/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 644 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: isEmpty(cons(x, xs)) -> false isEmpty(nil) -> true isZero(0) -> true isZero(s(x)) -> false head(cons(x, xs)) -> x tail(cons(x, xs)) -> xs tail(nil) -> nil append(nil, x) -> cons(x, nil) append(cons(y, ys), x) -> cons(y, append(ys, x)) p(s(s(x))) -> s(p(s(x))) p(s(0)) -> 0 p(0) -> 0 inc(s(x)) -> s(inc(x)) inc(0) -> s(0) addLists(xs, ys, zs) -> if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys))) if(true, true, b, xs, ys, xs2, ys2, zs, zs2) -> zs if(true, false, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError if(false, true, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError if(false, false, false, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs2, ys2, zs) if(false, false, true, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs, ys, zs2) addList(xs, ys) -> addLists(xs, ys, nil) 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(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(nil) -> nil encArg(true) -> true encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(differentLengthError) -> differentLengthError encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1)) encArg(cons_isZero(x_1)) -> isZero(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_addLists(x_1, x_2, x_3)) -> addLists(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7), encArg(x_8), encArg(x_9)) encArg(cons_addList(x_1, x_2)) -> addList(encArg(x_1), encArg(x_2)) encode_isEmpty(x_1) -> isEmpty(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_nil -> nil encode_true -> true encode_isZero(x_1) -> isZero(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_inc(x_1) -> inc(encArg(x_1)) encode_addLists(x_1, x_2, x_3) -> addLists(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7), encArg(x_8), encArg(x_9)) encode_differentLengthError -> differentLengthError encode_addList(x_1, x_2) -> addList(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: isEmpty(cons(x, xs)) -> false isEmpty(nil) -> true isZero(0) -> true isZero(s(x)) -> false head(cons(x, xs)) -> x tail(cons(x, xs)) -> xs tail(nil) -> nil append(nil, x) -> cons(x, nil) append(cons(y, ys), x) -> cons(y, append(ys, x)) p(s(s(x))) -> s(p(s(x))) p(s(0)) -> 0 p(0) -> 0 inc(s(x)) -> s(inc(x)) inc(0) -> s(0) addLists(xs, ys, zs) -> if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys))) if(true, true, b, xs, ys, xs2, ys2, zs, zs2) -> zs if(true, false, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError if(false, true, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError if(false, false, false, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs2, ys2, zs) if(false, false, true, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs, ys, zs2) addList(xs, ys) -> addLists(xs, ys, nil) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(nil) -> nil encArg(true) -> true encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(differentLengthError) -> differentLengthError encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1)) encArg(cons_isZero(x_1)) -> isZero(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_addLists(x_1, x_2, x_3)) -> addLists(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7), encArg(x_8), encArg(x_9)) encArg(cons_addList(x_1, x_2)) -> addList(encArg(x_1), encArg(x_2)) encode_isEmpty(x_1) -> isEmpty(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_nil -> nil encode_true -> true encode_isZero(x_1) -> isZero(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_inc(x_1) -> inc(encArg(x_1)) encode_addLists(x_1, x_2, x_3) -> addLists(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7), encArg(x_8), encArg(x_9)) encode_differentLengthError -> differentLengthError encode_addList(x_1, x_2) -> addList(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: isEmpty(cons(x, xs)) -> false isEmpty(nil) -> true isZero(0) -> true isZero(s(x)) -> false head(cons(x, xs)) -> x tail(cons(x, xs)) -> xs tail(nil) -> nil append(nil, x) -> cons(x, nil) append(cons(y, ys), x) -> cons(y, append(ys, x)) p(s(s(x))) -> s(p(s(x))) p(s(0)) -> 0 p(0) -> 0 inc(s(x)) -> s(inc(x)) inc(0) -> s(0) addLists(xs, ys, zs) -> if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys))) if(true, true, b, xs, ys, xs2, ys2, zs, zs2) -> zs if(true, false, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError if(false, true, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError if(false, false, false, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs2, ys2, zs) if(false, false, true, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs, ys, zs2) addList(xs, ys) -> addLists(xs, ys, nil) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(nil) -> nil encArg(true) -> true encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(differentLengthError) -> differentLengthError encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1)) encArg(cons_isZero(x_1)) -> isZero(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_addLists(x_1, x_2, x_3)) -> addLists(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7), encArg(x_8), encArg(x_9)) encArg(cons_addList(x_1, x_2)) -> addList(encArg(x_1), encArg(x_2)) encode_isEmpty(x_1) -> isEmpty(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_nil -> nil encode_true -> true encode_isZero(x_1) -> isZero(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_inc(x_1) -> inc(encArg(x_1)) encode_addLists(x_1, x_2, x_3) -> addLists(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7), encArg(x_8), encArg(x_9)) encode_differentLengthError -> differentLengthError encode_addList(x_1, x_2) -> addList(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: isEmpty(cons(x, xs)) -> false isEmpty(nil) -> true isZero(0) -> true isZero(s(x)) -> false head(cons(x, xs)) -> x tail(cons(x, xs)) -> xs tail(nil) -> nil append(nil, x) -> cons(x, nil) append(cons(y, ys), x) -> cons(y, append(ys, x)) p(s(s(x))) -> s(p(s(x))) p(s(0)) -> 0 p(0) -> 0 inc(s(x)) -> s(inc(x)) inc(0) -> s(0) addLists(xs, ys, zs) -> if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys))) if(true, true, b, xs, ys, xs2, ys2, zs, zs2) -> zs if(true, false, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError if(false, true, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError if(false, false, false, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs2, ys2, zs) if(false, false, true, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs, ys, zs2) addList(xs, ys) -> addLists(xs, ys, nil) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(nil) -> nil encArg(true) -> true encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(differentLengthError) -> differentLengthError encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1)) encArg(cons_isZero(x_1)) -> isZero(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_addLists(x_1, x_2, x_3)) -> addLists(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7), encArg(x_8), encArg(x_9)) encArg(cons_addList(x_1, x_2)) -> addList(encArg(x_1), encArg(x_2)) encode_isEmpty(x_1) -> isEmpty(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_nil -> nil encode_true -> true encode_isZero(x_1) -> isZero(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_inc(x_1) -> inc(encArg(x_1)) encode_addLists(x_1, x_2, x_3) -> addLists(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7), encArg(x_8), encArg(x_9)) encode_differentLengthError -> differentLengthError encode_addList(x_1, x_2) -> addList(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 append(cons(y, ys), x) ->^+ cons(y, append(ys, x)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [ys / cons(y, ys)]. 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: isEmpty(cons(x, xs)) -> false isEmpty(nil) -> true isZero(0) -> true isZero(s(x)) -> false head(cons(x, xs)) -> x tail(cons(x, xs)) -> xs tail(nil) -> nil append(nil, x) -> cons(x, nil) append(cons(y, ys), x) -> cons(y, append(ys, x)) p(s(s(x))) -> s(p(s(x))) p(s(0)) -> 0 p(0) -> 0 inc(s(x)) -> s(inc(x)) inc(0) -> s(0) addLists(xs, ys, zs) -> if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys))) if(true, true, b, xs, ys, xs2, ys2, zs, zs2) -> zs if(true, false, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError if(false, true, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError if(false, false, false, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs2, ys2, zs) if(false, false, true, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs, ys, zs2) addList(xs, ys) -> addLists(xs, ys, nil) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(nil) -> nil encArg(true) -> true encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(differentLengthError) -> differentLengthError encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1)) encArg(cons_isZero(x_1)) -> isZero(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_addLists(x_1, x_2, x_3)) -> addLists(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7), encArg(x_8), encArg(x_9)) encArg(cons_addList(x_1, x_2)) -> addList(encArg(x_1), encArg(x_2)) encode_isEmpty(x_1) -> isEmpty(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_nil -> nil encode_true -> true encode_isZero(x_1) -> isZero(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_inc(x_1) -> inc(encArg(x_1)) encode_addLists(x_1, x_2, x_3) -> addLists(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7), encArg(x_8), encArg(x_9)) encode_differentLengthError -> differentLengthError encode_addList(x_1, x_2) -> addList(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: isEmpty(cons(x, xs)) -> false isEmpty(nil) -> true isZero(0) -> true isZero(s(x)) -> false head(cons(x, xs)) -> x tail(cons(x, xs)) -> xs tail(nil) -> nil append(nil, x) -> cons(x, nil) append(cons(y, ys), x) -> cons(y, append(ys, x)) p(s(s(x))) -> s(p(s(x))) p(s(0)) -> 0 p(0) -> 0 inc(s(x)) -> s(inc(x)) inc(0) -> s(0) addLists(xs, ys, zs) -> if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys))) if(true, true, b, xs, ys, xs2, ys2, zs, zs2) -> zs if(true, false, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError if(false, true, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError if(false, false, false, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs2, ys2, zs) if(false, false, true, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs, ys, zs2) addList(xs, ys) -> addLists(xs, ys, nil) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(nil) -> nil encArg(true) -> true encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(differentLengthError) -> differentLengthError encArg(cons_isEmpty(x_1)) -> isEmpty(encArg(x_1)) encArg(cons_isZero(x_1)) -> isZero(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_inc(x_1)) -> inc(encArg(x_1)) encArg(cons_addLists(x_1, x_2, x_3)) -> addLists(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7), encArg(x_8), encArg(x_9)) encArg(cons_addList(x_1, x_2)) -> addList(encArg(x_1), encArg(x_2)) encode_isEmpty(x_1) -> isEmpty(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_nil -> nil encode_true -> true encode_isZero(x_1) -> isZero(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_inc(x_1) -> inc(encArg(x_1)) encode_addLists(x_1, x_2, x_3) -> addLists(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6), encArg(x_7), encArg(x_8), encArg(x_9)) encode_differentLengthError -> differentLengthError encode_addList(x_1, x_2) -> addList(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST