/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 573 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__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X)))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__prod(0, X) -> 0 a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__zero(0) -> true a__zero(s(X)) -> false a__p(s(X)) -> mark(X) mark(fact(X)) -> a__fact(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(zero(X)) -> a__zero(mark(X)) mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) mark(p(X)) -> a__p(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(true) -> true mark(false) -> false a__fact(X) -> fact(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__zero(X) -> zero(X) a__prod(X1, X2) -> prod(X1, X2) a__p(X) -> p(X) a__add(X1, X2) -> add(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(prod(x_1, x_2)) -> prod(encArg(x_1), encArg(x_2)) encArg(fact(x_1)) -> fact(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(zero(x_1)) -> zero(encArg(x_1)) encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_a__fact(x_1)) -> a__fact(encArg(x_1)) encArg(cons_a__add(x_1, x_2)) -> a__add(encArg(x_1), encArg(x_2)) encArg(cons_a__prod(x_1, x_2)) -> a__prod(encArg(x_1), encArg(x_2)) encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__zero(x_1)) -> a__zero(encArg(x_1)) encArg(cons_a__p(x_1)) -> a__p(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__fact(x_1) -> a__fact(encArg(x_1)) encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a__zero(x_1) -> a__zero(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_prod(x_1, x_2) -> prod(encArg(x_1), encArg(x_2)) encode_fact(x_1) -> fact(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_a__add(x_1, x_2) -> a__add(encArg(x_1), encArg(x_2)) encode_a__prod(x_1, x_2) -> a__prod(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_a__p(x_1) -> a__p(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_zero(x_1) -> zero(encArg(x_1)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) ---------------------------------------- (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__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X)))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__prod(0, X) -> 0 a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__zero(0) -> true a__zero(s(X)) -> false a__p(s(X)) -> mark(X) mark(fact(X)) -> a__fact(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(zero(X)) -> a__zero(mark(X)) mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) mark(p(X)) -> a__p(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(true) -> true mark(false) -> false a__fact(X) -> fact(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__zero(X) -> zero(X) a__prod(X1, X2) -> prod(X1, X2) a__p(X) -> p(X) a__add(X1, X2) -> add(X1, X2) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(prod(x_1, x_2)) -> prod(encArg(x_1), encArg(x_2)) encArg(fact(x_1)) -> fact(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(zero(x_1)) -> zero(encArg(x_1)) encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_a__fact(x_1)) -> a__fact(encArg(x_1)) encArg(cons_a__add(x_1, x_2)) -> a__add(encArg(x_1), encArg(x_2)) encArg(cons_a__prod(x_1, x_2)) -> a__prod(encArg(x_1), encArg(x_2)) encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__zero(x_1)) -> a__zero(encArg(x_1)) encArg(cons_a__p(x_1)) -> a__p(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__fact(x_1) -> a__fact(encArg(x_1)) encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a__zero(x_1) -> a__zero(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_prod(x_1, x_2) -> prod(encArg(x_1), encArg(x_2)) encode_fact(x_1) -> fact(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_a__add(x_1, x_2) -> a__add(encArg(x_1), encArg(x_2)) encode_a__prod(x_1, x_2) -> a__prod(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_a__p(x_1) -> a__p(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_zero(x_1) -> zero(encArg(x_1)) encode_add(x_1, x_2) -> add(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, INF). The TRS R consists of the following rules: a__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X)))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__prod(0, X) -> 0 a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__zero(0) -> true a__zero(s(X)) -> false a__p(s(X)) -> mark(X) mark(fact(X)) -> a__fact(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(zero(X)) -> a__zero(mark(X)) mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) mark(p(X)) -> a__p(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(true) -> true mark(false) -> false a__fact(X) -> fact(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__zero(X) -> zero(X) a__prod(X1, X2) -> prod(X1, X2) a__p(X) -> p(X) a__add(X1, X2) -> add(X1, X2) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(prod(x_1, x_2)) -> prod(encArg(x_1), encArg(x_2)) encArg(fact(x_1)) -> fact(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(zero(x_1)) -> zero(encArg(x_1)) encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_a__fact(x_1)) -> a__fact(encArg(x_1)) encArg(cons_a__add(x_1, x_2)) -> a__add(encArg(x_1), encArg(x_2)) encArg(cons_a__prod(x_1, x_2)) -> a__prod(encArg(x_1), encArg(x_2)) encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__zero(x_1)) -> a__zero(encArg(x_1)) encArg(cons_a__p(x_1)) -> a__p(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__fact(x_1) -> a__fact(encArg(x_1)) encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a__zero(x_1) -> a__zero(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_prod(x_1, x_2) -> prod(encArg(x_1), encArg(x_2)) encode_fact(x_1) -> fact(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_a__add(x_1, x_2) -> a__add(encArg(x_1), encArg(x_2)) encode_a__prod(x_1, x_2) -> a__prod(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_a__p(x_1) -> a__p(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_zero(x_1) -> zero(encArg(x_1)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) 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__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X)))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__prod(0, X) -> 0 a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__zero(0) -> true a__zero(s(X)) -> false a__p(s(X)) -> mark(X) mark(fact(X)) -> a__fact(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(zero(X)) -> a__zero(mark(X)) mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) mark(p(X)) -> a__p(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(true) -> true mark(false) -> false a__fact(X) -> fact(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__zero(X) -> zero(X) a__prod(X1, X2) -> prod(X1, X2) a__p(X) -> p(X) a__add(X1, X2) -> add(X1, X2) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(prod(x_1, x_2)) -> prod(encArg(x_1), encArg(x_2)) encArg(fact(x_1)) -> fact(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(zero(x_1)) -> zero(encArg(x_1)) encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_a__fact(x_1)) -> a__fact(encArg(x_1)) encArg(cons_a__add(x_1, x_2)) -> a__add(encArg(x_1), encArg(x_2)) encArg(cons_a__prod(x_1, x_2)) -> a__prod(encArg(x_1), encArg(x_2)) encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__zero(x_1)) -> a__zero(encArg(x_1)) encArg(cons_a__p(x_1)) -> a__p(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__fact(x_1) -> a__fact(encArg(x_1)) encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a__zero(x_1) -> a__zero(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_prod(x_1, x_2) -> prod(encArg(x_1), encArg(x_2)) encode_fact(x_1) -> fact(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_a__add(x_1, x_2) -> a__add(encArg(x_1), encArg(x_2)) encode_a__prod(x_1, x_2) -> a__prod(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_a__p(x_1) -> a__p(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_zero(x_1) -> zero(encArg(x_1)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) 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(prod(X1, X2)) ->^+ a__prod(mark(X1), mark(X2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / prod(X1, X2)]. 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__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X)))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__prod(0, X) -> 0 a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__zero(0) -> true a__zero(s(X)) -> false a__p(s(X)) -> mark(X) mark(fact(X)) -> a__fact(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(zero(X)) -> a__zero(mark(X)) mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) mark(p(X)) -> a__p(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(true) -> true mark(false) -> false a__fact(X) -> fact(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__zero(X) -> zero(X) a__prod(X1, X2) -> prod(X1, X2) a__p(X) -> p(X) a__add(X1, X2) -> add(X1, X2) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(prod(x_1, x_2)) -> prod(encArg(x_1), encArg(x_2)) encArg(fact(x_1)) -> fact(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(zero(x_1)) -> zero(encArg(x_1)) encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_a__fact(x_1)) -> a__fact(encArg(x_1)) encArg(cons_a__add(x_1, x_2)) -> a__add(encArg(x_1), encArg(x_2)) encArg(cons_a__prod(x_1, x_2)) -> a__prod(encArg(x_1), encArg(x_2)) encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__zero(x_1)) -> a__zero(encArg(x_1)) encArg(cons_a__p(x_1)) -> a__p(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__fact(x_1) -> a__fact(encArg(x_1)) encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a__zero(x_1) -> a__zero(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_prod(x_1, x_2) -> prod(encArg(x_1), encArg(x_2)) encode_fact(x_1) -> fact(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_a__add(x_1, x_2) -> a__add(encArg(x_1), encArg(x_2)) encode_a__prod(x_1, x_2) -> a__prod(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_a__p(x_1) -> a__p(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_zero(x_1) -> zero(encArg(x_1)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) 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__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X)))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__prod(0, X) -> 0 a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__zero(0) -> true a__zero(s(X)) -> false a__p(s(X)) -> mark(X) mark(fact(X)) -> a__fact(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(zero(X)) -> a__zero(mark(X)) mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) mark(p(X)) -> a__p(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(true) -> true mark(false) -> false a__fact(X) -> fact(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__zero(X) -> zero(X) a__prod(X1, X2) -> prod(X1, X2) a__p(X) -> p(X) a__add(X1, X2) -> add(X1, X2) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(prod(x_1, x_2)) -> prod(encArg(x_1), encArg(x_2)) encArg(fact(x_1)) -> fact(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(zero(x_1)) -> zero(encArg(x_1)) encArg(add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_a__fact(x_1)) -> a__fact(encArg(x_1)) encArg(cons_a__add(x_1, x_2)) -> a__add(encArg(x_1), encArg(x_2)) encArg(cons_a__prod(x_1, x_2)) -> a__prod(encArg(x_1), encArg(x_2)) encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__zero(x_1)) -> a__zero(encArg(x_1)) encArg(cons_a__p(x_1)) -> a__p(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__fact(x_1) -> a__fact(encArg(x_1)) encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_a__zero(x_1) -> a__zero(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_prod(x_1, x_2) -> prod(encArg(x_1), encArg(x_2)) encode_fact(x_1) -> fact(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_a__add(x_1, x_2) -> a__add(encArg(x_1), encArg(x_2)) encode_a__prod(x_1, x_2) -> a__prod(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_a__p(x_1) -> a__p(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_zero(x_1) -> zero(encArg(x_1)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST