/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 1 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 335 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: active(fib(N)) -> mark(sel(N, fib1(s(0), s(0)))) active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y)))) active(add(0, X)) -> mark(X) active(add(s(X), Y)) -> mark(s(add(X, Y))) active(sel(0, cons(X, XS))) -> mark(X) active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) active(fib(X)) -> fib(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(fib1(X1, X2)) -> fib1(active(X1), X2) active(fib1(X1, X2)) -> fib1(X1, active(X2)) active(s(X)) -> s(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(add(X1, X2)) -> add(active(X1), X2) active(add(X1, X2)) -> add(X1, active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) fib1(mark(X1), X2) -> mark(fib1(X1, X2)) fib1(X1, mark(X2)) -> mark(fib1(X1, X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) add(mark(X1), X2) -> mark(add(X1, X2)) add(X1, mark(X2)) -> mark(add(X1, X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(0) -> ok(0) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(add(X1, X2)) -> add(proper(X1), proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) add(ok(X1), ok(X2)) -> ok(add(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) 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(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_fib(x_1)) -> fib(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_fib1(x_1, x_2)) -> fib1(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_fib(x_1) -> fib(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_fib1(x_1, x_2) -> fib1(encArg(x_1), encArg(x_2)) 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_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) ---------------------------------------- (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: active(fib(N)) -> mark(sel(N, fib1(s(0), s(0)))) active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y)))) active(add(0, X)) -> mark(X) active(add(s(X), Y)) -> mark(s(add(X, Y))) active(sel(0, cons(X, XS))) -> mark(X) active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) active(fib(X)) -> fib(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(fib1(X1, X2)) -> fib1(active(X1), X2) active(fib1(X1, X2)) -> fib1(X1, active(X2)) active(s(X)) -> s(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(add(X1, X2)) -> add(active(X1), X2) active(add(X1, X2)) -> add(X1, active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) fib1(mark(X1), X2) -> mark(fib1(X1, X2)) fib1(X1, mark(X2)) -> mark(fib1(X1, X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) add(mark(X1), X2) -> mark(add(X1, X2)) add(X1, mark(X2)) -> mark(add(X1, X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(0) -> ok(0) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(add(X1, X2)) -> add(proper(X1), proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) add(ok(X1), ok(X2)) -> ok(add(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_fib(x_1)) -> fib(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_fib1(x_1, x_2)) -> fib1(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_fib(x_1) -> fib(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_fib1(x_1, x_2) -> fib1(encArg(x_1), encArg(x_2)) 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_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) 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: active(fib(N)) -> mark(sel(N, fib1(s(0), s(0)))) active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y)))) active(add(0, X)) -> mark(X) active(add(s(X), Y)) -> mark(s(add(X, Y))) active(sel(0, cons(X, XS))) -> mark(X) active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) active(fib(X)) -> fib(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(fib1(X1, X2)) -> fib1(active(X1), X2) active(fib1(X1, X2)) -> fib1(X1, active(X2)) active(s(X)) -> s(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(add(X1, X2)) -> add(active(X1), X2) active(add(X1, X2)) -> add(X1, active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) fib1(mark(X1), X2) -> mark(fib1(X1, X2)) fib1(X1, mark(X2)) -> mark(fib1(X1, X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) add(mark(X1), X2) -> mark(add(X1, X2)) add(X1, mark(X2)) -> mark(add(X1, X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(0) -> ok(0) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(add(X1, X2)) -> add(proper(X1), proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) add(ok(X1), ok(X2)) -> ok(add(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_fib(x_1)) -> fib(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_fib1(x_1, x_2)) -> fib1(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_fib(x_1) -> fib(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_fib1(x_1, x_2) -> fib1(encArg(x_1), encArg(x_2)) 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_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) 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: active(fib(N)) -> mark(sel(N, fib1(s(0), s(0)))) active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y)))) active(add(0, X)) -> mark(X) active(add(s(X), Y)) -> mark(s(add(X, Y))) active(sel(0, cons(X, XS))) -> mark(X) active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) active(fib(X)) -> fib(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(fib1(X1, X2)) -> fib1(active(X1), X2) active(fib1(X1, X2)) -> fib1(X1, active(X2)) active(s(X)) -> s(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(add(X1, X2)) -> add(active(X1), X2) active(add(X1, X2)) -> add(X1, active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) fib1(mark(X1), X2) -> mark(fib1(X1, X2)) fib1(X1, mark(X2)) -> mark(fib1(X1, X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) add(mark(X1), X2) -> mark(add(X1, X2)) add(X1, mark(X2)) -> mark(add(X1, X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(0) -> ok(0) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(add(X1, X2)) -> add(proper(X1), proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) add(ok(X1), ok(X2)) -> ok(add(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_fib(x_1)) -> fib(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_fib1(x_1, x_2)) -> fib1(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_fib(x_1) -> fib(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_fib1(x_1, x_2) -> fib1(encArg(x_1), encArg(x_2)) 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_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) 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 fib1(ok(X1), ok(X2)) ->^+ ok(fib1(X1, X2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / ok(X1), X2 / ok(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: active(fib(N)) -> mark(sel(N, fib1(s(0), s(0)))) active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y)))) active(add(0, X)) -> mark(X) active(add(s(X), Y)) -> mark(s(add(X, Y))) active(sel(0, cons(X, XS))) -> mark(X) active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) active(fib(X)) -> fib(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(fib1(X1, X2)) -> fib1(active(X1), X2) active(fib1(X1, X2)) -> fib1(X1, active(X2)) active(s(X)) -> s(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(add(X1, X2)) -> add(active(X1), X2) active(add(X1, X2)) -> add(X1, active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) fib1(mark(X1), X2) -> mark(fib1(X1, X2)) fib1(X1, mark(X2)) -> mark(fib1(X1, X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) add(mark(X1), X2) -> mark(add(X1, X2)) add(X1, mark(X2)) -> mark(add(X1, X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(0) -> ok(0) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(add(X1, X2)) -> add(proper(X1), proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) add(ok(X1), ok(X2)) -> ok(add(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_fib(x_1)) -> fib(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_fib1(x_1, x_2)) -> fib1(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_fib(x_1) -> fib(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_fib1(x_1, x_2) -> fib1(encArg(x_1), encArg(x_2)) 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_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) 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: active(fib(N)) -> mark(sel(N, fib1(s(0), s(0)))) active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y)))) active(add(0, X)) -> mark(X) active(add(s(X), Y)) -> mark(s(add(X, Y))) active(sel(0, cons(X, XS))) -> mark(X) active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) active(fib(X)) -> fib(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(fib1(X1, X2)) -> fib1(active(X1), X2) active(fib1(X1, X2)) -> fib1(X1, active(X2)) active(s(X)) -> s(active(X)) active(cons(X1, X2)) -> cons(active(X1), X2) active(add(X1, X2)) -> add(active(X1), X2) active(add(X1, X2)) -> add(X1, active(X2)) fib(mark(X)) -> mark(fib(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) fib1(mark(X1), X2) -> mark(fib1(X1, X2)) fib1(X1, mark(X2)) -> mark(fib1(X1, X2)) s(mark(X)) -> mark(s(X)) cons(mark(X1), X2) -> mark(cons(X1, X2)) add(mark(X1), X2) -> mark(add(X1, X2)) add(X1, mark(X2)) -> mark(add(X1, X2)) proper(fib(X)) -> fib(proper(X)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2)) proper(s(X)) -> s(proper(X)) proper(0) -> ok(0) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(add(X1, X2)) -> add(proper(X1), proper(X2)) fib(ok(X)) -> ok(fib(X)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2)) s(ok(X)) -> ok(s(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) add(ok(X1), ok(X2)) -> ok(add(X1, X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(0) -> 0 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_fib(x_1)) -> fib(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_fib1(x_1, x_2)) -> fib1(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_fib(x_1) -> fib(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_fib1(x_1, x_2) -> fib1(encArg(x_1), encArg(x_2)) 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_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST