/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 762 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: gcd(x, y) -> gcd2(x, y, 0) gcd2(x, y, i) -> if1(le(x, 0), le(y, 0), le(x, y), le(y, x), x, y, inc(i)) if1(true, b1, b2, b3, x, y, i) -> pair(result(y), neededIterations(i)) if1(false, b1, b2, b3, x, y, i) -> if2(b1, b2, b3, x, y, i) if2(true, b2, b3, x, y, i) -> pair(result(x), neededIterations(i)) if2(false, b2, b3, x, y, i) -> if3(b2, b3, x, y, i) if3(false, b3, x, y, i) -> gcd2(minus(x, y), y, i) if3(true, b3, x, y, i) -> if4(b3, x, y, i) if4(false, x, y, i) -> gcd2(x, minus(y, x), i) if4(true, x, y, i) -> pair(result(x), neededIterations(i)) inc(0) -> 0 inc(s(i)) -> s(inc(i)) le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) minus(x, 0) -> x minus(0, y) -> 0 minus(s(x), s(y)) -> minus(x, y) a -> b a -> c 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(true) -> true encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(result(x_1)) -> result(encArg(x_1)) encArg(neededIterations(x_1)) -> neededIterations(encArg(x_1)) encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(b) -> b encArg(c) -> c encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_gcd2(x_1, x_2, x_3)) -> gcd2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5, x_6)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if3(x_1, x_2, x_3, x_4, x_5)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if4(x_1, x_2, x_3, x_4)) -> if4(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_a) -> a encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_gcd2(x_1, x_2, x_3) -> gcd2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_if1(x_1, x_2, x_3, x_4, x_5, x_6, x_7) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_inc(x_1) -> inc(encArg(x_1)) encode_true -> true encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_result(x_1) -> result(encArg(x_1)) encode_neededIterations(x_1) -> neededIterations(encArg(x_1)) encode_false -> false encode_if2(x_1, x_2, x_3, x_4, x_5, x_6) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if3(x_1, x_2, x_3, x_4, x_5) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if4(x_1, x_2, x_3, x_4) -> if4(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s(x_1) -> s(encArg(x_1)) encode_a -> a encode_b -> b encode_c -> c ---------------------------------------- (2) 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: gcd(x, y) -> gcd2(x, y, 0) gcd2(x, y, i) -> if1(le(x, 0), le(y, 0), le(x, y), le(y, x), x, y, inc(i)) if1(true, b1, b2, b3, x, y, i) -> pair(result(y), neededIterations(i)) if1(false, b1, b2, b3, x, y, i) -> if2(b1, b2, b3, x, y, i) if2(true, b2, b3, x, y, i) -> pair(result(x), neededIterations(i)) if2(false, b2, b3, x, y, i) -> if3(b2, b3, x, y, i) if3(false, b3, x, y, i) -> gcd2(minus(x, y), y, i) if3(true, b3, x, y, i) -> if4(b3, x, y, i) if4(false, x, y, i) -> gcd2(x, minus(y, x), i) if4(true, x, y, i) -> pair(result(x), neededIterations(i)) inc(0) -> 0 inc(s(i)) -> s(inc(i)) le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) minus(x, 0) -> x minus(0, y) -> 0 minus(s(x), s(y)) -> minus(x, y) a -> b a -> c The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(result(x_1)) -> result(encArg(x_1)) encArg(neededIterations(x_1)) -> neededIterations(encArg(x_1)) encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(b) -> b encArg(c) -> c encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_gcd2(x_1, x_2, x_3)) -> gcd2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5, x_6)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if3(x_1, x_2, x_3, x_4, x_5)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if4(x_1, x_2, x_3, x_4)) -> if4(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_a) -> a encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_gcd2(x_1, x_2, x_3) -> gcd2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_if1(x_1, x_2, x_3, x_4, x_5, x_6, x_7) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_inc(x_1) -> inc(encArg(x_1)) encode_true -> true encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_result(x_1) -> result(encArg(x_1)) encode_neededIterations(x_1) -> neededIterations(encArg(x_1)) encode_false -> false encode_if2(x_1, x_2, x_3, x_4, x_5, x_6) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if3(x_1, x_2, x_3, x_4, x_5) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if4(x_1, x_2, x_3, x_4) -> if4(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s(x_1) -> s(encArg(x_1)) encode_a -> a encode_b -> b encode_c -> c 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, INF). The TRS R consists of the following rules: gcd(x, y) -> gcd2(x, y, 0) gcd2(x, y, i) -> if1(le(x, 0), le(y, 0), le(x, y), le(y, x), x, y, inc(i)) if1(true, b1, b2, b3, x, y, i) -> pair(result(y), neededIterations(i)) if1(false, b1, b2, b3, x, y, i) -> if2(b1, b2, b3, x, y, i) if2(true, b2, b3, x, y, i) -> pair(result(x), neededIterations(i)) if2(false, b2, b3, x, y, i) -> if3(b2, b3, x, y, i) if3(false, b3, x, y, i) -> gcd2(minus(x, y), y, i) if3(true, b3, x, y, i) -> if4(b3, x, y, i) if4(false, x, y, i) -> gcd2(x, minus(y, x), i) if4(true, x, y, i) -> pair(result(x), neededIterations(i)) inc(0) -> 0 inc(s(i)) -> s(inc(i)) le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) minus(x, 0) -> x minus(0, y) -> 0 minus(s(x), s(y)) -> minus(x, y) a -> b a -> c The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(result(x_1)) -> result(encArg(x_1)) encArg(neededIterations(x_1)) -> neededIterations(encArg(x_1)) encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(b) -> b encArg(c) -> c encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_gcd2(x_1, x_2, x_3)) -> gcd2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5, x_6)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if3(x_1, x_2, x_3, x_4, x_5)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if4(x_1, x_2, x_3, x_4)) -> if4(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_a) -> a encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_gcd2(x_1, x_2, x_3) -> gcd2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_if1(x_1, x_2, x_3, x_4, x_5, x_6, x_7) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_inc(x_1) -> inc(encArg(x_1)) encode_true -> true encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_result(x_1) -> result(encArg(x_1)) encode_neededIterations(x_1) -> neededIterations(encArg(x_1)) encode_false -> false encode_if2(x_1, x_2, x_3, x_4, x_5, x_6) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if3(x_1, x_2, x_3, x_4, x_5) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if4(x_1, x_2, x_3, x_4) -> if4(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s(x_1) -> s(encArg(x_1)) encode_a -> a encode_b -> b encode_c -> c Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: 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: gcd(x, y) -> gcd2(x, y, 0) gcd2(x, y, i) -> if1(le(x, 0), le(y, 0), le(x, y), le(y, x), x, y, inc(i)) if1(true, b1, b2, b3, x, y, i) -> pair(result(y), neededIterations(i)) if1(false, b1, b2, b3, x, y, i) -> if2(b1, b2, b3, x, y, i) if2(true, b2, b3, x, y, i) -> pair(result(x), neededIterations(i)) if2(false, b2, b3, x, y, i) -> if3(b2, b3, x, y, i) if3(false, b3, x, y, i) -> gcd2(minus(x, y), y, i) if3(true, b3, x, y, i) -> if4(b3, x, y, i) if4(false, x, y, i) -> gcd2(x, minus(y, x), i) if4(true, x, y, i) -> pair(result(x), neededIterations(i)) inc(0) -> 0 inc(s(i)) -> s(inc(i)) le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) minus(x, 0) -> x minus(0, y) -> 0 minus(s(x), s(y)) -> minus(x, y) a -> b a -> c The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(result(x_1)) -> result(encArg(x_1)) encArg(neededIterations(x_1)) -> neededIterations(encArg(x_1)) encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(b) -> b encArg(c) -> c encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_gcd2(x_1, x_2, x_3)) -> gcd2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5, x_6)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if3(x_1, x_2, x_3, x_4, x_5)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if4(x_1, x_2, x_3, x_4)) -> if4(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_a) -> a encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_gcd2(x_1, x_2, x_3) -> gcd2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_if1(x_1, x_2, x_3, x_4, x_5, x_6, x_7) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_inc(x_1) -> inc(encArg(x_1)) encode_true -> true encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_result(x_1) -> result(encArg(x_1)) encode_neededIterations(x_1) -> neededIterations(encArg(x_1)) encode_false -> false encode_if2(x_1, x_2, x_3, x_4, x_5, x_6) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if3(x_1, x_2, x_3, x_4, x_5) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if4(x_1, x_2, x_3, x_4) -> if4(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s(x_1) -> s(encArg(x_1)) encode_a -> a encode_b -> b encode_c -> c Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence le(s(x), s(y)) ->^+ le(x, y) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [x / s(x), y / s(y)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following 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: gcd(x, y) -> gcd2(x, y, 0) gcd2(x, y, i) -> if1(le(x, 0), le(y, 0), le(x, y), le(y, x), x, y, inc(i)) if1(true, b1, b2, b3, x, y, i) -> pair(result(y), neededIterations(i)) if1(false, b1, b2, b3, x, y, i) -> if2(b1, b2, b3, x, y, i) if2(true, b2, b3, x, y, i) -> pair(result(x), neededIterations(i)) if2(false, b2, b3, x, y, i) -> if3(b2, b3, x, y, i) if3(false, b3, x, y, i) -> gcd2(minus(x, y), y, i) if3(true, b3, x, y, i) -> if4(b3, x, y, i) if4(false, x, y, i) -> gcd2(x, minus(y, x), i) if4(true, x, y, i) -> pair(result(x), neededIterations(i)) inc(0) -> 0 inc(s(i)) -> s(inc(i)) le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) minus(x, 0) -> x minus(0, y) -> 0 minus(s(x), s(y)) -> minus(x, y) a -> b a -> c The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(result(x_1)) -> result(encArg(x_1)) encArg(neededIterations(x_1)) -> neededIterations(encArg(x_1)) encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(b) -> b encArg(c) -> c encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_gcd2(x_1, x_2, x_3)) -> gcd2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5, x_6)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if3(x_1, x_2, x_3, x_4, x_5)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if4(x_1, x_2, x_3, x_4)) -> if4(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_a) -> a encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_gcd2(x_1, x_2, x_3) -> gcd2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_if1(x_1, x_2, x_3, x_4, x_5, x_6, x_7) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_inc(x_1) -> inc(encArg(x_1)) encode_true -> true encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_result(x_1) -> result(encArg(x_1)) encode_neededIterations(x_1) -> neededIterations(encArg(x_1)) encode_false -> false encode_if2(x_1, x_2, x_3, x_4, x_5, x_6) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if3(x_1, x_2, x_3, x_4, x_5) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if4(x_1, x_2, x_3, x_4) -> if4(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s(x_1) -> s(encArg(x_1)) encode_a -> a encode_b -> b encode_c -> c Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: 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: gcd(x, y) -> gcd2(x, y, 0) gcd2(x, y, i) -> if1(le(x, 0), le(y, 0), le(x, y), le(y, x), x, y, inc(i)) if1(true, b1, b2, b3, x, y, i) -> pair(result(y), neededIterations(i)) if1(false, b1, b2, b3, x, y, i) -> if2(b1, b2, b3, x, y, i) if2(true, b2, b3, x, y, i) -> pair(result(x), neededIterations(i)) if2(false, b2, b3, x, y, i) -> if3(b2, b3, x, y, i) if3(false, b3, x, y, i) -> gcd2(minus(x, y), y, i) if3(true, b3, x, y, i) -> if4(b3, x, y, i) if4(false, x, y, i) -> gcd2(x, minus(y, x), i) if4(true, x, y, i) -> pair(result(x), neededIterations(i)) inc(0) -> 0 inc(s(i)) -> s(inc(i)) le(s(x), 0) -> false le(0, y) -> true le(s(x), s(y)) -> le(x, y) minus(x, 0) -> x minus(0, y) -> 0 minus(s(x), s(y)) -> minus(x, y) a -> b a -> c The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(result(x_1)) -> result(encArg(x_1)) encArg(neededIterations(x_1)) -> neededIterations(encArg(x_1)) encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(b) -> b encArg(c) -> c encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_gcd2(x_1, x_2, x_3)) -> gcd2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if1(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5, x_6)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if3(x_1, x_2, x_3, x_4, x_5)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if4(x_1, x_2, x_3, x_4)) -> if4(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_a) -> a encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_gcd2(x_1, x_2, x_3) -> gcd2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_if1(x_1, x_2, x_3, x_4, x_5, x_6, x_7) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_inc(x_1) -> inc(encArg(x_1)) encode_true -> true encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_result(x_1) -> result(encArg(x_1)) encode_neededIterations(x_1) -> neededIterations(encArg(x_1)) encode_false -> false encode_if2(x_1, x_2, x_3, x_4, x_5, x_6) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if3(x_1, x_2, x_3, x_4, x_5) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if4(x_1, x_2, x_3, x_4) -> if4(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s(x_1) -> s(encArg(x_1)) encode_a -> a encode_b -> b encode_c -> c Rewrite Strategy: INNERMOST