KILLED proof of input_1Jyf2iMVE1.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 6 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 200 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 37 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 378 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 147 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 3988 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 1061 ms] (36) CpxRNTS (37) CompletionProof [UPPER BOUND(ID), 0 ms] (38) CpxTypedWeightedCompleteTrs (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (44) CdtProblem (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 14 ms] (62) CdtProblem (63) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 404 ms] (70) CdtProblem (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtInstantiationProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem ---------------------------------------- (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: le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) double(0) -> 0 double(s(x)) -> s(s(double(x))) log(0) -> logError log(s(x)) -> loop(s(x), s(0), 0) loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) if(true, x, y, z) -> z if(false, x, y, z) -> loop(x, double(y), s(z)) 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: le(s(x), 0') -> false le(0', y) -> true le(s(x), s(y)) -> le(x, y) double(0') -> 0' double(s(x)) -> s(s(double(x))) log(0') -> logError log(s(x)) -> loop(s(x), s(0'), 0') loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) if(true, x, y, z) -> z if(false, x, y, z) -> loop(x, double(y), s(z)) 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: le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) double(0) -> 0 double(s(x)) -> s(s(double(x))) log(0) -> logError log(s(x)) -> loop(s(x), s(0), 0) loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) if(true, x, y, z) -> z if(false, x, y, z) -> loop(x, double(y), s(z)) 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: le(s(x), 0) -> false [1] le(0, y) -> true [1] le(s(x), s(y)) -> le(x, y) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] log(0) -> logError [1] log(s(x)) -> loop(s(x), s(0), 0) [1] loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) [1] if(true, x, y, z) -> z [1] if(false, x, y, z) -> loop(x, double(y), s(z)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: le(s(x), 0) -> false [1] le(0, y) -> true [1] le(s(x), s(y)) -> le(x, y) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] log(0) -> logError [1] log(s(x)) -> loop(s(x), s(0), 0) [1] loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) [1] if(true, x, y, z) -> z [1] if(false, x, y, z) -> loop(x, double(y), s(z)) [1] The TRS has the following type information: le :: s:0:logError -> s:0:logError -> false:true s :: s:0:logError -> s:0:logError 0 :: s:0:logError false :: false:true true :: false:true double :: s:0:logError -> s:0:logError log :: s:0:logError -> s:0:logError logError :: s:0:logError loop :: s:0:logError -> s:0:logError -> s:0:logError -> s:0:logError if :: false:true -> s:0:logError -> s:0:logError -> s:0:logError -> s:0:logError Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: log_1 loop_3 if_4 (c) The following functions are completely defined: le_2 double_1 Due to the following rules being added: le(v0, v1) -> null_le [0] double(v0) -> null_double [0] And the following fresh constants: null_le, null_double ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: le(s(x), 0) -> false [1] le(0, y) -> true [1] le(s(x), s(y)) -> le(x, y) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] log(0) -> logError [1] log(s(x)) -> loop(s(x), s(0), 0) [1] loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) [1] if(true, x, y, z) -> z [1] if(false, x, y, z) -> loop(x, double(y), s(z)) [1] le(v0, v1) -> null_le [0] double(v0) -> null_double [0] The TRS has the following type information: le :: s:0:logError:null_double -> s:0:logError:null_double -> false:true:null_le s :: s:0:logError:null_double -> s:0:logError:null_double 0 :: s:0:logError:null_double false :: false:true:null_le true :: false:true:null_le double :: s:0:logError:null_double -> s:0:logError:null_double log :: s:0:logError:null_double -> s:0:logError:null_double logError :: s:0:logError:null_double loop :: s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double if :: false:true:null_le -> s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double null_le :: false:true:null_le null_double :: s:0:logError:null_double Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: le(s(x), 0) -> false [1] le(0, y) -> true [1] le(s(x), s(y)) -> le(x, y) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] log(0) -> logError [1] log(s(x)) -> loop(s(x), s(0), 0) [1] loop(0, s(y), z) -> if(true, 0, s(y), z) [2] loop(s(x'), s(y), z) -> if(le(x', y), s(x'), s(y), z) [2] loop(x, s(y), z) -> if(null_le, x, s(y), z) [1] if(true, x, y, z) -> z [1] if(false, x, 0, z) -> loop(x, 0, s(z)) [2] if(false, x, s(x''), z) -> loop(x, s(s(double(x''))), s(z)) [2] if(false, x, y, z) -> loop(x, null_double, s(z)) [1] le(v0, v1) -> null_le [0] double(v0) -> null_double [0] The TRS has the following type information: le :: s:0:logError:null_double -> s:0:logError:null_double -> false:true:null_le s :: s:0:logError:null_double -> s:0:logError:null_double 0 :: s:0:logError:null_double false :: false:true:null_le true :: false:true:null_le double :: s:0:logError:null_double -> s:0:logError:null_double log :: s:0:logError:null_double -> s:0:logError:null_double logError :: s:0:logError:null_double loop :: s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double if :: false:true:null_le -> s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double -> s:0:logError:null_double null_le :: false:true:null_le null_double :: s:0:logError:null_double Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 false => 1 true => 2 logError => 1 null_le => 0 null_double => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: double(z') -{ 1 }-> 0 :|: z' = 0 double(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 double(z') -{ 1 }-> 1 + (1 + double(x)) :|: z' = 1 + x, x >= 0 if(z', z'', z1, z2) -{ 1 }-> z :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x if(z', z'', z1, z2) -{ 2 }-> loop(x, 0, 1 + z) :|: z1 = 0, z >= 0, z2 = z, x >= 0, z'' = x, z' = 1 if(z', z'', z1, z2) -{ 1 }-> loop(x, 0, 1 + z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1 if(z', z'', z1, z2) -{ 2 }-> loop(x, 1 + (1 + double(x'')), 1 + z) :|: z >= 0, z1 = 1 + x'', z2 = z, x >= 0, z'' = x, z' = 1, x'' >= 0 le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0 le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 log(z') -{ 1 }-> loop(1 + x, 1 + 0, 0) :|: z' = 1 + x, x >= 0 log(z') -{ 1 }-> 1 :|: z' = 0 loop(z', z'', z1) -{ 2 }-> if(le(x', y), 1 + x', 1 + y, z) :|: z1 = z, z >= 0, z' = 1 + x', x' >= 0, y >= 0, z'' = 1 + y loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + y, z) :|: z1 = z, z >= 0, y >= 0, z'' = 1 + y, z' = 0 loop(z', z'', z1) -{ 1 }-> if(0, x, 1 + y, z) :|: z1 = z, z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: double(z') -{ 1 }-> 0 :|: z' = 0 double(z') -{ 0 }-> 0 :|: z' >= 0 double(z') -{ 1 }-> 1 + (1 + double(z' - 1)) :|: z' - 1 >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 1 + (1 + double(z1 - 1)), 1 + z2) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0 log(z') -{ 1 }-> 1 :|: z' = 0 loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0 loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { double } { le } { if, loop } { log } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: double(z') -{ 1 }-> 0 :|: z' = 0 double(z') -{ 0 }-> 0 :|: z' >= 0 double(z') -{ 1 }-> 1 + (1 + double(z' - 1)) :|: z' - 1 >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 1 + (1 + double(z1 - 1)), 1 + z2) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0 log(z') -{ 1 }-> 1 :|: z' = 0 loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0 loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0 Function symbols to be analyzed: {double}, {le}, {if,loop}, {log} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: double(z') -{ 1 }-> 0 :|: z' = 0 double(z') -{ 0 }-> 0 :|: z' >= 0 double(z') -{ 1 }-> 1 + (1 + double(z' - 1)) :|: z' - 1 >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 1 + (1 + double(z1 - 1)), 1 + z2) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0 log(z') -{ 1 }-> 1 :|: z' = 0 loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0 loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0 Function symbols to be analyzed: {double}, {le}, {if,loop}, {log} ---------------------------------------- (21) 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' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: double(z') -{ 1 }-> 0 :|: z' = 0 double(z') -{ 0 }-> 0 :|: z' >= 0 double(z') -{ 1 }-> 1 + (1 + double(z' - 1)) :|: z' - 1 >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 1 + (1 + double(z1 - 1)), 1 + z2) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0 log(z') -{ 1 }-> 1 :|: z' = 0 loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0 loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0 Function symbols to be analyzed: {double}, {le}, {if,loop}, {log} Previous analysis results are: double: runtime: ?, size: O(n^1) [2*z'] ---------------------------------------- (23) 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' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: double(z') -{ 1 }-> 0 :|: z' = 0 double(z') -{ 0 }-> 0 :|: z' >= 0 double(z') -{ 1 }-> 1 + (1 + double(z' - 1)) :|: z' - 1 >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 1 + (1 + double(z1 - 1)), 1 + z2) :|: z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0 log(z') -{ 1 }-> 1 :|: z' = 0 loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0 loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0 Function symbols to be analyzed: {le}, {if,loop}, {log} Previous analysis results are: double: runtime: O(n^1) [1 + z'], size: O(n^1) [2*z'] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: double(z') -{ 1 }-> 0 :|: z' = 0 double(z') -{ 0 }-> 0 :|: z' >= 0 double(z') -{ 1 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 + z1 }-> loop(z'', 1 + (1 + s'), 1 + z2) :|: s' >= 0, s' <= 2 * (z1 - 1), z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0 log(z') -{ 1 }-> 1 :|: z' = 0 loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0 loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0 Function symbols to be analyzed: {le}, {if,loop}, {log} Previous analysis results are: double: runtime: O(n^1) [1 + z'], size: O(n^1) [2*z'] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: le after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: double(z') -{ 1 }-> 0 :|: z' = 0 double(z') -{ 0 }-> 0 :|: z' >= 0 double(z') -{ 1 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 + z1 }-> loop(z'', 1 + (1 + s'), 1 + z2) :|: s' >= 0, s' <= 2 * (z1 - 1), z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0 log(z') -{ 1 }-> 1 :|: z' = 0 loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0 loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0 Function symbols to be analyzed: {le}, {if,loop}, {log} Previous analysis results are: double: runtime: O(n^1) [1 + z'], size: O(n^1) [2*z'] le: runtime: ?, size: O(1) [2] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: le after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z'' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: double(z') -{ 1 }-> 0 :|: z' = 0 double(z') -{ 0 }-> 0 :|: z' >= 0 double(z') -{ 1 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 + z1 }-> loop(z'', 1 + (1 + s'), 1 + z2) :|: s' >= 0, s' <= 2 * (z1 - 1), z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0 log(z') -{ 1 }-> 1 :|: z' = 0 loop(z', z'', z1) -{ 2 }-> if(le(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0 loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0 Function symbols to be analyzed: {if,loop}, {log} Previous analysis results are: double: runtime: O(n^1) [1 + z'], size: O(n^1) [2*z'] le: runtime: O(n^1) [2 + z''], size: O(1) [2] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: double(z') -{ 1 }-> 0 :|: z' = 0 double(z') -{ 0 }-> 0 :|: z' >= 0 double(z') -{ 1 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 + z1 }-> loop(z'', 1 + (1 + s'), 1 + z2) :|: s' >= 0, s' <= 2 * (z1 - 1), z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0 le(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0 log(z') -{ 1 }-> 1 :|: z' = 0 loop(z', z'', z1) -{ 3 + z'' }-> if(s1, 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s1 >= 0, s1 <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0 loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0 Function symbols to be analyzed: {if,loop}, {log} Previous analysis results are: double: runtime: O(n^1) [1 + z'], size: O(n^1) [2*z'] le: runtime: O(n^1) [2 + z''], size: O(1) [2] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: loop after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: double(z') -{ 1 }-> 0 :|: z' = 0 double(z') -{ 0 }-> 0 :|: z' >= 0 double(z') -{ 1 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 + z1 }-> loop(z'', 