/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 181 ms] (4) CpxRelTRS (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, 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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) CompleteCoflocoProof [FINISHED, 2211 ms] (14) BOUNDS(1, n^2) (15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRelTRS (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (18) typed CpxTrs (19) OrderProof [LOWER BOUND(ID), 1 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 254 ms] (22) BEST (23) proven lower bound (24) LowerBoundPropagationProof [FINISHED, 0 ms] (25) BOUNDS(n^1, INF) (26) typed CpxTrs (27) RewriteLemmaProof [LOWER BOUND(ID), 63 ms] (28) typed CpxTrs (29) RewriteLemmaProof [LOWER BOUND(ID), 90 ms] (30) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] quot(0, s(y)) -> 0 [1] quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y))) [1] encArg(0) -> 0 [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0] encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) [0] encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) [0] encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) [0] encode_true -> true [0] encode_false -> false [0] encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) [0] 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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] quot(0, s(y)) -> 0 [1] quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y))) [1] encArg(0) -> 0 [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0] encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) [0] encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) [0] encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) [0] encode_true -> true [0] encode_false -> false [0] encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) [0] The TRS has the following type information: minus :: 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot 0 :: 0:s:true:false:cons_minus:cons_le:cons_quot s :: 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot le :: 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot true :: 0:s:true:false:cons_minus:cons_le:cons_quot false :: 0:s:true:false:cons_minus:cons_le:cons_quot quot :: 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot encArg :: 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot cons_minus :: 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot cons_le :: 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot cons_quot :: 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot encode_minus :: 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot encode_0 :: 0:s:true:false:cons_minus:cons_le:cons_quot encode_s :: 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot encode_le :: 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot encode_true :: 0:s:true:false:cons_minus:cons_le:cons_quot encode_false :: 0:s:true:false:cons_minus:cons_le:cons_quot encode_quot :: 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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: encArg(v0) -> null_encArg [0] encode_minus(v0, v1) -> null_encode_minus [0] encode_0 -> null_encode_0 [0] encode_s(v0) -> null_encode_s [0] encode_le(v0, v1) -> null_encode_le [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] encode_quot(v0, v1) -> null_encode_quot [0] minus(v0, v1) -> null_minus [0] le(v0, v1) -> null_le [0] quot(v0, v1) -> null_quot [0] And the following fresh constants: null_encArg, null_encode_minus, null_encode_0, null_encode_s, null_encode_le, null_encode_true, null_encode_false, null_encode_quot, null_minus, null_le, null_quot ---------------------------------------- (10) 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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] quot(0, s(y)) -> 0 [1] quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y))) [1] encArg(0) -> 0 [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0] encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) [0] encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) [0] encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) [0] encode_true -> true [0] encode_false -> false [0] encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) [0] encArg(v0) -> null_encArg [0] encode_minus(v0, v1) -> null_encode_minus [0] encode_0 -> null_encode_0 [0] encode_s(v0) -> null_encode_s [0] encode_le(v0, v1) -> null_encode_le [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] encode_quot(v0, v1) -> null_encode_quot [0] minus(v0, v1) -> null_minus [0] le(v0, v1) -> null_le [0] quot(v0, v1) -> null_quot [0] The TRS has the following type information: minus :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot 0 :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot s :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot le :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot true :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot false :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot quot :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot encArg :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot cons_minus :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot cons_le :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot cons_quot :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot encode_minus :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot encode_0 :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot encode_s :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot encode_le :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot encode_true :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot encode_false :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot encode_quot :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot -> 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot null_encArg :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot null_encode_minus :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot null_encode_0 :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot null_encode_s :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot null_encode_le :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot null_encode_true :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot null_encode_false :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot null_encode_quot :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot null_minus :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot null_le :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot null_quot :: 0:s:true:false:cons_minus:cons_le:cons_quot:null_encArg:null_encode_minus:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_quot:null_minus:null_le:null_quot Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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 => 2 false => 1 null_encArg => 0 null_encode_minus => 0 null_encode_0 => 0 null_encode_s => 0 null_encode_le => 0 null_encode_true => 0 null_encode_false => 0 null_encode_quot => 0 null_minus => 0 null_le => 0 null_quot => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 0 }-> quot(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_le(z, z') -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_minus(z, z') -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_quot(z, z') -{ 0 }-> quot(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quot(z, z') -{ 1 }-> 1 + quot(minus(1 + x, 1 + y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (13) CompleteCoflocoProof (FINISHED) Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: eq(start(V1, V),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]). eq(start(V1, V),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]). eq(start(V1, V),0,[quot(V1, V, Out)],[V1 >= 0,V >= 0]). eq(start(V1, V),0,[encArg(V1, Out)],[V1 >= 0]). eq(start(V1, V),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]). eq(start(V1, V),0,[fun1(Out)],[]). eq(start(V1, V),0,[fun2(V1, Out)],[V1 >= 0]). eq(start(V1, V),0,[fun3(V1, V, Out)],[V1 >= 0,V >= 0]). eq(start(V1, V),0,[fun4(Out)],[]). eq(start(V1, V),0,[fun5(Out)],[]). eq(start(V1, V),0,[fun6(V1, V, Out)],[V1 >= 0,V >= 0]). eq(minus(V1, V, Out),1,[],[Out = V2,V2 >= 0,V1 = V2,V = 0]). eq(minus(V1, V, Out),1,[minus(V3, V4, Ret)],[Out = Ret,V = 1 + V4,V3 >= 0,V4 >= 0,V1 = 1 + V3]). eq(le(V1, V, Out),1,[],[Out = 2,V5 >= 0,V1 = 0,V = V5]). eq(le(V1, V, Out),1,[],[Out = 1,V6 >= 0,V1 = 1 + V6,V = 0]). eq(le(V1, V, Out),1,[le(V7, V8, Ret1)],[Out = Ret1,V = 1 + V8,V7 >= 0,V8 >= 0,V1 = 1 + V7]). eq(quot(V1, V, Out),1,[],[Out = 0,V = 1 + V9,V9 >= 0,V1 = 0]). eq(quot(V1, V, Out),1,[minus(1 + V11, 1 + V10, Ret10),quot(Ret10, 1 + V10, Ret11)],[Out = 1 + Ret11,V = 1 + V10,V11 >= 0,V10 >= 0,V1 = 1 + V11]). eq(encArg(V1, Out),0,[],[Out = 0,V1 = 0]). eq(encArg(V1, Out),0,[encArg(V12, Ret12)],[Out = 1 + Ret12,V1 = 1 + V12,V12 >= 0]). eq(encArg(V1, Out),0,[],[Out = 2,V1 = 2]). eq(encArg(V1, Out),0,[],[Out = 1,V1 = 1]). eq(encArg(V1, Out),0,[encArg(V13, Ret0),encArg(V14, Ret13),minus(Ret0, Ret13, Ret2)],[Out = Ret2,V13 >= 0,V1 = 1 + V13 + V14,V14 >= 0]). eq(encArg(V1, Out),0,[encArg(V15, Ret01),encArg(V16, Ret14),le(Ret01, Ret14, Ret3)],[Out = Ret3,V15 >= 0,V1 = 1 + V15 + V16,V16 >= 0]). eq(encArg(V1, Out),0,[encArg(V18, Ret02),encArg(V17, Ret15),quot(Ret02, Ret15, Ret4)],[Out = Ret4,V18 >= 0,V1 = 1 + V17 + V18,V17 >= 0]). eq(fun(V1, V, Out),0,[encArg(V20, Ret03),encArg(V19, Ret16),minus(Ret03, Ret16, Ret5)],[Out = Ret5,V20 >= 0,V19 >= 0,V1 = V20,V = V19]). eq(fun1(Out),0,[],[Out = 0]). eq(fun2(V1, Out),0,[encArg(V21, Ret17)],[Out = 1 + Ret17,V21 >= 0,V1 = V21]). eq(fun3(V1, V, Out),0,[encArg(V22, Ret04),encArg(V23, Ret18),le(Ret04, Ret18, Ret6)],[Out = Ret6,V22 >= 0,V23 >= 0,V1 = V22,V = V23]). eq(fun4(Out),0,[],[Out = 2]). eq(fun5(Out),0,[],[Out = 1]). eq(fun6(V1, V, Out),0,[encArg(V25, Ret05),encArg(V24, Ret19),quot(Ret05, Ret19, Ret7)],[Out = Ret7,V25 >= 0,V24 >= 0,V1 = V25,V = V24]). eq(encArg(V1, Out),0,[],[Out = 0,V26 >= 0,V1 = V26]). eq(fun(V1, V, Out),0,[],[Out = 0,V28 >= 0,V27 >= 0,V1 = V28,V = V27]). eq(fun2(V1, Out),0,[],[Out = 0,V29 >= 0,V1 = V29]). eq(fun3(V1, V, Out),0,[],[Out = 0,V30 >= 0,V31 >= 0,V1 = V30,V = V31]). eq(fun4(Out),0,[],[Out = 0]). eq(fun5(Out),0,[],[Out = 0]). eq(fun6(V1, V, Out),0,[],[Out = 0,V32 >= 0,V33 >= 0,V1 = V32,V = V33]). eq(minus(V1, V, Out),0,[],[Out = 0,V34 >= 0,V35 >= 0,V1 = V34,V = V35]). eq(le(V1, V, Out),0,[],[Out = 0,V37 >= 0,V36 >= 0,V1 = V37,V = V36]). eq(quot(V1, V, Out),0,[],[Out = 0,V38 >= 0,V39 >= 0,V1 = V38,V = V39]). input_output_vars(minus(V1,V,Out),[V1,V],[Out]). input_output_vars(le(V1,V,Out),[V1,V],[Out]). input_output_vars(quot(V1,V,Out),[V1,V],[Out]). input_output_vars(encArg(V1,Out),[V1],[Out]). input_output_vars(fun(V1,V,Out),[V1,V],[Out]). input_output_vars(fun1(Out),[],[Out]). input_output_vars(fun2(V1,Out),[V1],[Out]). input_output_vars(fun3(V1,V,Out),[V1,V],[Out]). input_output_vars(fun4(Out),[],[Out]). input_output_vars(fun5(Out),[],[Out]). input_output_vars(fun6(V1,V,Out),[V1,V],[Out]). CoFloCo proof output: Preprocessing Cost Relations ===================================== #### Computed strongly connected components 0. recursive : [le/3] 1. recursive : [minus/3] 2. recursive : [quot/3] 3. recursive [non_tail,multiple] : [encArg/2] 4. non_recursive : [fun/3] 5. non_recursive : [fun1/1] 6. non_recursive : [fun2/2] 7. non_recursive : [fun3/3] 8. non_recursive : [fun4/1] 9. non_recursive : [fun5/1] 10. non_recursive : [fun6/3] 11. non_recursive : [start/2] #### Obtained direct recursion through partial evaluation 0. SCC is partially evaluated into le/3 1. SCC is partially evaluated into minus/3 2. SCC is partially evaluated into quot/3 3. SCC is partially evaluated into encArg/2 4. SCC is partially evaluated into fun/3 5. SCC is completely evaluated into other SCCs 6. SCC is partially evaluated into fun2/2 7. SCC is partially evaluated into fun3/3 8. SCC is partially evaluated into fun4/1 9. SCC is partially evaluated into fun5/1 10. SCC is partially evaluated into fun6/3 11. SCC is partially evaluated into start/2 Control-Flow Refinement of Cost Relations ===================================== ### Specialization of cost equations le/3 * CE 18 is refined into CE [41] * CE 16 is refined into CE [42] * CE 15 is refined into CE [43] * CE 17 is refined into CE [44] ### Cost equations --> "Loop" of le/3 * CEs [44] --> Loop 24 * CEs [41] --> Loop 25 * CEs [42] --> Loop 26 * CEs [43] --> Loop 27 ### Ranking functions of CR le(V1,V,Out) * RF of phase [24]: [V,V1] #### Partial ranking functions of CR le(V1,V,Out) * Partial RF of phase [24]: - RF of loop [24:1]: V V1 ### Specialization of cost equations minus/3 * CE 14 is refined into CE [45] * CE 12 is refined into CE [46] * CE 13 is refined into CE [47] ### Cost equations --> "Loop" of minus/3 * CEs [47] --> Loop 28 * CEs [45] --> Loop 29 * CEs [46] --> Loop 30 ### Ranking functions of CR minus(V1,V,Out) * RF of phase [28]: [V,V1] #### Partial ranking functions of CR minus(V1,V,Out) * Partial RF of phase [28]: - RF of loop [28:1]: V V1 ### Specialization of cost equations quot/3 * CE 19 is refined into CE [48] * CE 21 is refined into CE [49] * CE 20 is refined into CE [50,51] ### Cost equations --> "Loop" of quot/3 * CEs [51] --> Loop 31 * CEs [50] --> Loop 32 * CEs [48,49] --> Loop 33 ### Ranking functions of CR quot(V1,V,Out) * RF of phase [31]: [V1,V1-V+1] #### Partial ranking functions of CR quot(V1,V,Out) * Partial RF of phase [31]: - RF of loop [31:1]: V1 V1-V+1 ### Specialization of cost equations encArg/2 * CE 22 is refined into CE [52] * CE 24 is refined into CE [53] * CE 25 is refined into CE [54] * CE 26 is refined into CE [55,56,57] * CE 27 is refined into CE [58,59,60,61,62] * CE 28 is refined into CE [63,64,65] * CE 23 is refined into CE [66] ### Cost equations --> "Loop" of encArg/2 * CEs [66] --> Loop 34 * CEs [65] --> Loop 35 * CEs [57] --> Loop 36 * CEs [55] --> Loop 37 * CEs [62] --> Loop 38 * CEs [58] --> Loop 39 * CEs [61,64] --> Loop 40 * CEs [59] --> Loop 41 * CEs [56,60,63] --> Loop 42 * CEs [52] --> Loop 43 * CEs [53] --> Loop 44 * CEs [54] --> Loop 45 ### Ranking functions of CR encArg(V1,Out) * RF of phase [34,35,36,37,38,39,40,41,42]: [V1] #### Partial ranking functions of CR encArg(V1,Out) * Partial RF of phase [34,35,36,37,38,39,40,41,42]: - RF of loop [34:1,35:1,35:2,36:1,36:2,37:1,37:2,38:1,38:2,39:1,39:2,40:1,40:2,41:1,41:2,42:1,42:2]: V1 ### Specialization of cost equations fun/3 * CE 29 is refined into CE [67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85] * CE 30 is refined into CE [86] ### Cost equations --> "Loop" of fun/3 * CEs [71] --> Loop 46 * CEs [70,83] --> Loop 47 * CEs [76] --> Loop 48 * CEs [67,69,72,74,79] --> Loop 49 * CEs [68,73,75,77,78,80,81,82,84,85,86] --> Loop 50 ### Ranking functions of CR fun(V1,V,Out) #### Partial ranking functions of CR fun(V1,V,Out) ### Specialization of cost equations fun2/2 * CE 31 is refined into CE [87,88,89] * CE 32 is refined into CE [90] ### Cost equations --> "Loop" of fun2/2 * CEs [89] --> Loop 51 * CEs [90] --> Loop 52 * CEs [87,88] --> Loop 53 ### Ranking functions of CR fun2(V1,Out) #### Partial ranking functions of CR fun2(V1,Out) ### Specialization of cost equations fun3/3 * CE 33 is refined into CE [91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116] * CE 34 is refined into CE [117] ### Cost equations --> "Loop" of fun3/3 * CEs [96,99,113] --> Loop 54 * CEs [98] --> Loop 55 * CEs [97,114] --> Loop 56 * CEs [92,94,101,103,105,109] --> Loop 57 * CEs [91,95,100,106,108,111,115] --> Loop 58 * CEs [93,102,104,107,110,112,116,117] --> Loop 59 ### Ranking functions of CR fun3(V1,V,Out) #### Partial ranking functions of CR fun3(V1,V,Out) ### Specialization of cost equations fun4/1 * CE 35 is refined into CE [118] * CE 36 is refined into CE [119] ### Cost equations --> "Loop" of fun4/1 * CEs [118] --> Loop 60 * CEs [119] --> Loop 61 ### Ranking functions of CR fun4(Out) #### Partial ranking functions of CR fun4(Out) ### Specialization of cost equations fun5/1 * CE 37 is refined into CE [120] * CE 38 is refined into CE [121] ### Cost equations --> "Loop" of fun5/1 * CEs [120] --> Loop 62 * CEs [121] --> Loop 63 ### Ranking functions of CR fun5(Out) #### Partial ranking functions of CR fun5(Out) ### Specialization of cost equations fun6/3 * CE 39 is refined into CE [122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138] * CE 40 is refined into CE [139] ### Cost equations --> "Loop" of fun6/3 * CEs [123] --> Loop 64 * CEs [126,127] --> Loop 65 * CEs [125,137] --> Loop 66 * CEs [124,130,131,133,134] --> Loop 67 * CEs [122,128,129,132,135,136,138,139] --> Loop 68 ### Ranking functions of CR fun6(V1,V,Out) #### Partial ranking functions of CR fun6(V1,V,Out) ### Specialization of cost equations start/2 * CE 1 is refined into CE [140,141,142] * CE 2 is refined into CE [143,144,145,146,147] * CE 3 is refined into CE [148,149,150] * CE 4 is refined into CE [151,152,153] * CE 5 is refined into CE [154,155,156] * CE 6 is refined into CE [157] * CE 7 is refined into CE [158,159,160] * CE 8 is refined into CE [161,162,163] * CE 9 is refined into CE [164,165] * CE 10 is refined into CE [166,167] * CE 11 is refined into CE [168,169,170] ### Cost equations --> "Loop" of start/2 * CEs [140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170] --> Loop 69 ### Ranking functions of CR start(V1,V) #### Partial ranking functions of CR start(V1,V) Computing Bounds ===================================== #### Cost of chains of