KILLED proof of input_zwjJBTCuSC.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 102 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 90 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 329 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 140 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 33 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 33 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 4666 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 2116 ms] (44) CpxRNTS (45) CompletionProof [UPPER BOUND(ID), 0 ms] (46) CpxTypedWeightedCompleteTrs (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (52) CdtProblem (53) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 29 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (120) 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: 1024 -> 1024_1(0) 1024_1(x) -> if(lt(x, 10), x) if(true, x) -> double(1024_1(s(x))) if(false, x) -> s(0) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) double(0) -> 0 double(s(x)) -> s(s(double(x))) 10 -> double(s(double(s(s(0))))) 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: 1024' -> 1024_1(0') 1024_1(x) -> if(lt(x, 10'), x) if(true, x) -> double(1024_1(s(x))) if(false, x) -> s(0') lt(0', s(y)) -> true lt(x, 0') -> false lt(s(x), s(y)) -> lt(x, y) double(0') -> 0' double(s(x)) -> s(s(double(x))) 10' -> double(s(double(s(s(0'))))) 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: 1024 -> 1024_1(0) 1024_1(x) -> if(lt(x, 10), x) if(true, x) -> double(1024_1(s(x))) if(false, x) -> s(0) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) double(0) -> 0 double(s(x)) -> s(s(double(x))) 10 -> double(s(double(s(s(0))))) 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: 1024 -> 1024_1(0) [1] 1024_1(x) -> if(lt(x, 10), x) [1] if(true, x) -> double(1024_1(s(x))) [1] if(false, x) -> s(0) [1] lt(0, s(y)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] 10 -> double(s(double(s(s(0))))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: 1024 => 1024' 10 => 10' ---------------------------------------- (8) 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: 1024' -> 1024_1(0) [1] 1024_1(x) -> if(lt(x, 10'), x) [1] if(true, x) -> double(1024_1(s(x))) [1] if(false, x) -> s(0) [1] lt(0, s(y)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] 10' -> double(s(double(s(s(0))))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 1024' -> 1024_1(0) [1] 1024_1(x) -> if(lt(x, 10'), x) [1] if(true, x) -> double(1024_1(s(x))) [1] if(false, x) -> s(0) [1] lt(0, s(y)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] 10' -> double(s(double(s(s(0))))) [1] The TRS has the following type information: 1024' :: 0:s 1024_1 :: 0:s -> 0:s 0 :: 0:s if :: true:false -> 0:s -> 0:s lt :: 0:s -> 0:s -> true:false 10' :: 0:s true :: true:false double :: 0:s -> 0:s s :: 0:s -> 0:s false :: true:false Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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: 1024' (c) The following functions are completely defined: lt_2 10' double_1 1024_1_1 if_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (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: 1024' -> 1024_1(0) [1] 1024_1(x) -> if(lt(x, 10'), x) [1] if(true, x) -> double(1024_1(s(x))) [1] if(false, x) -> s(0) [1] lt(0, s(y)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] 10' -> double(s(double(s(s(0))))) [1] The TRS has the following type information: 1024' :: 0:s 1024_1 :: 0:s -> 0:s 0 :: 0:s if :: true:false -> 0:s -> 0:s lt :: 0:s -> 0:s -> true:false 10' :: 0:s true :: true:false double :: 0:s -> 0:s s :: 0:s -> 0:s false :: true:false Rewrite Strategy: INNERMOST ---------------------------------------- (13) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (14) 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: 1024' -> 1024_1(0) [1] 1024_1(x) -> if(lt(x, double(s(double(s(s(0)))))), x) [2] if(true, x) -> double(if(lt(s(x), 10'), s(x))) [2] if(false, x) -> s(0) [1] lt(0, s(y)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] 10' -> double(s(s(s(double(s(0)))))) [2] The TRS has the following type information: 1024' :: 0:s 1024_1 :: 0:s -> 0:s 0 :: 0:s if :: true:false -> 0:s -> 0:s lt :: 0:s -> 0:s -> true:false 10' :: 0:s true :: true:false double :: 0:s -> 0:s s :: 0:s -> 0:s false :: true:false Rewrite Strategy: INNERMOST ---------------------------------------- (15) 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 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 2 }-> double(1 + (1 + (1 + double(1 + 0)))) :|: 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 2 }-> if(lt(x, double(1 + double(1 + (1 + 0)))), x) :|: x >= 0, z = x double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(x)) :|: x >= 0, z = 1 + x if(z, z') -{ 2 }-> double(if(lt(1 + x, 10'), 1 + x)) :|: z' = x, z = 1, x >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' = x, x >= 0, z = 0 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 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 2 }-> double(1 + (1 + (1 + double(1 + 0)))) :|: 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 2 }-> if(lt(z, double(1 + double(1 + (1 + 0)))), z) :|: z >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 