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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 672 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: fstsplit(0, x) -> nil fstsplit(s(n), nil) -> nil fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t)) sndsplit(0, x) -> x sndsplit(s(n), nil) -> nil sndsplit(s(n), cons(h, t)) -> sndsplit(n, t) empty(nil) -> true empty(cons(h, t)) -> false leq(0, m) -> true leq(s(n), 0) -> false leq(s(n), s(m)) -> leq(n, m) length(nil) -> 0 length(cons(h, t)) -> s(length(t)) app(nil, x) -> x app(cons(h, t), x) -> cons(h, app(t, x)) map_f(pid, nil) -> nil map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t)) process(store, m) -> if1(store, m, leq(m, length(store))) if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store))) if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store)))) if2(store, m, false) -> process(app(map_f(self, nil), sndsplit(m, store)), m) if3(store, m, false) -> process(sndsplit(m, app(map_f(self, nil), store)), m) 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(nil) -> nil encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(true) -> true encArg(false) -> false encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(self) -> self encArg(cons_fstsplit(x_1, x_2)) -> fstsplit(encArg(x_1), encArg(x_2)) encArg(cons_sndsplit(x_1, x_2)) -> sndsplit(encArg(x_1), encArg(x_2)) encArg(cons_empty(x_1)) -> empty(encArg(x_1)) encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_map_f(x_1, x_2)) -> map_f(encArg(x_1), encArg(x_2)) encArg(cons_process(x_1, x_2)) -> process(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if3(x_1, x_2, x_3)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_fstsplit(x_1, x_2) -> fstsplit(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sndsplit(x_1, x_2) -> sndsplit(encArg(x_1), encArg(x_2)) encode_empty(x_1) -> empty(encArg(x_1)) encode_true -> true encode_false -> false encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_map_f(x_1, x_2) -> map_f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_process(x_1, x_2) -> process(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if3(x_1, x_2, x_3) -> if3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_self -> self ---------------------------------------- (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: fstsplit(0, x) -> nil fstsplit(s(n), nil) -> nil fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t)) sndsplit(0, x) -> x sndsplit(s(n), nil) -> nil sndsplit(s(n), cons(h, t)) -> sndsplit(n, t) empty(nil) -> true empty(cons(h, t)) -> false leq(0, m) -> true leq(s(n), 0) -> false leq(s(n), s(m)) -> leq(n, m) length(nil) -> 0 length(cons(h, t)) -> s(length(t)) app(nil, x) -> x app(cons(h, t), x) -> cons(h, app(t, x)) map_f(pid, nil) -> nil map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t)) process(store, m) -> if1(store, m, leq(m, length(store))) if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store))) if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store)))) if2(store, m, false) -> process(app(map_f(self, nil), sndsplit(m, store)), m) if3(store, m, false) -> process(sndsplit(m, app(map_f(self, nil), store)), m) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(nil) -> nil encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(true) -> true encArg(false) -> false encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(self) -> self encArg(cons_fstsplit(x_1, x_2)) -> fstsplit(encArg(x_1), encArg(x_2)) encArg(cons_sndsplit(x_1, x_2)) -> sndsplit(encArg(x_1), encArg(x_2)) encArg(cons_empty(x_1)) -> empty(encArg(x_1)) encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_map_f(x_1, x_2)) -> map_f(encArg(x_1), encArg(x_2)) encArg(cons_process(x_1, x_2)) -> process(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if3(x_1, x_2, x_3)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_fstsplit(x_1, x_2) -> fstsplit(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sndsplit(x_1, x_2) -> sndsplit(encArg(x_1), encArg(x_2)) encode_empty(x_1) -> empty(encArg(x_1)) encode_true -> true encode_false -> false encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_map_f(x_1, x_2) -> map_f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_process(x_1, x_2) -> process(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if3(x_1, x_2, x_3) -> if3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_self -> self 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: fstsplit(0, x) -> nil fstsplit(s(n), nil) -> nil fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t)) sndsplit(0, x) -> x sndsplit(s(n), nil) -> nil sndsplit(s(n), cons(h, t)) -> sndsplit(n, t) empty(nil) -> true empty(cons(h, t)) -> false leq(0, m) -> true leq(s(n), 0) -> false leq(s(n), s(m)) -> leq(n, m) length(nil) -> 0 length(cons(h, t)) -> s(length(t)) app(nil, x) -> x app(cons(h, t), x) -> cons(h, app(t, x)) map_f(pid, nil) -> nil map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t)) process(store, m) -> if1(store, m, leq(m, length(store))) if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store))) if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store)))) if2(store, m, false) -> process(app(map_f(self, nil), sndsplit(m, store)), m) if3(store, m, false) -> process(sndsplit(m, app(map_f(self, nil), store)), m) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(nil) -> nil encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(true) -> true encArg(false) -> false encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(self) -> self encArg(cons_fstsplit(x_1, x_2)) -> fstsplit(encArg(x_1), encArg(x_2)) encArg(cons_sndsplit(x_1, x_2)) -> sndsplit(encArg(x_1), encArg(x_2)) encArg(cons_empty(x_1)) -> empty(encArg(x_1)) encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_map_f(x_1, x_2)) -> map_f(encArg(x_1), encArg(x_2)) encArg(cons_process(x_1, x_2)) -> process(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if3(x_1, x_2, x_3)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_fstsplit(x_1, x_2) -> fstsplit(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sndsplit(x_1, x_2) -> sndsplit(encArg(x_1), encArg(x_2)) encode_empty(x_1) -> empty(encArg(x_1)) encode_true -> true encode_false -> false encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_map_f(x_1, x_2) -> map_f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_process(x_1, x_2) -> process(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if3(x_1, x_2, x_3) -> if3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_self -> self 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: fstsplit(0, x) -> nil fstsplit(s(n), nil) -> nil fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t)) sndsplit(0, x) -> x sndsplit(s(n), nil) -> nil sndsplit(s(n), cons(h, t)) -> sndsplit(n, t) empty(nil) -> true empty(cons(h, t)) -> false leq(0, m) -> true leq(s(n), 0) -> false leq(s(n), s(m)) -> leq(n, m) length(nil) -> 0 length(cons(h, t)) -> s(length(t)) app(nil, x) -> x app(cons(h, t), x) -> cons(h, app(t, x)) map_f(pid, nil) -> nil map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t)) process(store, m) -> if1(store, m, leq(m, length(store))) if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store))) if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store)))) if2(store, m, false) -> process(app(map_f(self, nil), sndsplit(m, store)), m) if3(store, m, false) -> process(sndsplit(m, app(map_f(self, nil), store)), m) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(nil) -> nil encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(true) -> true encArg(false) -> false encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(self) -> self encArg(cons_fstsplit(x_1, x_2)) -> fstsplit(encArg(x_1), encArg(x_2)) encArg(cons_sndsplit(x_1, x_2)) -> sndsplit(encArg(x_1), encArg(x_2)) encArg(cons_empty(x_1)) -> empty(encArg(x_1)) encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_map_f(x_1, x_2)) -> map_f(encArg(x_1), encArg(x_2)) encArg(cons_process(x_1, x_2)) -> process(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if3(x_1, x_2, x_3)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_fstsplit(x_1, x_2) -> fstsplit(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sndsplit(x_1, x_2) -> sndsplit(encArg(x_1), encArg(x_2)) encode_empty(x_1) -> empty(encArg(x_1)) encode_true -> true encode_false -> false encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_map_f(x_1, x_2) -> map_f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_process(x_1, x_2) -> process(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if3(x_1, x_2, x_3) -> if3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_self -> self 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 map_f(pid, cons(h, t)) ->^+ app(f(pid, h), map_f(pid, t)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [t / cons(h, t)]. 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: fstsplit(0, x) -> nil fstsplit(s(n), nil) -> nil fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t)) sndsplit(0, x) -> x sndsplit(s(n), nil) -> nil sndsplit(s(n), cons(h, t)) -> sndsplit(n, t) empty(nil) -> true empty(cons(h, t)) -> false leq(0, m) -> true leq(s(n), 0) -> false leq(s(n), s(m)) -> leq(n, m) length(nil) -> 0 length(cons(h, t)) -> s(length(t)) app(nil, x) -> x app(cons(h, t), x) -> cons(h, app(t, x)) map_f(pid, nil) -> nil map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t)) process(store, m) -> if1(store, m, leq(m, length(store))) if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store))) if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store)))) if2(store, m, false) -> process(app(map_f(self, nil), sndsplit(m, store)), m) if3(store, m, false) -> process(sndsplit(m, app(map_f(self, nil), store)), m) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(nil) -> nil encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(true) -> true encArg(false) -> false encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(self) -> self encArg(cons_fstsplit(x_1, x_2)) -> fstsplit(encArg(x_1), encArg(x_2)) encArg(cons_sndsplit(x_1, x_2)) -> sndsplit(encArg(x_1), encArg(x_2)) encArg(cons_empty(x_1)) -> empty(encArg(x_1)) encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_map_f(x_1, x_2)) -> map_f(encArg(x_1), encArg(x_2)) encArg(cons_process(x_1, x_2)) -> process(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if3(x_1, x_2, x_3)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_fstsplit(x_1, x_2) -> fstsplit(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sndsplit(x_1, x_2) -> sndsplit(encArg(x_1), encArg(x_2)) encode_empty(x_1) -> empty(encArg(x_1)) encode_true -> true encode_false -> false encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_map_f(x_1, x_2) -> map_f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_process(x_1, x_2) -> process(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if3(x_1, x_2, x_3) -> if3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_self -> self 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: fstsplit(0, x) -> nil fstsplit(s(n), nil) -> nil fstsplit(s(n), cons(h, t)) -> cons(h, fstsplit(n, t)) sndsplit(0, x) -> x sndsplit(s(n), nil) -> nil sndsplit(s(n), cons(h, t)) -> sndsplit(n, t) empty(nil) -> true empty(cons(h, t)) -> false leq(0, m) -> true leq(s(n), 0) -> false leq(s(n), s(m)) -> leq(n, m) length(nil) -> 0 length(cons(h, t)) -> s(length(t)) app(nil, x) -> x app(cons(h, t), x) -> cons(h, app(t, x)) map_f(pid, nil) -> nil map_f(pid, cons(h, t)) -> app(f(pid, h), map_f(pid, t)) process(store, m) -> if1(store, m, leq(m, length(store))) if1(store, m, true) -> if2(store, m, empty(fstsplit(m, store))) if1(store, m, false) -> if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store)))) if2(store, m, false) -> process(app(map_f(self, nil), sndsplit(m, store)), m) if3(store, m, false) -> process(sndsplit(m, app(map_f(self, nil), store)), m) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(nil) -> nil encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(true) -> true encArg(false) -> false encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(self) -> self encArg(cons_fstsplit(x_1, x_2)) -> fstsplit(encArg(x_1), encArg(x_2)) encArg(cons_sndsplit(x_1, x_2)) -> sndsplit(encArg(x_1), encArg(x_2)) encArg(cons_empty(x_1)) -> empty(encArg(x_1)) encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2)) encArg(cons_length(x_1)) -> length(encArg(x_1)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_map_f(x_1, x_2)) -> map_f(encArg(x_1), encArg(x_2)) encArg(cons_process(x_1, x_2)) -> process(encArg(x_1), encArg(x_2)) encArg(cons_if1(x_1, x_2, x_3)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if3(x_1, x_2, x_3)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_fstsplit(x_1, x_2) -> fstsplit(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sndsplit(x_1, x_2) -> sndsplit(encArg(x_1), encArg(x_2)) encode_empty(x_1) -> empty(encArg(x_1)) encode_true -> true encode_false -> false encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2)) encode_length(x_1) -> length(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_map_f(x_1, x_2) -> map_f(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_process(x_1, x_2) -> process(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if3(x_1, x_2, x_3) -> if3(encArg(x_1), encArg(x_2), encArg(x_3)) encode_self -> self Rewrite Strategy: INNERMOST