le(V1,V,Out): * Chain [[24],27]: 1*it(24)+1 Such that:it(24) =< V1 with precondition: [Out=2,V1>=1,V>=V1] * Chain [[24],26]: 1*it(24)+1 Such that:it(24) =< V with precondition: [Out=1,V>=1,V1>=V+1] * Chain [[24],25]: 1*it(24)+0 Such that:it(24) =< V with precondition: [Out=0,V1>=1,V>=1] * Chain [27]: 1 with precondition: [V1=0,Out=2,V>=0] * Chain [26]: 1 with precondition: [V=0,Out=1,V1>=1] * Chain [25]: 0 with precondition: [Out=0,V1>=0,V>=0] #### Cost of chains of minus(V1,V,Out): * Chain [[28],30]: 1*it(28)+1 Such that:it(28) =< V with precondition: [V1=Out+V,V>=1,V1>=V] * Chain [[28],29]: 1*it(28)+0 Such that:it(28) =< V with precondition: [Out=0,V1>=1,V>=1] * Chain [30]: 1 with precondition: [V=0,V1=Out,V1>=0] * Chain [29]: 0 with precondition: [Out=0,V1>=0,V>=0] #### Cost of chains of quot(V1,V,Out): * Chain [[31],33]: 2*it(31)+1*s(5)+1 Such that:it(31) =< V1-V+1 aux(3) =< V1 it(31) =< aux(3) s(5) =< aux(3) with precondition: [V>=1,Out>=1,V1+1>=Out+V] * Chain [[31],32,33]: 3*it(31)+1*s(6)+2 Such that:s(6) =< V aux(4) =< V1 it(31) =< aux(4) with precondition: [V>=1,Out>=2,V1+1>=Out+V] * Chain [33]: 1 with precondition: [Out=0,V1>=0,V>=0] * Chain [32,33]: 1*s(6)+2 Such that:s(6) =< V with precondition: [Out=1,V1>=1,V>=1] #### Cost of chains of encArg(V1,Out): * Chain [45]: 0 with precondition: [V1=1,Out=1] * Chain [44]: 0 with precondition: [V1=2,Out=2] * Chain [43]: 0 with precondition: [Out=0,V1>=0] * Chain [multiple([34,35,36,37,38,39,40,41,42],[[45],[44],[43]])]: 6*it(35)+3*it(37)+1*it(41)+7*s(31)+2*s(33)+2*s(35)+2*s(37)+0 Such that:aux(10) =< 2*V1 aux(26) =< V1 aux(27) =< 2/3*V1 aux(28) =< 4/5*V1 it(35) =< aux(26) it(37) =< aux(26) it(41) =< aux(26) it(35) =< aux(27) it(41) =< aux(28) aux(18) =< aux(10)-1 aux(12) =< aux(10)-2 s(38) =< it(37)*aux(18) s(36) =< it(35)*aux(12) s(33) =< it(35)*aux(12) s(32) =< it(35)*aux(10) s(37) =< s(38) s(35) =< s(36) s(31) =< s(32) with precondition: [V1>=1,Out>=0,2*V1>=Out] #### Cost of chains of fun(V1,V,Out): * Chain [50]: 12*s(59)+6*s(60)+2*s(61)+4*s(66)+4*s(68)+4*s(69)+14*s(70)+24*s(75)+12*s(76)+4*s(77)+8*s(82)+8*s(84)+8*s(85)+28*s(86)+3*s(87)+2*s(122)+1 Such that:aux(32) =< 2 aux(33) =< V1 aux(34) =< 2*V1 aux(35) =< 2/3*V1 aux(36) =< 4/5*V1 aux(37) =< V aux(38) =< 2*V aux(39) =< 2/3*V aux(40) =< 4/5*V s(122) =< aux(32) s(87) =< aux(38) s(75) =< aux(37) s(76) =< aux(37) s(77) =< aux(37) s(75) =< aux(39) s(77) =< aux(40) s(78) =< aux(38)-1 s(79) =< aux(38)-2 s(80) =< s(76)*s(78) s(81) =< s(75)*s(79) s(82) =< s(75)*s(79) s(83) =< s(75)*aux(38) s(84) =< s(80) s(85) =< s(81) s(86) =< s(83) s(59) =< aux(33) s(60) =< aux(33) s(61) =< aux(33) s(59) =< aux(35) s(61) =< aux(36) s(62) =< aux(34)-1 s(63) =< aux(34)-2 s(64) =< s(60)*s(62) s(65) =< s(59)*s(63) s(66) =< s(59)*s(63) s(67) =< s(59)*aux(34) s(68) =< s(64) s(69) =< s(65) s(70) =< s(67) with precondition: [Out=0,V1>=0,V>=0] * Chain [49]: 18*s(163)+9*s(164)+3*s(165)+6*s(170)+6*s(172)+6*s(173)+21*s(174)+18*s(179)+9*s(180)+3*s(181)+6*s(186)+6*s(188)+6*s(189)+21*s(190)+1*s(223)+1 Such that:aux(42) =< V1 aux(43) =< 2*V1 aux(44) =< 2/3*V1 aux(45) =< 4/5*V1 aux(46) =< V aux(47) =< 2*V aux(48) =< 2/3*V aux(49) =< 4/5*V s(179) =< aux(46) s(180) =< aux(46) s(181) =< aux(46) s(179) =< aux(48) s(181) =< aux(49) s(182) =< aux(47)-1 s(183) =< aux(47)-2 s(184) =< s(180)*s(182) s(185) =< s(179)*s(183) s(186) =< s(179)*s(183) s(187) =< s(179)*aux(47) s(188) =< s(184) s(189) =< s(185) s(190) =< s(187) s(163) =< aux(42) s(164) =< aux(42) s(165) =< aux(42) s(163) =< aux(44) s(165) =< aux(45) s(166) =< aux(43)-1 s(167) =< aux(43)-2 s(168) =< s(164)*s(166) s(169) =< s(163)*s(167) s(170) =< s(163)*s(167) s(171) =< s(163)*aux(43) s(172) =< s(168) s(173) =< s(169) s(174) =< s(171) s(223) =< aux(47) with precondition: [V1>=1,V>=0,Out>=0,2*V1>=Out] * Chain [48]: 6*s(260)+3*s(261)+1*s(262)+2*s(267)+2*s(269)+2*s(270)+7*s(271)+1*s(272)+1 Such that:s(272) =< 2 s(256) =< V s(257) =< 2*V s(258) =< 2/3*V s(259) =< 4/5*V s(260) =< s(256) s(261) =< s(256) s(262) =< s(256) s(260) =< s(258) s(262) =< s(259) s(263) =< s(257)-1 s(264) =< s(257)-2 s(265) =< s(261)*s(263) s(266) =< s(260)*s(264) s(267) =< s(260)*s(264) s(268) =< s(260)*s(257) s(269) =< s(265) s(270) =< s(266) s(271) =< s(268) with precondition: [V1=2,1>=Out,V>=1,Out>=0] * Chain [47]: 6*s(277)+3*s(278)+1*s(279)+2*s(284)+2*s(286)+2*s(287)+7*s(288)+2*s(289)+0 Such that:s(273) =< V1 s(274) =< 2*V1 s(275) =< 2/3*V1 s(276) =< 4/5*V1 aux(50) =< 2 s(289) =< aux(50) s(277) =< s(273) s(278) =< s(273) s(279) =< s(273) s(277) =< s(275) s(279) =< s(276) s(280) =< s(274)-1 s(281) =< s(274)-2 s(282) =< s(278)*s(280) s(283) =< s(277)*s(281) s(284) =< s(277)*s(281) s(285) =< s(277)*s(274) s(286) =< s(282) s(287) =< s(283) s(288) =< s(285) with precondition: [V=2,Out=0,V1>=0] * Chain [46]: 6*s(295)+3*s(296)+1*s(297)+2*s(302)+2*s(304)+2*s(305)+7*s(306)+1*s(307)+1 Such that:s(307) =< 2 s(291) =< V1 s(292) =< 2*V1 s(293) =< 2/3*V1 s(294) =< 4/5*V1 s(295) =< s(291) s(296) =< s(291) s(297) =< s(291) s(295) =< s(293) s(297) =< s(294) s(298) =< s(292)-1 s(299) =< s(292)-2 s(300) =< s(296)*s(298) s(301) =< s(295)*s(299) s(302) =< s(295)*s(299) s(303) =< s(295)*s(292) s(304) =< s(300) s(305) =< s(301) s(306) =< s(303) with precondition: [V=2,Out>=0,2*V1>=Out+2] #### Cost of chains of fun2(V1,Out): * Chain [53]: 6*s(415)+3*s(416)+1*s(417)+2*s(422)+2*s(424)+2*s(425)+7*s(426)+0 Such that:s(411) =< V1 s(412) =< 2*V1 s(413) =< 2/3*V1 s(414) =< 4/5*V1 s(415) =< s(411) s(416) =< s(411) s(417) =< s(411) s(415) =< s(413) s(417) =< s(414) s(418) =< s(412)-1 s(419) =< s(412)-2 s(420) =< s(416)*s(418) s(421) =< s(415)*s(419) s(422) =< s(415)*s(419) s(423) =< s(415)*s(412) s(424) =< s(420) s(425) =< s(421) s(426) =< s(423) with precondition: [V1>=1,Out>=1,2*V1+1>=Out] * Chain [52]: 0 with precondition: [Out=0,V1>=0] * Chain [51]: 0 with precondition: [Out=1,V1>=0] #### Cost