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 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { double } { lt } { 10' } { if } { 1024_1 } { 1024' } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 2 }-> double(1 + (1 + (1 + double(1 + 0)))) :|: 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 2 }-> if(lt(z, double(1 + double(1 + (1 + 0)))), z) :|: z >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 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 Function symbols to be analyzed: {double}, {lt}, {10'}, {if}, {1024_1}, {1024'} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 2 }-> double(1 + (1 + (1 + double(1 + 0)))) :|: 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 2 }-> if(lt(z, double(1 + double(1 + (1 + 0)))), z) :|: z >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 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 Function symbols to be analyzed: {double}, {lt}, {10'}, {if}, {1024_1}, {1024'} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: double 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: 10' -{ 2 }-> double(1 + (1 + (1 + double(1 + 0)))) :|: 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 2 }-> if(lt(z, double(1 + double(1 + (1 + 0)))), z) :|: z >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 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 Function symbols to be analyzed: {double}, {lt}, {10'}, {if}, {1024_1}, {1024'} Previous analysis results are: double: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: double after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 2 }-> double(1 + (1 + (1 + double(1 + 0)))) :|: 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 2 }-> if(lt(z, double(1 + double(1 + (1 + 0)))), z) :|: z >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 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 Function symbols to be analyzed: {lt}, {10'}, {if}, {1024_1}, {1024'} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1))) 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 7 + s }-> if(lt(z, s'), z) :|: s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0 if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 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 Function symbols to be analyzed: {lt}, {10'}, {if}, {1024_1}, {1024'} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (29) 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 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1))) 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 7 + s }-> if(lt(z, s'), z) :|: s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0 if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 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 Function symbols to be analyzed: {lt}, {10'}, {if}, {1024_1}, {1024'} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] lt: runtime: ?, size: O(1) [1] ---------------------------------------- (31) 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' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1))) 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 7 + s }-> if(lt(z, s'), z) :|: s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0 if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 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 Function symbols to be analyzed: {10'}, {if}, {1024_1}, {1024'} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] lt: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1))) 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 9 + s + s' }-> if(s3, z) :|: s3 >= 0, s3 <= 1, s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0 if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0 lt(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= 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 Function symbols to be analyzed: {10'}, {if}, {1024_1}, {1024'} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] lt: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: 10' after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 10 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1))) 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 9 + s + s' }-> if(s3, z) :|: s3 >= 0, s3 <= 1, s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0 if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0 lt(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= 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 Function symbols to be analyzed: {10'}, {if}, {1024_1}, {1024'} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] lt: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: ?, size: O(1) [10] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: 10' after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 10 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1))) 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 9 + s + s' }-> if(s3, z) :|: s3 >= 0, s3 <= 1, s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0 if(z, z') -{ 2 }-> double(if(lt(1 + z', 10'), 1 + z')) :|: z = 1, z' >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0 lt(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= 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 Function symbols