/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), ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 559 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: a__primes -> a__sieve(a__from(s(s(0)))) a__from(X) -> cons(mark(X), from(s(X))) a__head(cons(X, Y)) -> mark(X) a__tail(cons(X, Y)) -> mark(Y) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))) a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) mark(primes) -> a__primes mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(true) -> true mark(false) -> false mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) a__primes -> primes a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__filter(X1, X2) -> filter(X1, X2) 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(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(divides(x_1, x_2)) -> divides(encArg(x_1), encArg(x_2)) encArg(filter(x_1, x_2)) -> filter(encArg(x_1), encArg(x_2)) encArg(sieve(x_1)) -> sieve(encArg(x_1)) encArg(primes) -> primes encArg(head(x_1)) -> head(encArg(x_1)) encArg(tail(x_1)) -> tail(encArg(x_1)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__primes) -> a__primes encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__head(x_1)) -> a__head(encArg(x_1)) encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1)) encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__filter(x_1, x_2)) -> a__filter(encArg(x_1), encArg(x_2)) encArg(cons_a__sieve(x_1)) -> a__sieve(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__primes -> a__primes encode_a__sieve(x_1) -> a__sieve(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_a__head(x_1) -> a__head(encArg(x_1)) encode_a__tail(x_1) -> a__tail(encArg(x_1)) encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_false -> false encode_a__filter(x_1, x_2) -> a__filter(encArg(x_1), encArg(x_2)) encode_divides(x_1, x_2) -> divides(encArg(x_1), encArg(x_2)) encode_filter(x_1, x_2) -> filter(encArg(x_1), encArg(x_2)) encode_sieve(x_1) -> sieve(encArg(x_1)) encode_primes -> primes encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) ---------------------------------------- (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: a__primes -> a__sieve(a__from(s(s(0)))) a__from(X) -> cons(mark(X), from(s(X))) a__head(cons(X, Y)) -> mark(X) a__tail(cons(X, Y)) -> mark(Y) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))) a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) mark(primes) -> a__primes mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(true) -> true mark(false) -> false mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) a__primes -> primes a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__filter(X1, X2) -> filter(X1, X2) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(divides(x_1, x_2)) -> divides(encArg(x_1), encArg(x_2)) encArg(filter(x_1, x_2)) -> filter(encArg(x_1), encArg(x_2)) encArg(sieve(x_1)) -> sieve(encArg(x_1)) encArg(primes) -> primes encArg(head(x_1)) -> head(encArg(x_1)) encArg(tail(x_1)) -> tail(encArg(x_1)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__primes) -> a__primes encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__head(x_1)) -> a__head(encArg(x_1)) encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1)) encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__filter(x_1, x_2)) -> a__filter(encArg(x_1), encArg(x_2)) encArg(cons_a__sieve(x_1)) -> a__sieve(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__primes -> a__primes encode_a__sieve(x_1) -> a__sieve(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_a__head(x_1) -> a__head(encArg(x_1)) encode_a__tail(x_1) -> a__tail(encArg(x_1)) encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_false -> false encode_a__filter(x_1, x_2) -> a__filter(encArg(x_1), encArg(x_2)) encode_divides(x_1, x_2) -> divides(encArg(x_1), encArg(x_2)) encode_filter(x_1, x_2) -> filter(encArg(x_1), encArg(x_2)) encode_sieve(x_1) -> sieve(encArg(x_1)) encode_primes -> primes encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) 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: a__primes -> a__sieve(a__from(s(s(0)))) a__from(X) -> cons(mark(X), from(s(X))) a__head(cons(X, Y)) -> mark(X) a__tail(cons(X, Y)) -> mark(Y) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))) a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) mark(primes) -> a__primes mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(true) -> true mark(false) -> false mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) a__primes -> primes a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__filter(X1, X2) -> filter(X1, X2) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(divides(x_1, x_2)) -> divides(encArg(x_1), encArg(x_2)) encArg(filter(x_1, x_2)) -> filter(encArg(x_1), encArg(x_2)) encArg(sieve(x_1)) -> sieve(encArg(x_1)) encArg(primes) -> primes encArg(head(x_1)) -> head(encArg(x_1)) encArg(tail(x_1)) -> tail(encArg(x_1)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__primes) -> a__primes encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__head(x_1)) -> a__head(encArg(x_1)) encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1)) encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__filter(x_1, x_2)) -> a__filter(encArg(x_1), encArg(x_2)) encArg(cons_a__sieve(x_1)) -> a__sieve(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__primes -> a__primes encode_a__sieve(x_1) -> a__sieve(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_a__head(x_1) -> a__head(encArg(x_1)) encode_a__tail(x_1) -> a__tail(encArg(x_1)) encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_false -> false encode_a__filter(x_1, x_2) -> a__filter(encArg(x_1), encArg(x_2)) encode_divides(x_1, x_2) -> divides(encArg(x_1), encArg(x_2)) encode_filter(x_1, x_2) -> filter(encArg(x_1), encArg(x_2)) encode_sieve(x_1) -> sieve(encArg(x_1)) encode_primes -> primes encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) 