1 + (1 + s'), 1 + z2) :|: s' >= 0, s' <= 2 * (z1 - 1), z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0 le(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0 log(z') -{ 1 }-> 1 :|: z' = 0 loop(z', z'', z1) -{ 3 + z'' }-> if(s1, 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s1 >= 0, s1 <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0 loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0 Function symbols to be analyzed: {if,loop}, {log} Previous analysis results are: double: runtime: O(n^1) [1 + z'], size: O(n^1) [2*z'] le: runtime: O(n^1) [2 + z''], size: O(1) [2] if: runtime: ?, size: INF loop: runtime: ?, size: INF ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: double(z') -{ 1 }-> 0 :|: z' = 0 double(z') -{ 0 }-> 0 :|: z' >= 0 double(z') -{ 1 + z' }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z' - 1), z' - 1 >= 0 if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0 if(z', z'', z1, z2) -{ 2 }-> loop(z'', 0, 1 + z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1 if(z', z'', z1, z2) -{ 1 }-> loop(z'', 0, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1 if(z', z'', z1, z2) -{ 2 + z1 }-> loop(z'', 1 + (1 + s'), 1 + z2) :|: s' >= 0, s' <= 2 * (z1 - 1), z2 >= 0, z'' >= 0, z' = 1, z1 - 1 >= 0 le(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z') -{ 1 }-> loop(1 + (z' - 1), 1 + 0, 0) :|: z' - 1 >= 0 log(z') -{ 1 }-> 1 :|: z' = 0 loop(z', z'', z1) -{ 3 + z'' }-> if(s1, 1 + (z' - 1), 1 + (z'' - 1), z1) :|: s1 >= 0, s1 <= 2, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0 loop(z', z'', z1) -{ 2 }-> if(2, 0, 1 + (z'' - 1), z1) :|: z1 >= 0, z'' - 1 >= 0, z' = 0 loop(z', z'', z1) -{ 1 }-> if(0, z', 1 + (z'' - 1), z1) :|: z1 >= 0, z' >= 0, z'' - 1 >= 0 Function symbols to be analyzed: {if,loop}, {log} Previous analysis results are: double: runtime: O(n^1) [1 + z'], size: O(n^1) [2*z'] le: runtime: O(n^1) [2 + z''], size: O(1) [2] if: runtime: INF, size: INF loop: runtime: ?, size: INF ---------------------------------------- (37) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: le(v0, v1) -> null_le [0] double(v0) -> null_double [0] log(v0) -> null_log [0] loop(v0, v1, v2) -> null_loop [0] if(v0, v1, v2, v3) -> null_if [0] And the following fresh constants: null_le, null_double, null_log, null_loop, null_if ---------------------------------------- (38) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: le(s(x), 0) -> false [1] le(0, y) -> true [1] le(s(x), s(y)) -> le(x, y) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] log(0) -> logError [1] log(s(x)) -> loop(s(x), s(0), 0) [1] loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z) [1] if(true, x, y, z) -> z [1] if(false, x, y, z) -> loop(x, double(y), s(z)) [1] le(v0, v1) -> null_le [0] double(v0) -> null_double [0] log(v0) -> null_log [0] loop(v0, v1, v2) -> null_loop [0] if(v0, v1, v2, v3) -> null_if [0] The TRS has the following type information: le :: s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if -> false:true:null_le s :: s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if 0 :: s:0:logError:null_double:null_log:null_loop:null_if false :: false:true:null_le true :: false:true:null_le double :: s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if log :: s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if logError :: s:0:logError:null_double:null_log:null_loop:null_if loop :: s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if if :: false:true:null_le -> s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if -> s:0:logError:null_double:null_log:null_loop:null_if null_le :: false:true:null_le null_double :: s:0:logError:null_double:null_log:null_loop:null_if null_log :: s:0:logError:null_double:null_log:null_loop:null_if null_loop :: s:0:logError:null_double:null_log:null_loop:null_if null_if :: s:0:logError:null_double:null_log:null_loop:null_if Rewrite Strategy: INNERMOST ---------------------------------------- (39) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 false => 1 true => 2 logError => 1 null_le => 0 null_double => 0 null_log => 0 null_loop => 0 null_if => 0 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: double(z') -{ 1 }-> 0 :|: z' = 0 double(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 double(z') -{ 1 }-> 1 + (1 + double(x)) :|: z' = 1 + x, x >= 0 if(z', z'', z1, z2) -{ 1 }-> z :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x if(z', z'', z1, z2) -{ 1 }-> loop(x, double(y), 1 + z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1 if(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0 le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0 le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 log(z') -{ 1 }-> loop(1 + x, 1 + 0, 0) :|: z' = 1 + x, x >= 0 log(z') -{ 1 }-> 1 :|: z' = 0 log(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 loop(z', z'', z1) -{ 1 }-> if(le(x, 1 + y), x, 1 + y, z) :|: z1 = z, z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y loop(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (41) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) log(0) -> logError log(s(z0)) -> loop(s(z0), s(0), 0) loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2)) Tuples: LE(s(z0), 0) -> c LE(0, z0) -> c1 LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(0) -> c3 DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOG(0) -> c5 LOG(s(z0)) -> c6(LOOP(s(z0), s(0), 0)) LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1))) IF(true, z0, z1, z2) -> c8 IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1)) S tuples: LE(s(z0), 0) -> c LE(0, z0) -> c1 LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(0) -> c3 DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOG(0) -> c5 LOG(s(z0)) -> c6(LOOP(s(z0), s(0), 0)) LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1))) IF(true, z0, z1, z2) -> c8 IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1)) K tuples:none Defined Rule Symbols: le_2, double_1, log_1, loop_3, if_4 Defined Pair Symbols: LE_2, DOUBLE_1, LOG_1, LOOP_3, IF_4 Compound Symbols: c, c1, c2_1, c3, c4_1, c5, c6_1, c7_2, c8, c9_2 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: LOG(s(z0)) -> c6(LOOP(s(z0), s(0), 0)) Removed 5 trailing nodes: IF(true, z0, z1, z2) -> c8 LE(0, z0) -> c1 DOUBLE(0) -> c3 LE(s(z0), 0) -> c LOG(0) -> c5 ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) log(0) -> logError log(s(z0)) -> loop(s(z0), s(0), 0) loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2)) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1))) IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1))) IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1)) K tuples:none Defined Rule Symbols: le_2, double_1, log_1, loop_3, if_4 Defined Pair Symbols: LE_2, DOUBLE_1, LOOP_3, IF_4 Compound Symbols: c2_1, c4_1, c7_2, c9_2 ---------------------------------------- (45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: log(0) -> logError log(s(z0)) -> loop(s(z0), s(0), 0) loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1))) IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1))) IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1)) K tuples:none Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: LE_2, DOUBLE_1, LOOP_3, IF_4 Compound Symbols: c2_1, c4_1, c7_2, c9_2 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1))) by LOOP(0, s(x1), x2) -> c7(IF(true, 0, s(x1), x2), LE(0, s(x1))) LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1)) LOOP(0, s(x1), x2) -> c7(IF(true, 0, s(x1), x2), LE(0, s(x1))) LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1)) LOOP(0, s(x1), x2) -> c7(IF(true, 0, s(x1), x2), LE(0, s(x1))) LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) K tuples:none Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: LE_2, DOUBLE_1, IF_4, LOOP_3 Compound Symbols: c2_1, c4_1, c9_2, c7_2 ---------------------------------------- (49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: LOOP(0, s(x1), x2) -> c7(IF(true, 0, s(x1), x2), LE(0, s(x1))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1)) LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1)) LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) K tuples:none Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: LE_2, DOUBLE_1, IF_4, LOOP_3 Compound Symbols: c2_1, c4_1, c9_2, c7_2 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1)) by IF(false, x0, 0, x2) -> c9(LOOP(x0, 0, s(x2)), DOUBLE(0)) IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) IF(false, x0, 0, x2) -> c9(LOOP(x0, 0, s(x2)), DOUBLE(0)) IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) IF(false, x0, 0, x2) -> c9(LOOP(x0, 0, s(x2)), DOUBLE(0)) IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) K tuples:none Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: LE_2, DOUBLE_1, LOOP_3, IF_4 Compound Symbols: c2_1, c4_1, c7_2, c9_2 ---------------------------------------- (53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF(false, x0, 0, x2) -> c9(LOOP(x0, 0, s(x2)), DOUBLE(0)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) K tuples:none Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: LE_2, DOUBLE_1, LOOP_3, IF_4 Compound Symbols: c2_1, c4_1, c7_2, c9_2 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LOOP(s(z0), s(z1), x2) -> c7(IF(le(z0, z1), s(z0), s(z1), x2), LE(s(z0), s(z1))) by LOOP(s(0), s(z0), x2) -> c7(IF(true, s(0), s(z0), x2), LE(s(0), s(z0))) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(0), x2) -> c7(IF(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) LOOP(s(0), s(z0), x2) -> c7(IF(true, s(0), s(z0), x2), LE(s(0), s(z0))) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(0), x2) -> c7(IF(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) LOOP(s(0), s(z0), x2) -> c7(IF(true, s(0), s(z0), x2), LE(s(0), s(z0))) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(0), x2) -> c7(IF(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) K tuples:none Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: LE_2, DOUBLE_1, IF_4, LOOP_3 Compound Symbols: c2_1, c4_1, c9_2, c7_2, c7_1 ---------------------------------------- (57) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: LOOP(s(s(z0)), s(0), x2) -> c7(IF(false, s(s(z0)), s(0), x2), LE(s(s(z0)), s(0))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) LOOP(s(0), s(z0), x2) -> c7(IF(true, s(0), s(z0), x2), LE(s(0), s(z0))) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) LOOP(s(0), s(z0), x2) -> c7(IF(true, s(0), s(z0), x2), LE(s(0), s(z0))) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) K tuples:none Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: LE_2, DOUBLE_1, IF_4, LOOP_3 Compound Symbols: c2_1, c4_1, c9_2, c7_2, c7_1 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0))) K tuples:none Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: LE_2, DOUBLE_1, IF_4, LOOP_3 Compound Symbols: c2_1, c4_1, c9_2, c7_2, c7_1 ---------------------------------------- (61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0))) We considered the (Usable) Rules: le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DOUBLE(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 POL(LE(x_1, x_2)) = 0 POL(LOOP(x_1, x_2, x_3)) = x_1 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2)) = x_1 + x_2 POL(double(x_1)) = [1] + x_1 POL(false) = 0 POL(le(x_1, x_2)) = 0 POL(s(x_1)) = [1] POL(true) = 0 ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) K tuples: LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: LE_2, DOUBLE_1, IF_4, LOOP_3 Compound Symbols: c2_1, c4_1, c9_2, c7_2, c7_1 ---------------------------------------- (63) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, x0, s(z0), x2) -> c9(LOOP(x0, s(s(double(z0))), s(x2)), DOUBLE(s(z0))) by IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(double(s(x1)))), s(x2)), DOUBLE(s(s(x1)))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0))) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(double(s(x1)))), s(x2)), DOUBLE(s(s(x1)))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(double(s(x1)))), s(x2)), DOUBLE(s(s(x1)))) K tuples: LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: LE_2, DOUBLE_1, LOOP_3, IF_4 Compound Symbols: c2_1, c4_1, c7_2, c7_1, c9_2 ---------------------------------------- (65) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: LOOP(s(0), s(z0), x2) -> c7(LE(s(0), s(z0))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(double(s(x1)))), s(x2)), DOUBLE(s(s(x1)))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(double(s(x1)))), s(x2)), DOUBLE(s(s(x1)))) K tuples: LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: LE_2, DOUBLE_1, LOOP_3, IF_4 Compound Symbols: c2_1, c4_1, c7_2, c7_1, c9_2 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(double(s(x1)))), s(x2)), DOUBLE(s(s(x1)))) by IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0)))) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0)))) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0)))) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) K tuples: LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: LE_2, DOUBLE_1, LOOP_3, IF_4 Compound Symbols: c2_1, c4_1, c7_2, c7_1, c9_2, c9_1 ---------------------------------------- (69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) We