to be analyzed: {if}, {1024_1}, {1024'} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] lt: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: O(1) [10], size: O(1) [10] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1))) 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 9 + s + s' }-> if(s3, z) :|: s3 >= 0, s3 <= 1, s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0 if(z, z') -{ 14 + s5 }-> double(if(s6, 1 + z')) :|: s5 >= 0, s5 <= 10, s6 >= 0, s6 <= 1, z = 1, z' >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0 lt(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= 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 Function symbols to be analyzed: {if}, {1024_1}, {1024'} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] lt: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: O(1) [10], size: O(1) [10] ---------------------------------------- (41) 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: ? ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1))) 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 9 + s + s' }-> if(s3, z) :|: s3 >= 0, s3 <= 1, s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0 if(z, z') -{ 14 + s5 }-> double(if(s6, 1 + z')) :|: s5 >= 0, s5 <= 10, s6 >= 0, s6 <= 1, z = 1, z' >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0 lt(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= 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 Function symbols to be analyzed: {if}, {1024_1}, {1024'} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] lt: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: O(1) [10], size: O(1) [10] if: runtime: ?, size: INF ---------------------------------------- (43) 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: ? ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 8 + s1 }-> s2 :|: s1 >= 0, s1 <= 2 * (1 + 0), s2 >= 0, s2 <= 2 * (1 + (1 + (1 + s1))) 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 9 + s + s' }-> if(s3, z) :|: s3 >= 0, s3 <= 1, s >= 0, s <= 2 * (1 + (1 + 0)), s' >= 0, s' <= 2 * (1 + s), z >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0 if(z, z') -{ 14 + s5 }-> double(if(s6, 1 + z')) :|: s5 >= 0, s5 <= 10, s6 >= 0, s6 <= 1, z = 1, z' >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' >= 0, z = 0 lt(z, z') -{ 2 + z' }-> s4 :|: s4 >= 0, s4 <= 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 Function symbols to be analyzed: {if}, {1024_1}, {1024'} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] lt: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: O(1) [10], size: O(1) [10] if: runtime: INF, size: INF ---------------------------------------- (45) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (46) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 1024' -> 1024_1(0) [1] 1024_1(x) -> if(lt(x, 10'), x) [1] if(true, x) -> double(1024_1(s(x))) [1] if(false, x) -> s(0) [1] lt(0, s(y)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] 10' -> double(s(double(s(s(0))))) [1] The TRS has the following type information: 1024' :: 0:s 1024_1 :: 0:s -> 0:s 0 :: 0:s if :: true:false -> 0:s -> 0:s lt :: 0:s -> 0:s -> true:false 10' :: 0:s true :: true:false double :: 0:s -> 0:s s :: 0:s -> 0:s false :: true:false Rewrite Strategy: INNERMOST ---------------------------------------- (47) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 1 }-> double(1 + double(1 + (1 + 0))) :|: 1024' -{ 1 }-> 1024_1(0) :|: 1024_1(z) -{ 1 }-> if(lt(x, 10'), x) :|: x >= 0, z = x double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(x)) :|: x >= 0, z = 1 + x if(z, z') -{ 1 }-> double(1024_1(1 + x)) :|: z' = x, z = 1, x >= 0 if(z, z') -{ 1 }-> 1 + 0 :|: z' = x, x >= 0, z = 0 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 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (49) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: 1024 -> 1024_1(0) 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) 10 -> double(s(double(s(s(0))))) Tuples: 1024' -> c(1024_1'(0)) 1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10') IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0))) IF(false, z0) -> c3 LT(0, s(z0)) -> c4 LT(z0, 0) -> c5 LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(0) -> c7 DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c9(DOUBLE(s(double(s(s(0))))), DOUBLE(s(s(0)))) S tuples: 1024' -> c(1024_1'(0)) 1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10') IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0))) IF(false, z0) -> c3 LT(0, s(z0)) -> c4 LT(z0, 0) -> c5 LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(0) -> c7 DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c9(DOUBLE(s(double(s(s(0))))), DOUBLE(s(s(0)))) K tuples:none Defined Rule Symbols: 1024, 1024_1_1, if_2, lt_2, double_1, 10 Defined Pair Symbols: 1024', 1024_1'_1, IF_2, LT_2, DOUBLE_1, 10' Compound Symbols: c_1, c1_3, c2_2, c3, c4, c5, c6_1, c7, c8_1, c9_2 ---------------------------------------- (51) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: 1024' -> c(1024_1'(0)) Removed 4 trailing nodes: DOUBLE(0) -> c7 IF(false, z0) -> c3 LT(0, s(z0)) -> c4 LT(z0, 0) -> c5 ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: 1024 -> 1024_1(0) 