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: a__primes -> a__sieve(a__from(s(s(0)))) a__from(X) -> cons(mark(X), from(s(X))) a__head(cons(X, Y)) -> mark(X) a__tail(cons(X, Y)) -> mark(Y) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))) a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) mark(primes) -> a__primes mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(true) -> true mark(false) -> false mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) a__primes -> primes a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__filter(X1, X2) -> filter(X1, X2) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(divides(x_1, x_2)) -> divides(encArg(x_1), encArg(x_2)) encArg(filter(x_1, x_2)) -> filter(encArg(x_1), encArg(x_2)) encArg(sieve(x_1)) -> sieve(encArg(x_1)) encArg(primes) -> primes encArg(head(x_1)) -> head(encArg(x_1)) encArg(tail(x_1)) -> tail(encArg(x_1)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__primes) -> a__primes encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__head(x_1)) -> a__head(encArg(x_1)) encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1)) encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__filter(x_1, x_2)) -> a__filter(encArg(x_1), encArg(x_2)) encArg(cons_a__sieve(x_1)) -> a__sieve(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__primes -> a__primes encode_a__sieve(x_1) -> a__sieve(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_a__head(x_1) -> a__head(encArg(x_1)) encode_a__tail(x_1) -> a__tail(encArg(x_1)) encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_false -> false encode_a__filter(x_1, x_2) -> a__filter(encArg(x_1), encArg(x_2)) encode_divides(x_1, x_2) -> divides(encArg(x_1), encArg(x_2)) encode_filter(x_1, x_2) -> filter(encArg(x_1), encArg(x_2)) encode_sieve(x_1) -> sieve(encArg(x_1)) encode_primes -> primes encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) 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 mark(from(X)) ->^+ a__from(mark(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / from(X)]. 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: a__primes -> a__sieve(a__from(s(s(0)))) a__from(X) -> cons(mark(X), from(s(X))) a__head(cons(X, Y)) -> mark(X) a__tail(cons(X, Y)) -> mark(Y) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))) a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) mark(primes) -> a__primes mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(true) -> true mark(false) -> false mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) a__primes -> primes a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__filter(X1, X2) -> filter(X1, X2) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(divides(x_1, x_2)) -> divides(encArg(x_1), encArg(x_2)) encArg(filter(x_1, x_2)) -> filter(encArg(x_1), encArg(x_2)) encArg(sieve(x_1)) -> sieve(encArg(x_1)) encArg(primes) -> primes encArg(head(x_1)) -> head(encArg(x_1)) encArg(tail(x_1)) -> tail(encArg(x_1)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__primes) -> a__primes encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__head(x_1)) -> a__head(encArg(x_1)) encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1)) encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__filter(x_1, x_2)) -> a__filter(encArg(x_1), encArg(x_2)) encArg(cons_a__sieve(x_1)) -> a__sieve(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__primes -> a__primes encode_a__sieve(x_1) -> a__sieve(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_a__head(x_1) -> a__head(encArg(x_1)) encode_a__tail(x_1) -> a__tail(encArg(x_1)) encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_false -> false encode_a__filter(x_1, x_2) -> a__filter(encArg(x_1), encArg(x_2)) encode_divides(x_1, x_2) -> divides(encArg(x_1), encArg(x_2)) encode_filter(x_1, x_2) -> filter(encArg(x_1), encArg(x_2)) encode_sieve(x_1) -> sieve(encArg(x_1)) encode_primes -> primes encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) 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: a__primes -> a__sieve(a__from(s(s(0)))) a__from(X) -> cons(mark(X), from(s(X))) a__head(cons(X, Y)) -> mark(X) a__tail(cons(X, Y)) -> mark(Y) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))) a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) mark(primes) -> a__primes mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(true) -> true mark(false) -> false mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) a__primes -> primes a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__filter(X1, X2) -> filter(X1, X2) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(divides(x_1, x_2)) -> divides(encArg(x_1), encArg(x_2)) encArg(filter(x_1, x_2)) -> filter(encArg(x_1), encArg(x_2)) encArg(sieve(x_1)) -> sieve(encArg(x_1)) encArg(primes) -> primes encArg(head(x_1)) -> head(encArg(x_1)) encArg(tail(x_1)) -> tail(encArg(x_1)) encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__primes) -> a__primes encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__head(x_1)) -> a__head(encArg(x_1)) encArg(cons_a__tail(x_1)) -> a__tail(encArg(x_1)) encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__filter(x_1, x_2)) -> a__filter(encArg(x_1), encArg(x_2)) encArg(cons_a__sieve(x_1)) -> a__sieve(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__primes -> a__primes encode_a__sieve(x_1) -> a__sieve(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_a__head(x_1) -> a__head(encArg(x_1)) encode_a__tail(x_1) -> a__tail(encArg(x_1)) encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_true -> true encode_false -> false encode_a__filter(x_1, x_2) -> a__filter(encArg(x_1), encArg(x_2)) encode_divides(x_1, x_2) -> divides(encArg(x_1), encArg(x_2)) encode_filter(x_1, x_2) -> filter(encArg(x_1), encArg(x_2)) encode_sieve(x_1) -> sieve(encArg(x_1)) encode_primes -> primes encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST