))) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0))) K tuples:none Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, LT_2, IF_5, FIBITER_4 Compound Symbols: c1_1, c4_1, c8_2, c6_2, c6_1 ---------------------------------------- (55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0))) We considered the (Usable) Rules:none And the Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0)))) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FIBITER(x_1, x_2, x_3, x_4)) = [1] POL(IF(x_1, x_2, x_3, x_4, x_5)) = [1] POL(LT(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1, x_2)) = x_1 + x_2 POL(false) = [3] POL(lt(x_1, x_2)) = [3] + [3]x_1 POL(plus(x_1, x_2)) = [2] + [3]x_2 POL(s(x_1)) = 0 POL(true) = [3] ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0)))) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0)))) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) K tuples: FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, LT_2, IF_5, FIBITER_4 Compound Symbols: c1_1, c4_1, c8_2, c6_2, c6_1 ---------------------------------------- (57) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) by IF(true, s(s(x0)), s(0), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)), PLUS(x1, x2)) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0)))) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0))) IF(true, s(s(x0)), s(0), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)), PLUS(x1, x2)) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0)))) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(0), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)), PLUS(x1, x2)) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) K tuples: FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, LT_2, FIBITER_4, IF_5 Compound Symbols: c1_1, c4_1, c6_2, c6_1, c8_2 ---------------------------------------- (59) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0)))) FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) IF(true, s(s(x0)), s(0), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)), PLUS(x1, x2)) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(0), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)), PLUS(x1, x2)) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) K tuples: FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, LT_2, FIBITER_4, IF_5 Compound Symbols: c1_1, c4_1, c6_2, c6_1, c8_2 ---------------------------------------- (61) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) IF(true, s(s(x0)), s(0), x1, x2) -> c(PLUS(x1, x2)) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) IF(true, s(s(x0)), s(0), x1, x2) -> c(PLUS(x1, x2)) K tuples: FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, LT_2, FIBITER_4, IF_5 Compound Symbols: c1_1, c4_1, c6_2, c6_1, c8_2, c_1 ---------------------------------------- (63) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: IF(true, s(s(x0)), s(0), x1, x2) -> c(PLUS(x1, x2)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) K tuples: FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, LT_2, FIBITER_4, IF_5 Compound Symbols: c1_1, c4_1, c6_2, c6_1, c8_2, c_1 ---------------------------------------- (65) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) S tuples: PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) K tuples: FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: PLUS_2, LT_2, FIBITER_4, IF_5 Compound Symbols: c1_1, c4_1, c6_2, c6_1, c8_2, c_1 ---------------------------------------- (67) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) by PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) S tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) K tuples: FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, FIBITER_4, IF_5, PLUS_2 Compound Symbols: c4_1, c6_2, c6_1, c8_2, c_1, c1_1 ---------------------------------------- (69) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) by FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0), LT(s(s(0)), s(s(x0)))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0), LT(s(s(0)), s(s(x0)))) S tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0), LT(s(s(0)), s(s(x0)))) K tuples: FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, FIBITER_4, IF_5, PLUS_2 Compound Symbols: c4_1, c6_1, c8_2, c_1, c1_1, c6_2 ---------------------------------------- (71) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) S tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) K tuples: FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, FIBITER_4, IF_5, PLUS_2 Compound Symbols: c4_1, c6_1, c8_2, c_1, c1_1, c6_2, c2_1 ---------------------------------------- (73) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) S tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) K tuples: FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, FIBITER_4, IF_5, PLUS_2 Compound Symbols: c4_1, c6_1, c8_2, c_1, c1_1, c6_2, c2_1 ---------------------------------------- (75) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) by FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) S tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, IF_5, PLUS_2, FIBITER_4 Compound Symbols: c4_1, c8_2, c_1, c1_1, c6_2, c2_1, c6_1 ---------------------------------------- (77) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) by IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)), PLUS(x1, x2)) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)), PLUS(x1, x2)) S tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)), PLUS(x1, x2)) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, IF_5, PLUS_2, FIBITER_4 Compound Symbols: c4_1, c_1, c1_1, c6_2, c2_1, c6_1, c8_2 ---------------------------------------- (79) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) S tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, IF_5, PLUS_2, FIBITER_4 Compound Symbols: c4_1, c_1, c1_1, c6_2, c2_1, c6_1, c8_2, c3_1 ---------------------------------------- (81) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) S tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, IF_5, PLUS_2, FIBITER_4 Compound Symbols: c4_1, c_1, c1_1, c6_2, c2_1, c6_1, c8_2, c3_1 ---------------------------------------- (83) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) by FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0), LT(s(s(s(0))), s(s(x0)))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0), LT(s(s(s(0))), s(s(x0)))) S tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0), LT(s(s(s(0))), s(s(x0)))) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, IF_5, PLUS_2, FIBITER_4 Compound Symbols: c4_1, c_1, c1_1, c2_1, c6_1, c8_2, c3_1, c6_2 ---------------------------------------- (85) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) S tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, IF_5, PLUS_2, FIBITER_4 Compound Symbols: c4_1, c_1, c1_1, c2_1, c6_1, c8_2, c3_1, c6_2, c5_1 ---------------------------------------- (87) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) S tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, IF_5, PLUS_2, FIBITER_4 Compound Symbols: c4_1, c_1, c1_1, c2_1, c6_1, c8_2, c3_1, c6_2, c5_1 ---------------------------------------- (89) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) by FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) S tuples: LT(s(z0), s(z1)) -> c4(LT(z0, z1)) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: LT_2, IF_5, PLUS_2, FIBITER_4 Compound Symbols: c4_1, c_1, c1_1, c2_1, c6_1, c8_2, c3_1, c6_2, c5_1 ---------------------------------------- (91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LT(s(z0), s(z1)) -> c4(LT(z0, z1)) by LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1))) S tuples: PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1))) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: IF_5, PLUS_2, FIBITER_4, LT_2 Compound Symbols: c_1, c1_1, c2_1, c6_1, c8_2, c3_1, c6_2, c5_1, c4_1 ---------------------------------------- (93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1)) by PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1))) PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1)) S tuples: IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1))) PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1)) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: IF_5, FIBITER_4, LT_2, PLUS_2 Compound Symbols: c_1, c2_1, c6_1, c8_2, c3_1, c6_2, c5_1, c4_1, c1_1 ---------------------------------------- (95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) by IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2)) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1))) PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1)) IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2)) S tuples: IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1))) PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1)) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2)) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: IF_5, FIBITER_4, LT_2, PLUS_2 Compound Symbols: c_1, c2_1, c6_1, c8_2, c3_1, c6_2, c5_1, c4_1, c1_1 ---------------------------------------- (97) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1))) by LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1)))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1)) IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2)) LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1)))) S tuples: IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1)) LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1)))) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2)) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: IF_5, FIBITER_4, PLUS_2, LT_2 Compound Symbols: c_1, c2_1, c6_1, c8_2, c3_1, c6_2, c5_1, c1_1, c4_1 ---------------------------------------- (99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0)))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0)))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1)) IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2)) LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1)))) S tuples: IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1)) LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1)))) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2)) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: IF_5, FIBITER_4, PLUS_2, LT_2 Compound Symbols: c_1, c2_1, c8_2, c3_1, c6_2, c5_1, c6_1, c1_1, c4_1 ---------------------------------------- (101) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) by FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c5(LT(s(s(s(0))), s(s(s(y1))))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1)) IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2)) LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1)))) FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c5(LT(s(s(s(0))), s(s(s(y1))))) S tuples: IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1)) LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1)))) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2)) FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c5(LT(s(s(s(0))), s(s(s(y1))))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: IF_5, FIBITER_4, PLUS_2, LT_2 Compound Symbols: c_1, c2_1, c8_2, c3_1, c6_2, c5_1, c6_1, c1_1, c4_1 ---------------------------------------- (103) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) by FIBITER(s(s(s(y1))), s(s(s(s(z1)))), z2, z3) -> c6(LT(s(s(s(s(z1)))), s(s(s(y1))))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1)) IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2)) LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1)))) FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c5(LT(s(s(s(0))), s(s(s(y1))))) FIBITER(s(s(s(y1))), s(s(s(s(z1)))), z2, z3) -> c6(LT(s(s(s(s(z1)))), s(s(s(y1))))) S tuples: IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1)) LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1)))) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2)) FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c5(LT(s(s(s(0))), s(s(s(y1))))) FIBITER(s(s(s(y1))), s(s(s(s(z1)))), z2, z3) -> c6(LT(s(s(s(s(z1)))), s(s(s(y1))))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: IF_5, FIBITER_4, PLUS_2, LT_2 Compound Symbols: c_1, c2_1, c8_2, c3_1, c6_2, c5_1, c6_1, c1_1, c4_1 ---------------------------------------- (105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) by FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c6(LT(s(s(s(0))), s(s(s(y1))))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) Tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1)) IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2)) LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1)))) FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c5(LT(s(s(s(0))), s(s(s(y1))))) FIBITER(s(s(s(y1))), s(s(s(s(z1)))), z2, z3) -> c6(LT(s(s(s(s(z1)))), s(s(s(y1))))) FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c6(LT(s(s(s(0))), s(s(s(y1))))) S tuples: IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3)) FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0)))) PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1)) LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1)))) K tuples: IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0)) IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2))) FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0)) IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2)) FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c5(LT(s(s(s(0))), s(s(s(y1))))) FIBITER(s(s(s(y1))), s(s(s(s(z1)))), z2, z3) -> c6(LT(s(s(s(s(z1)))), s(s(s(y1))))) FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c6(LT(s(s(s(0))), s(s(s(y1))))) Defined Rule Symbols: lt_2, plus_2 Defined Pair Symbols: IF_5, FIBITER_4, PLUS_2, LT_2 Compound Symbols: c_1, c2_1, c8_2, c3_1, c6_2, c5_1, c1_1, c4_1, c6_1