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) 10 -> double(s(double(s(s(0))))) Tuples: 1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10') IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c9(DOUBLE(s(double(s(s(0))))), DOUBLE(s(s(0)))) S tuples: 1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10') IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c9(DOUBLE(s(double(s(s(0))))), DOUBLE(s(s(0)))) K tuples:none Defined Rule Symbols: 1024, 1024_1_1, if_2, lt_2, double_1, 10 Defined Pair Symbols: 1024_1'_1, IF_2, LT_2, DOUBLE_1, 10' Compound Symbols: c1_3, c2_2, c6_1, c8_1, c9_2 ---------------------------------------- (53) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: 1024 -> 1024_1(0) 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) 10 -> double(s(double(s(s(0))))) Tuples: 1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10') IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(double(s(s(0)))))) 10' -> c(DOUBLE(s(s(0)))) S tuples: 1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10') IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(double(s(s(0)))))) 10' -> c(DOUBLE(s(s(0)))) K tuples:none Defined Rule Symbols: 1024, 1024_1_1, if_2, lt_2, double_1, 10 Defined Pair Symbols: 1024_1'_1, IF_2, LT_2, DOUBLE_1, 10' Compound Symbols: c1_3, c2_2, c6_1, c8_1, c_1 ---------------------------------------- (55) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: 1024 -> 1024_1(0) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: 1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10') IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(double(s(s(0)))))) 10' -> c(DOUBLE(s(s(0)))) S tuples: 1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10') IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(double(s(s(0)))))) 10' -> c(DOUBLE(s(s(0)))) K tuples:none Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: 1024_1'_1, IF_2, LT_2, DOUBLE_1, 10' Compound Symbols: c1_3, c2_2, c6_1, c8_1, c_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace 1024_1'(z0) -> c1(IF(lt(z0, 10), z0), LT(z0, 10), 10') by 1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10') ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(double(s(s(0)))))) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10') S tuples: IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0))) LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(double(s(s(0)))))) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10') K tuples:none Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: IF_2, LT_2, DOUBLE_1, 10', 1024_1'_1 Compound Symbols: c2_2, c6_1, c8_1, c_1, c1_3 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(true, z0) -> c2(DOUBLE(1024_1(s(z0))), 1024_1'(s(z0))) by IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(double(s(s(0)))))) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10') IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(double(s(s(0)))))) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10') IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0))) K tuples:none Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c1_3, c2_2 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace 10' -> c(DOUBLE(s(double(s(s(0)))))) by 10' -> c(DOUBLE(s(s(s(double(s(0))))))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10') IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(double(s(0))))))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10') IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(double(s(0))))))) K tuples:none Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c1_3, c2_2 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace 1024_1'(x0) -> c1(IF(lt(x0, double(s(double(s(s(0)))))), x0), LT(x0, 10), 10') by 1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c1(LT(x0, 10), 10') ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(double(s(0))))))) 1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c1(LT(x0, 10), 10') S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(double(s(0))))))) 1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c1(LT(x0, 10), 10') K tuples:none Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', IF_2, 1024_1'_1 Compound Symbols: c6_1, c8_1, c_1, c2_2, c1_3, c1_2 ---------------------------------------- (65) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(double(s(0))))))) 1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(double(s(0))))))) 1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') K tuples:none Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', IF_2, 1024_1'_1 Compound Symbols: c6_1, c8_1, c_1, c2_2, c1_3, c3_1 ---------------------------------------- (67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') We considered the (Usable) Rules:none And the Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(double(s(0))))))) 1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(10) = 0 POL(10') = 0 POL(1024_1(x_1)) = [1] + x_1 POL(1024_1'(x_1)) = [1] POL(DOUBLE(x_1)) = 0 POL(IF(x_1, x_2)) = [1] POL(LT(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(double(x_1)) = 0 POL(false) = [1] POL(if(x_1, x_2)) = [1] + x_2 POL(lt(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(true) = [1] ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(double(s(0))))))) 1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(double(s(0))))))) 1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', IF_2, 1024_1'_1 Compound Symbols: c6_1, c8_1, c_1, c2_2, c1_3, c3_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), 10), s(x0))), 1024_1'(s(x0))) by IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(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) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(double(s(0))))))) 1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(double(s(0))))))) 1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0))) K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c1_3, c3_1, c2_2 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace 10' -> c(DOUBLE(s(s(s(double(s(0))))))) by 10' -> c(DOUBLE(s(s(s(s(s(double(0)))))))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(s(s(double(0)))))))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(s(s(double(0)))))))) K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c1_3, c3_1, c2_2 ---------------------------------------- (73) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace 1024_1'(x0) -> c1(IF(lt(x0, s(s(double(double(s(s(0))))))), x0), LT(x0, 10), 10') by 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(s(s(double(0)))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(s(s(double(0)))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c1_3, c3_1, c2_2 ---------------------------------------- (75) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace 1024_1'(x0) -> c1(IF(lt(x0, double(s(s(s(double(s(0))))))), x0), LT(x0, 10), 10') by 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(s(s(double(0)))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(s(s(double(0)))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c3_1, c2_2, c1_3 ---------------------------------------- (77) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace 1024_1'(x0) -> c3(LT(x0, 10)) by 1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0))))))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c3(10') IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(s(s(double(0)))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') 1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0))))))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0))) 10' -> c(DOUBLE(s(s(s(s(s(double(0)))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c3_1, c2_2, c1_3 ---------------------------------------- (79) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(true, x0) -> c2(DOUBLE(if(lt(s(x0), double(s(double(s(s(0)))))), s(x0))), 1024_1'(s(x0))) by IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c3(10') 10' -> c(DOUBLE(s(s(s(s(s(double(0)))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') 1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0))))))) IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(double(0)))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0))) K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c3_1, c1_3, c2_2 ---------------------------------------- (81) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace 10' -> c(DOUBLE(s(s(s(s(s(double(0)))))))) by 10' -> c(DOUBLE(s(s(s(s(s(0))))))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c3(10') 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') 1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0))))))) IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c3_1, c1_3, c2_2 ---------------------------------------- (83) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') by 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c3(10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') 1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0))))))) IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c3_1, c1_3, c2_2 ---------------------------------------- (85) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, double(s(double(s(s(0)))))), 10') by 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(x0) -> c3(10') 1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0))))))) IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(x0) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c3_1, c2_2, c1_3 ---------------------------------------- (87) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace 1024_1'(x0) -> c3(10') by 1024_1'(s(x0)) -> c3(10') ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0))))))) IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c3(10') S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(x0)) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c3_1, c2_2, c1_3 ---------------------------------------- (89) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace 1024_1'(z0) -> c3(LT(z0, double(s(double(s(s(0))))))) by 1024_1'(z0) -> c3(LT(z0, s(s(double(double(s(s(0)))))))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c3(10') 1024_1'(z0) -> c3(LT(z0, s(s(double(double(s(s(0)))))))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(x0)) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', IF_2, 1024_1'_1 Compound Symbols: c6_1, c8_1, c_1, c2_2, c1_3, c3_1 ---------------------------------------- (91) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(true, z0) -> c2(DOUBLE(if(lt(s(z0), s(s(double(double(s(s(0))))))), s(z0))), 1024_1'(s(z0))) by IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c3(10') 1024_1'(z0) -> c3(LT(z0, s(s(double(double(s(s(0)))))))) IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0))) K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(x0)) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c1_3, c3_1, c2_2 ---------------------------------------- (93) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace 1024_1'(z0) -> c1(IF(lt(z0, s(s(double(double(s(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') by 1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c3(10') 1024_1'(z0) -> c3(LT(z0, s(s(double(double(s(s(0)))))))) IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0))) 1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0))) 1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(x0)) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c1_3, c3_1, c2_2 ---------------------------------------- (95) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace 1024_1'(z0) -> c1(IF(lt(z0, double(s(s(s(double(s(0))))))), z0), LT(z0, s(s(double(double(s(s(0))))))), 10') by 1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(x0)) -> c3(10') 1024_1'(z0) -> c3(LT(z0, s(s(double(double(s(s(0)))))))) IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0))) 1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0))) 1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(x0)) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c3_1, c2_2, c1_3 ---------------------------------------- (97) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace 1024_1'(z0) -> c3(LT(z0, s(s(double(double(s(s(0)))))))) by 1024_1'(s(x0)) -> c3(LT(s(x0), s(s(double(double(s(s(0)))))))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(x0)) -> c3(10') IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0))) 1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c3(LT(s(x0), s(s(double(double(s(s(0)))))))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0))) 1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(x0)) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c3_1, c2_2, c1_3 ---------------------------------------- (99) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(true, z0) -> c2(DOUBLE(if(lt(z0, s(double(double(s(s(0)))))), s(z0))), 1024_1'(s(z0))) by IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(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) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(x0)) -> c3(10') 1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c3(LT(s(x0), s(s(double(double(s(s(0)))))))) IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0)))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0)))) K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(x0)) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c3_1, c1_3, c2_2 ---------------------------------------- (101) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace 1024_1'(s(x0)) -> c3(10') by 1024_1'(s(s(x0))) -> c3(10') ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c3(LT(s(x0), s(s(double(double(s(s(0)))))))) IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0)))) 1024_1'(s(s(x0))) -> c3(10') S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0)))) K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(s(x0))) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c1_3, c3_1, c2_2 ---------------------------------------- (103) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace 1024_1'(s(x0)) -> c1(IF(lt(s(x0), s(s(double(double(s(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') by 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(x0)) -> c3(LT(s(x0), s(s(double(double(s(s(0)))))))) IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0)))) 1024_1'(s(s(x0))) -> c3(10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0)))) 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(s(x0))) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c1_3, c3_1, c2_2 ---------------------------------------- (105) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace 1024_1'(s(x0)) -> c1(IF(lt(s(x0), double(s(s(s(double(s(0))))))), s(x0)), LT(s(x0), s(s(double(double(s(s(0))))))), 10') by 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(x0)) -> c3(LT(s(x0), s(s(double(double(s(s(0)))))))) IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0)))) 1024_1'(s(s(x0))) -> c3(10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0)))) 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(s(x0))) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c3_1, c2_2, c1_3 ---------------------------------------- (107) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace 1024_1'(s(x0)) -> c3(LT(s(x0), s(s(double(double(s(s(0)))))))) by 1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0)))))))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0)))) 1024_1'(s(s(x0))) -> c3(10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0)))))))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0)))) 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(s(x0))) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', IF_2, 1024_1'_1 Compound Symbols: c6_1, c8_1, c_1, c2_2, c3_1, c1_3 ---------------------------------------- (109) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(true, s(x0)) -> c2(DOUBLE(if(lt(s(x0), s(double(double(s(s(0)))))), s(s(x0)))), 1024_1'(s(s(x0)))) by IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0))))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(s(x0))) -> c3(10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0)))))))) IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0))))) S tuples: LT(s(z0), s(z1)) -> c6(LT(z0, z1)) DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0))))) K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(s(x0))) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: LT_2, DOUBLE_1, 10', 1024_1'_1, IF_2 Compound Symbols: c6_1, c8_1, c_1, c3_1, c1_3, c2_2 ---------------------------------------- (111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LT(s(z0), s(z1)) -> c6(LT(z0, z1)) by LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(s(x0))) -> c3(10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0)))))))) IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0))))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) S tuples: DOUBLE(s(z0)) -> c8(DOUBLE(z0)) 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0))))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(s(x0))) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: DOUBLE_1, 10', 1024_1'_1, IF_2, LT_2 Compound Symbols: c8_1, c_1, c3_1, c1_3, c2_2, c6_1 ---------------------------------------- (113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DOUBLE(s(z0)) -> c8(DOUBLE(z0)) by DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(s(x0))) -> c3(10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0)))))))) IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0))))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) S tuples: 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0))))) LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(s(x0))) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: 10', 1024_1'_1, IF_2, LT_2, DOUBLE_1 Compound Symbols: c_1, c3_1, c1_3, c2_2, c6_1, c8_1 ---------------------------------------- (115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1))) by LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(s(x0))) -> c3(10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0)))))))) IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0))))) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) S tuples: 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0))))) DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(s(x0))) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: 10', 1024_1'_1, IF_2, DOUBLE_1, LT_2 Compound Symbols: c_1, c3_1, c1_3, c2_2, c8_1, c6_1 ---------------------------------------- (117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0))) by DOUBLE(s(s(s(y0)))) -> c8(DOUBLE(s(s(y0)))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(s(x0))) -> c3(10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0)))))))) IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0))))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) DOUBLE(s(s(s(y0)))) -> c8(DOUBLE(s(s(y0)))) S tuples: 10' -> c(DOUBLE(s(s(0)))) 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0))))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) DOUBLE(s(s(s(y0)))) -> c8(DOUBLE(s(s(y0)))) K tuples: 1024_1'(x0) -> c3(LT(x0, 10)) 1024_1'(s(s(x0))) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: 10', 1024_1'_1, IF_2, LT_2, DOUBLE_1 Compound Symbols: c_1, c3_1, c1_3, c2_2, c6_1, c8_1 ---------------------------------------- (119) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: 10' -> c(DOUBLE(s(s(0)))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: lt(0, s(z0)) -> true lt(z0, 0) -> false lt(s(z0), s(z1)) -> lt(z0, z1) 10 -> double(s(double(s(s(0))))) double(s(z0)) -> s(s(double(z0))) double(0) -> 0 1024_1(z0) -> if(lt(z0, 10), z0) if(true, z0) -> double(1024_1(s(z0))) if(false, z0) -> s(0) Tuples: 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(s(x0))) -> c3(10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c3(LT(s(s(x0)), s(s(double(double(s(s(0)))))))) IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0))))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) DOUBLE(s(s(s(y0)))) -> c8(DOUBLE(s(s(y0)))) S tuples: 10' -> c(DOUBLE(s(s(s(s(s(0))))))) 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), s(s(double(double(s(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') 1024_1'(s(s(x0))) -> c1(IF(lt(s(s(x0)), double(s(s(s(double(s(0))))))), s(s(x0))), LT(s(s(x0)), s(s(double(double(s(s(0))))))), 10') IF(true, s(s(x0))) -> c2(DOUBLE(if(lt(s(s(x0)), s(double(double(s(s(0)))))), s(s(s(x0))))), 1024_1'(s(s(s(x0))))) LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1)))) DOUBLE(s(s(s(y0)))) -> c8(DOUBLE(s(s(y0)))) K tuples: 1024_1'(s(s(x0))) -> c3(10') Defined Rule Symbols: lt_2, 10, double_1, 1024_1_1, if_2 Defined Pair Symbols: 10', 1024_1'_1, IF_2, LT_2, DOUBLE_1 Compound Symbols: c_1, c3_1, c1_3, c2_2, c6_1, c8_1