of chains of fun3(V1,V,Out): * Chain [59]: 12*s(431)+6*s(432)+2*s(433)+4*s(438)+4*s(440)+4*s(441)+14*s(442)+18*s(447)+9*s(448)+3*s(449)+6*s(454)+6*s(456)+6*s(457)+21*s(458)+3*s(459)+1*s(494)+0 Such that:s(494) =< 2 aux(63) =< V1 aux(64) =< 2*V1 aux(65) =< 2/3*V1 aux(66) =< 4/5*V1 aux(67) =< V aux(68) =< 2*V aux(69) =< 2/3*V aux(70) =< 4/5*V s(459) =< aux(68) s(447) =< aux(67) s(448) =< aux(67) s(449) =< aux(67) s(447) =< aux(69) s(449) =< aux(70) s(450) =< aux(68)-1 s(451) =< aux(68)-2 s(452) =< s(448)*s(450) s(453) =< s(447)*s(451) s(454) =< s(447)*s(451) s(455) =< s(447)*aux(68) s(456) =< s(452) s(457) =< s(453) s(458) =< s(455) s(431) =< aux(63) s(432) =< aux(63) s(433) =< aux(63) s(431) =< aux(65) s(433) =< aux(66) s(434) =< aux(64)-1 s(435) =< aux(64)-2 s(436) =< s(432)*s(434) s(437) =< s(431)*s(435) s(438) =< s(431)*s(435) s(439) =< s(431)*aux(64) s(440) =< s(436) s(441) =< s(437) s(442) =< s(439) with precondition: [Out=0,V1>=0,V>=0] * Chain [58]: 18*s(518)+9*s(519)+3*s(520)+6*s(525)+6*s(527)+6*s(528)+21*s(529)+24*s(534)+12*s(535)+4*s(536)+8*s(541)+8*s(543)+8*s(544)+28*s(545)+1*s(578)+2*s(611)+1 Such that:aux(72) =< 2 aux(73) =< V1 aux(74) =< 2*V1 aux(75) =< 2/3*V1 aux(76) =< 4/5*V1 aux(77) =< V aux(78) =< 2*V aux(79) =< 2/3*V aux(80) =< 4/5*V s(611) =< aux(72) s(534) =< aux(77) s(535) =< aux(77) s(536) =< aux(77) s(534) =< aux(79) s(536) =< aux(80) s(537) =< aux(78)-1 s(538) =< aux(78)-2 s(539) =< s(535)*s(537) s(540) =< s(534)*s(538) s(541) =< s(534)*s(538) s(542) =< s(534)*aux(78) s(543) =< s(539) s(544) =< s(540) s(545) =< s(542) s(518) =< aux(73) s(519) =< aux(73) s(520) =< aux(73) s(518) =< aux(75) s(520) =< aux(76) s(521) =< aux(74)-1 s(522) =< aux(74)-2 s(523) =< s(519)*s(521) s(524) =< s(518)*s(522) s(525) =< s(518)*s(522) s(526) =< s(518)*aux(74) s(527) =< s(523) s(528) =< s(524) s(529) =< s(526) s(578) =< aux(78) with precondition: [Out=2,V1>=0,V>=0] * Chain [57]: 18*s(633)+9*s(634)+3*s(635)+6*s(640)+6*s(642)+6*s(643)+21*s(644)+24*s(649)+12*s(650)+4*s(651)+8*s(656)+8*s(658)+8*s(659)+28*s(660)+1*s(693)+1*s(742)+1 Such that:s(742) =< 1 aux(82) =< V1 aux(83) =< 2*V1 aux(84) =< 2/3*V1 aux(85) =< 4/5*V1 aux(86) =< V aux(87) =< 2*V aux(88) =< 2/3*V aux(89) =< 4/5*V s(649) =< aux(86) s(650) =< aux(86) s(651) =< aux(86) s(649) =< aux(88) s(651) =< aux(89) s(652) =< aux(87)-1 s(653) =< aux(87)-2 s(654) =< s(650)*s(652) s(655) =< s(649)*s(653) s(656) =< s(649)*s(653) s(657) =< s(649)*aux(87) s(658) =< s(654) s(659) =< s(655) s(660) =< s(657) s(633) =< aux(82) s(634) =< aux(82) s(635) =< aux(82) s(633) =< aux(84) s(635) =< aux(85) s(636) =< aux(83)-1 s(637) =< aux(83)-2 s(638) =< s(634)*s(636) s(639) =< s(633)*s(637) s(640) =< s(633)*s(637) s(641) =< s(633)*aux(83) s(642) =< s(638) s(643) =< s(639) s(644) =< s(641) s(693) =< aux(83) with precondition: [Out=1,V1>=1,V>=0] * Chain [56]: 6*s(747)+3*s(748)+1*s(749)+2*s(754)+2*s(756)+2*s(757)+7*s(758)+2*s(759)+0 Such that:s(743) =< V1 s(744) =< 2*V1 s(745) =< 2/3*V1 s(746) =< 4/5*V1 aux(90) =< 2 s(759) =< aux(90) s(747) =< s(743) s(748) =< s(743) s(749) =< s(743) s(747) =< s(745) s(749) =< s(746) s(750) =< s(744)-1 s(751) =< s(744)-2 s(752) =< s(748)*s(750) s(753) =< s(747)*s(751) s(754) =< s(747)*s(751) s(755) =< s(747)*s(744) s(756) =< s(752) s(757) =< s(753) s(758) =< s(755) with precondition: [V=2,Out=0,V1>=0] * Chain [55]: 6*s(765)+3*s(766)+1*s(767)+2*s(772)+2*s(774)+2*s(775)+7*s(776)+1*s(777)+1 Such that:s(777) =< 2 s(761) =< V1 s(762) =< 2*V1 s(763) =< 2/3*V1 s(764) =< 4/5*V1 s(765) =< s(761) s(766) =< s(761) s(767) =< s(761) s(765) =< s(763) s(767) =< s(764) s(768) =< s(762)-1 s(769) =< s(762)-2 s(770) =< s(766)*s(768) s(771) =< s(765)*s(769) s(772) =< s(765)*s(769) s(773) =< s(765)*s(762) s(774) =< s(770) s(775) =< s(771) s(776) =< s(773) with precondition: [V=2,Out=1,2*V1>=3] * Chain [54]: 12*s(782)+6*s(783)+2*s(784)+4*s(789)+4*s(791)+4*s(792)+14*s(793)+1*s(810)+1 Such that:s(810) =< 2 aux(91) =< V1 aux(92) =< 2*V1 aux(93) =< 2/3*V1 aux(94) =< 4/5*V1 s(782) =< aux(91) s(783) =< aux(91) s(784) =< aux(91) s(782) =< aux(93) s(784) =< aux(94) s(785) =< aux(92)-1 s(786) =< aux(92)-2 s(787) =< s(783)*s(785) s(788) =< s(782)*s(786) s(789) =< s(782)*s(786) s(790) =< s(782)*aux(92) s(791) =< s(787) s(792) =< s(788) s(793) =< s(790) with precondition: [V=2,Out=2,V1>=0] #### Cost of chains of fun4(Out): * Chain [61]: 0 with precondition: [Out=0] * Chain [60]: 0 with precondition: [Out=2] #### Cost of chains of fun5(Out): * Chain [63]: 0 with precondition: [Out=0] * Chain [62]: 0 with precondition: [Out=1] #### Cost of chains of fun6(V1,V,Out): * Chain [68]: 12*s(970)+6*s(971)+2*s(972)+4*s(977)+4*s(979)+4*s(980)+14*s(981)+18*s(986)+9*s(987)+3*s(988)+6*s(993)+6*s(995)+6*s(996)+21*s(997)+1 Such that:aux(109) =< V1 aux(110) =< 2*V1 aux(111) =< 2/3*V1 aux(112) =< 4/5*V1 aux(113) =< V aux(114) =< 2*V aux(115) =< 2/3*V aux(116) =< 4/5*V s(986) =< aux(113) s(987) =< aux(113) s(988) =< aux(113) s(986) =< aux(115) s(988) =< aux(116) s(989) =< aux(114)-1 s(990) =< aux(114)-2 s(991) =< s(987)*s(989) s(992) =< s(986)*s(990) s(993) =< s(986)*s(990) s(994) =< s(986)*aux(114) s(995) =< s(991) s(996) =< s(992) s(997) =< s(994) s(970) =< aux(109) s(971) =< aux(109) s(972) =< aux(109) s(970) =< aux(111) s(972) =< aux(112) s(973) =< aux(110)-1 s(974) =< aux(110)-2 s(975) =< s(971)*s(973) s(976) =< s(970)*s(974) s(977) =< s(970)*s(974) s(978) =< s(970)*aux(110) s(979) =< s(975) s(980) =< s(976) s(981) =< s(978) with precondition: [Out=0,V1>=0,V>=0] * Chain [67]: 6*s(1050)+3*s(1051)+1*s(1052)+2*s(1057)+2*s(1059)+2*s(1060)+7*s(1061)+18*s(1066)+9*s(1067)+3*s(1068)+6*s(1073)+6*s(1075)+6*s(1076)+21*s(1077)+2*s(1078)+1*s(1079)+4*s(1081)+1*s(1098)+13*s(1115)+2*s(1120)+2 Such that:s(1120) =< 1 s(1046) =< V1 aux(117) =< 2*V1 aux(118) =< 2*V1+1 s(1048) =< 2/3*V1 s(1049) =< 4/5*V1 aux(122) =< 2 aux(123) =< V aux(124) =< 2*V aux(125) =< 2/3*V aux(126) =< 4/5*V s(1115) =< aux(122) s(1120) =< aux(122) s(1078) =< aux(118) s(1079) =< aux(118) s(1078) =< aux(117) s(1081) =< aux(117) s(1066) =< aux(123) s(1067) =< aux(123) s(1068) =< aux(123) s(1066) =< aux(125) s(1068) =< aux(126) s(1069) =< aux(124)-1 s(1070) =< aux(124)-2 s(1071) =< s(1067)*s(1069) s(1072) =< s(1066)*s(1070) s(1073) =< s(1066)*s(1070) s(1074) =< s(1066)*aux(124) s(1075) =< s(1071) s(1076) =< s(1072) s(1077) =< s(1074) s(1050) =< s(1046) s(1051) =< s(1046) s(1052) =< s(1046) s(1050) =< s(1048) s(1052) =< s(1049) s(1053) =< aux(117)-1 s(1054) =< aux(117)-2 s(1055) =< s(1051)*s(1053) s(1056) =< s(1050)*s(1054) s(1057) =< s(1050)*s(1054) s(1058) =< s(1050)*aux(117) s(1059) =< s(1055) s(1060) =< s(1056) s(1061) =< s(1058) s(1098) =< aux(124) with precondition: [V1>=1,V>=1,Out>=1,2*V1>=Out] * Chain [66]: 6*s(1128)+3*s(1129)+1*s(1130)+2*s(1135)+2*s(1137)+2*s(1138)+7*s(1139)+1 Such that:s(1124) =< V1 s(1125) =< 2*V1 s(1126) =< 2/3*V1 s(1127) =< 4/5*V1 s(1128) =< s(1124) s(1129) =< s(1124) s(1130) =< s(1124) s(1128) =< s(1126) s(1130) =< s(1127) s(1131) =< s(1125)-1 s(1132) =< s(1125)-2 s(1133) =< s(1129)*s(1131) s(1134) =< s(1128)*s(1132) s(1135) =< s(1128)*s(1132) s(1136) =< s(1128)*s(1125) s(1137) =< s(1133) s(1138) =< s(1134) s(1139) =< s(1136) with precondition: [V=2,Out=0,V1>=0] * Chain [65]: 12*s(1144)+6*s(1145)+2*s(1146)+4*s(1151)+4*s(1153)+4*s(1154)+14*s(1155)+2*s(1156)+6*s(1173)+2 Such that:aux(128) =< 2 aux(129) =< V1 aux(130) =< 2*V1 aux(131) =< 2/3*V1 aux(132) =< 4/5*V1 s(1156) =< aux(128) s(1144) =< aux(129) s(1145) =< aux(129) s(1146) =< aux(129) s(1144) =< aux(131) s(1146) =< aux(132) s(1147) =< aux(130)-1 s(1148) =< aux(130)-2 s(1149) =< s(1145)*s(1147) s(1150) =< s(1144)*s(1148) s(1151) =< s(1144)*s(1148) s(1152) =< s(1144)*aux(130) s(1153) =< s(1149) s(1154) =< s(1150) s(1155) =< s(1152) s(1173) =< aux(130) with precondition: [V=2,Out>=1,2*V1>=Out+1] * Chain [64]: 6*s(1181)+3*s(1182)+1*s(1183)+2*s(1188)+2*s(1190)+2*s(1191)+7*s(1192)+6*s(1197)+3*s(1198)+1*s(1199)+2*s(1204)+2*s(1206)+2*s(1207)+7*s(1208)+1*s(1209)+2 Such that:s(1177) =< V1 s(1178) =< 2*V1 s(1179) =< 2/3*V1 s(1180) =< 4/5*V1 s(1193) =< V s(1195) =< 2/3*V s(1196) =< 4/5*V aux(133) =< 2*V s(1209) =< aux(133) s(1197) =< s(1193) s(1198) =< s(1193) s(1199) =< s(1193) s(1197) =< s(1195) s(1199) =< s(1196) s(1200) =< aux(133)-1 s(1201) =< aux(133)-2 s(1202) =< s(1198)*s(1200) s(1203) =< s(1197)*s(1201) s(1204) =< s(1197)*s(1201) s(1205) =< s(1197)*aux(133) s(1206) =< s(1202) s(1207) =< s(1203) s(1208) =< s(1205) s(1181) =< s(1177) s(1182) =< s(1177) s(1183) =< s(1177) s(1181) =< s(1179) s(1183) =< s(1180) s(1184) =< s(1178)-1 s(1185) =< s(1178)-2 s(1186) =< s(1182)*s(1184) s(1187) =< s(1181)*s(1185) s(1188) =< s(1181)*s(1185) s(1189) =< s(1181)*s(1178) s(1190) =< s(1186) s(1191) =< s(1187) s(1192) =< s(1189) with precondition: [Out=1,V1>=1,V>=1] #### Cost of chains of start(V1,V): * Chain [69]: 84*s(1317)+89*s(1321)+2*s(1323)+168*s(1331)+28*s(1333)+56*s(1338)+56*s(1340)+56*s(1341)+196*s(1342)+28*s(1343)+156*s(1352)+26*s(1354)+52*s(1359)+52*s(1361)+52*s(1362)+182*s(1363)+10*s(1376)+1*s(1480)+11*s(1514)+2*s(1582)+2*s(1594)+1*s(1595)+2 Such that:s(1323) =< V1-V+1 s(1583) =< 2*V1+1 aux(143) =< 1 aux(144) =< 2 aux(145) =< V1 aux(146) =< 2*V1 aux(147) =< 2/3*V1 aux(148) =< 4/5*V1 aux(149) =< V aux(150) =< 2*V aux(151) =< 2/3*V aux(152) =< 4/5*V s(1480) =< aux(143) s(1582) =< aux(143) s(1343) =< aux(144) s(1321) =< aux(145) s(1317) =< aux(149) s(1352) =< aux(149) s(1354) =< aux(149) s(1352) =< aux(151) s(1354) =< aux(152) s(1355) =< aux(150)-1 s(1356) =< aux(150)-2 s(1357) =< s(1317)*s(1355) s(1358) =< s(1352)*s(1356) s(1359) =< s(1352)*s(1356) s(1360) =< s(1352)*aux(150) s(1361) =< s(1357) s(1362) =< s(1358) s(1363) =< s(1360) s(1331) =< aux(145) s(1333) =< aux(145) s(1331) =< aux(147) s(1333) =< aux(148) s(1334) =< aux(146)-1 s(1335) =< aux(146)-2 s(1336) =< s(1321)*s(1334) s(1337) =< s(1331)*s(1335) s(1338) =< s(1331)*s(1335) s(1339) =< s(1331)*aux(146) s(1340) =< s(1336) s(1341) =< s(1337) s(1342) =< s(1339) s(1514) =< aux(146) s(1582) =< aux(144) s(1594) =< s(1583) s(1595) =< s(1583) s(1594) =< aux(146) s(1376) =< aux(150) s(1323) =< aux(145) with precondition: [] Closed-form bounds of start(V1,V): ------------------------------------- * Chain [69] with precondition: [] - Upper bound: nat(V1)*285+61+nat(V1)*112*nat(nat(2*V1)+ -2)+nat(V1)*56*nat(nat(2*V1)+ -1)+nat(V1)*196*nat(2*V1)+nat(V)*266+nat(V)*104*nat(nat(2*V)+ -2)+nat(V)*52*nat(nat(2*V)+ -1)+nat(V)*182*nat(2*V)+nat(2*V1)*11+nat(2*V)*10+nat(2*V1+1)*3+nat(V1-V+1)*2 - Complexity: n^2 ### Maximum cost of start(V1,V): nat(V1)*285+61+nat(V1)*112*nat(nat(2*V1)+ -2)+nat(V1)*56*nat(nat(2*V1)+ -1)+nat(V1)*196*nat(2*V1)+nat(V)*266+nat(V)*104*nat(nat(2*V)+ -2)+nat(V)*52*nat(nat(2*V)+ -1)+nat(V)*182*nat(2*V)+nat(2*V1)*11+nat(2*V)*10+nat(2*V1+1)*3+nat(V1-V+1)*2 Asymptotic class: n^2 * Total analysis performed in 1969 ms. ---------------------------------------- (14) BOUNDS(1, n^2) ---------------------------------------- (15) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (16) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) quot(0', s(y)) -> 0' quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y))) The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (17) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (18) Obligation: Innermost TRS: Rules: minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) quot(0', s(y)) -> 0' quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) Types: minus :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot 0' :: 0':s:true:false:cons_minus:cons_le:cons_quot s :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot le :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot true :: 0':s:true:false:cons_minus:cons_le:cons_quot false :: 0':s:true:false:cons_minus:cons_le:cons_quot quot :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encArg :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot cons_minus :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot cons_le :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot cons_quot :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_minus :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_0 :: 0':s:true:false:cons_minus:cons_le:cons_quot encode_s :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_le :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_true :: 0':s:true:false:cons_minus:cons_le:cons_quot encode_false :: 0':s:true:false:cons_minus:cons_le:cons_quot encode_quot :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot hole_0':s:true:false:cons_minus:cons_le:cons_quot1_3 :: 0':s:true:false:cons_minus:cons_le:cons_quot gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3 :: Nat -> 0':s:true:false:cons_minus:cons_le:cons_quot ---------------------------------------- (19) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: minus, le, quot, encArg They will be analysed ascendingly in the following order: minus < quot minus < encArg le < encArg quot < encArg ---------------------------------------- (20) Obligation: Innermost TRS: Rules: minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) quot(0', s(y)) -> 0' quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) Types: minus :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot 0' :: 0':s:true:false:cons_minus:cons_le:cons_quot s :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot le :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot true :: 0':s:true:false:cons_minus:cons_le:cons_quot false :: 0':s:true:false:cons_minus:cons_le:cons_quot quot :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encArg :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot cons_minus :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot cons_le :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot cons_quot :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_minus :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_0 :: 0':s:true:false:cons_minus:cons_le:cons_quot encode_s :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_le :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_true :: 0':s:true:false:cons_minus:cons_le:cons_quot encode_false :: 0':s:true:false:cons_minus:cons_le:cons_quot encode_quot :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot hole_0':s:true:false:cons_minus:cons_le:cons_quot1_3 :: 0':s:true:false:cons_minus:cons_le:cons_quot gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3 :: Nat -> 0':s:true:false:cons_minus:cons_le:cons_quot Generator Equations: gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(0) <=> 0' gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(+(x, 1)) <=> s(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(x)) The following defined symbols remain to be analysed: minus, le, quot, encArg They will be analysed ascendingly in the following order: minus < quot minus < encArg le < encArg quot < encArg ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: minus(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n4_3), gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n4_3)) -> gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(0), rt in Omega(1 + n4_3) Induction Base: minus(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(0), gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(0)) ->_R^Omega(1) gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(0) Induction Step: minus(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(+(n4_3, 1)), gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(+(n4_3, 1))) ->_R^Omega(1) minus(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n4_3), gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n4_3)) ->_IH gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (22) Complex Obligation (BEST) ---------------------------------------- (23) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) quot(0', s(y)) -> 0' quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) Types: minus :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot 0' :: 0':s:true:false:cons_minus:cons_le:cons_quot s :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot le :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot true :: 0':s:true:false:cons_minus:cons_le:cons_quot false :: 0':s:true:false:cons_minus:cons_le:cons_quot quot :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encArg :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot cons_minus :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot cons_le :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot cons_quot :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_minus :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_0 :: 0':s:true:false:cons_minus:cons_le:cons_quot encode_s :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_le :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_true :: 0':s:true:false:cons_minus:cons_le:cons_quot encode_false :: 0':s:true:false:cons_minus:cons_le:cons_quot encode_quot :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot hole_0':s:true:false:cons_minus:cons_le:cons_quot1_3 :: 0':s:true:false:cons_minus:cons_le:cons_quot gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3 :: Nat -> 0':s:true:false:cons_minus:cons_le:cons_quot Generator Equations: gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(0) <=> 0' gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(+(x, 1)) <=> s(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(x)) The following defined symbols remain to be analysed: minus, le, quot, encArg They will be analysed ascendingly in the following order: minus < quot minus < encArg le < encArg quot < encArg ---------------------------------------- (24) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (25) BOUNDS(n^1, INF) ---------------------------------------- (26) Obligation: Innermost TRS: Rules: minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) quot(0', s(y)) -> 0' quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) Types: minus :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot 0' :: 0':s:true:false:cons_minus:cons_le:cons_quot s :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot le :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot true :: 0':s:true:false:cons_minus:cons_le:cons_quot false :: 0':s:true:false:cons_minus:cons_le:cons_quot quot :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encArg :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot cons_minus :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot cons_le :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot cons_quot :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_minus :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_0 :: 0':s:true:false:cons_minus:cons_le:cons_quot encode_s :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_le :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_true :: 0':s:true:false:cons_minus:cons_le:cons_quot encode_false :: 0':s:true:false:cons_minus:cons_le:cons_quot encode_quot :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot hole_0':s:true:false:cons_minus:cons_le:cons_quot1_3 :: 0':s:true:false:cons_minus:cons_le:cons_quot gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3 :: Nat -> 0':s:true:false:cons_minus:cons_le:cons_quot Lemmas: minus(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n4_3), gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n4_3)) -> gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(0), rt in Omega(1 + n4_3) Generator Equations: gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(0) <=> 0' gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(+(x, 1)) <=> s(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(x)) The following defined symbols remain to be analysed: le, quot, encArg They will be analysed ascendingly in the following order: le < encArg quot < encArg ---------------------------------------- (27) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: le(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n504_3), gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n504_3)) -> true, rt in Omega(1 + n504_3) Induction Base: le(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(0), gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(0)) ->_R^Omega(1) true Induction Step: le(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(+(n504_3, 1)), gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(+(n504_3, 1))) ->_R^Omega(1) le(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n504_3), gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n504_3)) ->_IH true We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (28) Obligation: Innermost TRS: Rules: minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) quot(0', s(y)) -> 0' quot(s(x), s(y)) -> s(quot(minus(s(x), s(y)), s(y))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2)) Types: minus :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot 0' :: 0':s:true:false:cons_minus:cons_le:cons_quot s :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot le :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot true :: 0':s:true:false:cons_minus:cons_le:cons_quot false :: 0':s:true:false:cons_minus:cons_le:cons_quot quot :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encArg :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot cons_minus :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot cons_le :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot cons_quot :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_minus :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_0 :: 0':s:true:false:cons_minus:cons_le:cons_quot encode_s :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_le :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot encode_true :: 0':s:true:false:cons_minus:cons_le:cons_quot encode_false :: 0':s:true:false:cons_minus:cons_le:cons_quot encode_quot :: 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot -> 0':s:true:false:cons_minus:cons_le:cons_quot hole_0':s:true:false:cons_minus:cons_le:cons_quot1_3 :: 0':s:true:false:cons_minus:cons_le:cons_quot gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3 :: Nat -> 0':s:true:false:cons_minus:cons_le:cons_quot Lemmas: minus(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n4_3), gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n4_3)) -> gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(0), rt in Omega(1 + n4_3) le(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n504_3), gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n504_3)) -> true, rt in Omega(1 + n504_3) Generator Equations: gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(0) <=> 0' gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(+(x, 1)) <=> s(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(x)) The following defined symbols remain to be analysed: quot, encArg They will be analysed ascendingly in the following order: quot < encArg ---------------------------------------- (29) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n1081_3)) -> gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n1081_3), rt in Omega(0) Induction Base: encArg(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(+(n1081_3, 1))) ->_R^Omega(0) s(encArg(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(n1081_3))) ->_IH s(gen_0':s:true:false:cons_minus:cons_le:cons_quot2_3(c1082_3)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (30) BOUNDS(1, INF)