considered the (Usable) Rules:none And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0)))) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DOUBLE(x_1)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_3*x_4 + x_2*x_4 + x_3^2 + x_2*x_3 + x_2^2 + x_2^3 + x_2^2*x_3 + x_2^2*x_4 + x_2*x_4^2 + x_2*x_3*x_4 + x_2*x_3^2 + x_3^3 + x_3^2*x_4 + x_3*x_4^2 + x_4^3 POL(LE(x_1, x_2)) = 0 POL(LOOP(x_1, x_2, x_3)) = [1] + x_1 + x_1*x_3 + x_1^2 + x_1^3 + x_1^2*x_3 + x_1*x_3^2 + x_3^3 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(double(x_1)) = 0 POL(false) = 0 POL(le(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(true) = [1] ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0)))) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0)))) K tuples: LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: LE_2, DOUBLE_1, LOOP_3, IF_4 Compound Symbols: c2_1, c4_1, c7_2, c7_1, c9_2, c9_1 ---------------------------------------- (71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LE(s(z0), s(z1)) -> c2(LE(z0, z1)) by LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0)))) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) S tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) K tuples: LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: DOUBLE_1, LOOP_3, IF_4, LE_2 Compound Symbols: c4_1, c7_2, c7_1, c9_2, c9_1, c2_1 ---------------------------------------- (73) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace LOOP(s(s(z0)), s(s(z1)), x2) -> c7(IF(le(z0, z1), s(s(z0)), s(s(z1)), x2), LE(s(s(z0)), s(s(z1)))) by LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0)))))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0)))) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0)))))) S tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0)))))) K tuples: LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: DOUBLE_1, LOOP_3, IF_4, LE_2 Compound Symbols: c4_1, c7_1, c9_2, c9_1, c2_1, c7_2 ---------------------------------------- (75) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace LOOP(s(x0), s(x1), x2) -> c7(LE(s(x0), s(x1))) by LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0)))))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0)))) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0)))))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0)))))) S tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0)))))) K tuples: IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0)))))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: DOUBLE_1, IF_4, LE_2, LOOP_3 Compound Symbols: c4_1, c9_2, c9_1, c2_1, c7_2, c7_1 ---------------------------------------- (77) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, s(s(x0)), s(s(z0)), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(double(z0))))), s(x2)), DOUBLE(s(s(z0)))) by IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1)))))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0)))))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1)))))) S tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1)))))) K tuples: IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0)))))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: DOUBLE_1, IF_4, LE_2, LOOP_3 Compound Symbols: c4_1, c9_1, c2_1, c7_2, c7_1, c9_2 ---------------------------------------- (79) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, s(s(x0)), s(s(x1)), x2) -> c9(DOUBLE(s(s(x1)))) by IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0)))))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1)))))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) S tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1)))))) K tuples: LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: DOUBLE_1, LE_2, LOOP_3, IF_4 Compound Symbols: c4_1, c2_1, c7_2, c7_1, c9_2, c9_1 ---------------------------------------- (81) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(x2)), LE(s(s(x0)), s(s(s(s(y0)))))) by LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1)))))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) S tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1)))))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) K tuples: LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: DOUBLE_1, LE_2, LOOP_3, IF_4 Compound Symbols: c4_1, c2_1, c7_1, c9_2, c9_1, c7_2 ---------------------------------------- (83) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(LOOP(s(s(x0)), s(s(s(s(double(s(s(x1))))))), s(s(x2))), DOUBLE(s(s(s(s(x1)))))) by IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(double(s(z1)))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(double(s(z1)))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) S tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(double(s(z1)))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) K tuples: LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: DOUBLE_1, LE_2, LOOP_3, IF_4 Compound Symbols: c4_1, c2_1, c7_1, c9_1, c7_2, c9_2 ---------------------------------------- (85) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(double(s(z1)))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) by IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) S tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) K tuples: LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: DOUBLE_1, LE_2, LOOP_3, IF_4 Compound Symbols: c4_1, c2_1, c7_1, c9_1, c7_2, c9_2 ---------------------------------------- (87) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace LOOP(s(s(x0)), s(s(s(s(y0)))), s(x2)) -> c7(LE(s(s(x0)), s(s(s(s(y0)))))) by LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))))) S tuples: DOUBLE(s(z0)) -> c4(DOUBLE(z0)) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) K tuples: IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: DOUBLE_1, LE_2, IF_4, LOOP_3 Compound Symbols: c4_1, c2_1, c9_1, c7_2, c9_2, c7_1 ---------------------------------------- (89) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DOUBLE(s(z0)) -> c4(DOUBLE(z0)) by DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))))) DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0))) K tuples: IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: LE_2, IF_4, LOOP_3, DOUBLE_1 Compound Symbols: c2_1, c9_1, c7_2, c9_2, c7_1, c4_1 ---------------------------------------- (91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) by LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))))) DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) S tuples: LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) K tuples: IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: IF_4, LOOP_3, DOUBLE_1, LE_2 Compound Symbols: c9_1, c7_2, c9_2, c7_1, c4_1, c2_1 ---------------------------------------- (93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LOOP(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))), s(s(x2))) -> c7(LE(s(s(x0)), s(s(s(s(s(s(s(s(y0)))))))))) by LOOP(s(s(s(y0))), s(s(s(s(s(s(s(s(z1)))))))), s(s(z2))) -> c7(LE(s(s(s(y0))), s(s(s(s(s(s(s(s(z1)))))))))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) LOOP(s(s(s(y0))), s(s(s(s(s(s(s(s(z1)))))))), s(s(z2))) -> c7(LE(s(s(s(y0))), s(s(s(s(s(s(s(s(z1)))))))))) S tuples: LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) K tuples: IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) LOOP(s(s(s(y0))), s(s(s(s(s(s(s(s(z1)))))))), s(s(z2))) -> c7(LE(s(s(s(y0))), s(s(s(s(s(s(s(s(z1)))))))))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: IF_4, LOOP_3, DOUBLE_1, LE_2 Compound Symbols: c9_1, c7_2, c9_2, c4_1, c2_1, c7_1 ---------------------------------------- (95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DOUBLE(s(s(y0))) -> c4(DOUBLE(s(y0))) by DOUBLE(s(s(s(y0)))) -> c4(DOUBLE(s(s(y0)))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false double(0) -> 0 double(s(z0)) -> s(s(double(z0))) Tuples: IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) LOOP(s(s(s(y0))), s(s(s(s(s(s(s(s(z1)))))))), s(s(z2))) -> c7(LE(s(s(s(y0))), s(s(s(s(s(s(s(s(z1)))))))))) DOUBLE(s(s(s(y0)))) -> c4(DOUBLE(s(s(y0)))) S tuples: LOOP(s(s(x0)), s(s(s(s(y0)))), s(s(x2))) -> c7(IF(le(x0, s(s(y0))), s(s(x0)), s(s(s(s(y0)))), s(s(x2))), LE(s(s(x0)), s(s(s(s(y0)))))) IF(false, s(s(z0)), s(s(s(s(z1)))), s(z2)) -> c9(LOOP(s(s(z0)), s(s(s(s(s(s(s(s(double(z1))))))))), s(s(z2))), DOUBLE(s(s(s(s(z1)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) DOUBLE(s(s(s(y0)))) -> c4(DOUBLE(s(s(y0)))) K tuples: IF(false, s(s(x0)), s(s(s(s(x1)))), s(x2)) -> c9(DOUBLE(s(s(s(s(x1)))))) LOOP(s(s(s(y0))), s(s(s(s(s(s(s(s(z1)))))))), s(s(z2))) -> c7(LE(s(s(s(y0))), s(s(s(s(s(s(s(s(z1)))))))))) Defined Rule Symbols: le_2, double_1 Defined Pair Symbols: IF_4, LOOP_3, LE_2, DOUBLE_1 Compound Symbols: c9_1, c7_2, c9_2, c2_1, c7_1, c4_1