/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 436 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 5 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: Term_sub(Case(m, xi, n), s) -> Frozen(m, Sum_sub(xi, s), n, s) Frozen(m, Sum_constant(Left), n, s) -> Term_sub(m, s) Frozen(m, Sum_constant(Right), n, s) -> Term_sub(n, s) Frozen(m, Sum_term_var(xi), n, s) -> Case(Term_sub(m, s), xi, Term_sub(n, s)) Term_sub(Term_app(m, n), s) -> Term_app(Term_sub(m, s), Term_sub(n, s)) Term_sub(Term_pair(m, n), s) -> Term_pair(Term_sub(m, s), Term_sub(n, s)) Term_sub(Term_inl(m), s) -> Term_inl(Term_sub(m, s)) Term_sub(Term_inr(m), s) -> Term_inr(Term_sub(m, s)) Term_sub(Term_var(x), Id) -> Term_var(x) Term_sub(Term_var(x), Cons_usual(y, m, s)) -> m Term_sub(Term_var(x), Cons_usual(y, m, s)) -> Term_sub(Term_var(x), s) Term_sub(Term_var(x), Cons_sum(xi, k, s)) -> Term_sub(Term_var(x), s) Term_sub(Term_sub(m, s), t) -> Term_sub(m, Concat(s, t)) Sum_sub(xi, Id) -> Sum_term_var(xi) Sum_sub(xi, Cons_sum(psi, k, s)) -> Sum_constant(k) Sum_sub(xi, Cons_sum(psi, k, s)) -> Sum_sub(xi, s) Sum_sub(xi, Cons_usual(y, m, s)) -> Sum_sub(xi, s) Concat(Concat(s, t), u) -> Concat(s, Concat(t, u)) Concat(Cons_usual(x, m, s), t) -> Cons_usual(x, Term_sub(m, t), Concat(s, t)) Concat(Cons_sum(xi, k, s), t) -> Cons_sum(xi, k, Concat(s, t)) Concat(Id, s) -> s 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(Case(x_1, x_2, x_3)) -> Case(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(Sum_constant(x_1)) -> Sum_constant(encArg(x_1)) encArg(Left) -> Left encArg(Right) -> Right encArg(Sum_term_var(x_1)) -> Sum_term_var(encArg(x_1)) encArg(Term_app(x_1, x_2)) -> Term_app(encArg(x_1), encArg(x_2)) encArg(Term_pair(x_1, x_2)) -> Term_pair(encArg(x_1), encArg(x_2)) encArg(Term_inl(x_1)) -> Term_inl(encArg(x_1)) encArg(Term_inr(x_1)) -> Term_inr(encArg(x_1)) encArg(Term_var(x_1)) -> Term_var(encArg(x_1)) encArg(Id) -> Id encArg(Cons_usual(x_1, x_2, x_3)) -> Cons_usual(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(Cons_sum(x_1, x_2, x_3)) -> Cons_sum(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_Term_sub(x_1, x_2)) -> Term_sub(encArg(x_1), encArg(x_2)) encArg(cons_Frozen(x_1, x_2, x_3, x_4)) -> Frozen(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_Sum_sub(x_1, x_2)) -> Sum_sub(encArg(x_1), encArg(x_2)) encArg(cons_Concat(x_1, x_2)) -> Concat(encArg(x_1), encArg(x_2)) encode_Term_sub(x_1, x_2) -> Term_sub(encArg(x_1), encArg(x_2)) encode_Case(x_1, x_2, x_3) -> Case(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Frozen(x_1, x_2, x_3, x_4) -> Frozen(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_Sum_sub(x_1, x_2) -> Sum_sub(encArg(x_1), encArg(x_2)) encode_Sum_constant(x_1) -> Sum_constant(encArg(x_1)) encode_Left -> Left encode_Right -> Right encode_Sum_term_var(x_1) -> Sum_term_var(encArg(x_1)) encode_Term_app(x_1, x_2) -> Term_app(encArg(x_1), encArg(x_2)) encode_Term_pair(x_1, x_2) -> Term_pair(encArg(x_1), encArg(x_2)) encode_Term_inl(x_1) -> Term_inl(encArg(x_1)) encode_Term_inr(x_1) -> Term_inr(encArg(x_1)) encode_Term_var(x_1) -> Term_var(encArg(x_1)) encode_Id -> Id encode_Cons_usual(x_1, x_2, x_3) -> Cons_usual(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Cons_sum(x_1, x_2, x_3) -> Cons_sum(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Concat(x_1, x_2) -> Concat(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: Term_sub(Case(m, xi, n), s) -> Frozen(m, Sum_sub(xi, s), n, s) Frozen(m, Sum_constant(Left), n, s) -> Term_sub(m, s) Frozen(m, Sum_constant(Right), n, s) -> Term_sub(n, s) Frozen(m, Sum_term_var(xi), n, s) -> Case(Term_sub(m, s), xi, Term_sub(n, s)) Term_sub(Term_app(m, n), s) -> Term_app(Term_sub(m, s), Term_sub(n, s)) Term_sub(Term_pair(m, n), s) -> Term_pair(Term_sub(m, s), Term_sub(n, s)) Term_sub(Term_inl(m), s) -> Term_inl(Term_sub(m, s)) Term_sub(Term_inr(m), s) -> Term_inr(Term_sub(m, s)) Term_sub(Term_var(x), Id) -> Term_var(x) Term_sub(Term_var(x), Cons_usual(y, m, s)) -> m Term_sub(Term_var(x), Cons_usual(y, m, s)) -> Term_sub(Term_var(x), s) Term_sub(Term_var(x), Cons_sum(xi, k, s)) -> Term_sub(Term_var(x), s) Term_sub(Term_sub(m, s), t) -> Term_sub(m, Concat(s, t)) Sum_sub(xi, Id) -> Sum_term_var(xi) Sum_sub(xi, Cons_sum(psi, k, s)) -> Sum_constant(k) Sum_sub(xi, Cons_sum(psi, k, s)) -> Sum_sub(xi, s) Sum_sub(xi, Cons_usual(y, m, s)) -> Sum_sub(xi, s) Concat(Concat(s, t), u) -> Concat(s, Concat(t, u)) Concat(Cons_usual(x, m, s), t) -> Cons_usual(x, Term_sub(m, t), Concat(s, t)) Concat(Cons_sum(xi, k, s), t) -> Cons_sum(xi, k, Concat(s, t)) Concat(Id, s) -> s The (relative) TRS S consists of the following rules: encArg(Case(x_1, x_2, x_3)) -> Case(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(Sum_constant(x_1)) -> Sum_constant(encArg(x_1)) encArg(Left) -> Left encArg(Right) -> Right encArg(Sum_term_var(x_1)) -> Sum_term_var(encArg(x_1)) encArg(Term_app(x_1, x_2)) -> Term_app(encArg(x_1), encArg(x_2)) encArg(Term_pair(x_1, x_2)) -> Term_pair(encArg(x_1), encArg(x_2)) encArg(Term_inl(x_1)) -> Term_inl(encArg(x_1)) encArg(Term_inr(x_1)) -> Term_inr(encArg(x_1)) encArg(Term_var(x_1)) -> Term_var(encArg(x_1)) encArg(Id) -> Id encArg(Cons_usual(x_1, x_2, x_3)) -> Cons_usual(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(Cons_sum(x_1, x_2, x_3)) -> Cons_sum(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_Term_sub(x_1, x_2)) -> Term_sub(encArg(x_1), encArg(x_2)) encArg(cons_Frozen(x_1, x_2, x_3, x_4)) -> Frozen(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_Sum_sub(x_1, x_2)) -> Sum_sub(encArg(x_1), encArg(x_2)) encArg(cons_Concat(x_1, x_2)) -> Concat(encArg(x_1), encArg(x_2)) encode_Term_sub(x_1, x_2) -> Term_sub(encArg(x_1), encArg(x_2)) encode_Case(x_1, x_2, x_3) -> Case(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Frozen(x_1, x_2, x_3, x_4) -> Frozen(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_Sum_sub(x_1, x_2) -> Sum_sub(encArg(x_1), encArg(x_2)) encode_Sum_constant(x_1) -> Sum_constant(encArg(x_1)) encode_Left -> Left encode_Right -> Right encode_Sum_term_var(x_1) -> Sum_term_var(encArg(x_1)) encode_Term_app(x_1, x_2) -> Term_app(encArg(x_1), encArg(x_2)) encode_Term_pair(x_1, x_2) -> Term_pair(encArg(x_1), encArg(x_2)) encode_Term_inl(x_1) -> Term_inl(encArg(x_1)) encode_Term_inr(x_1) -> Term_inr(encArg(x_1)) encode_Term_var(x_1) -> Term_var(encArg(x_1)) encode_Id -> Id encode_Cons_usual(x_1, x_2, x_3) -> Cons_usual(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Cons_sum(x_1, x_2, x_3) -> Cons_sum(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Concat(x_1, x_2) -> Concat(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: Term_sub(Case(m, xi, n), s) -> Frozen(m, Sum_sub(xi, s), n, s) Frozen(m, Sum_constant(Left), n, s) -> Term_sub(m, s) Frozen(m, Sum_constant(Right), n, s) -> Term_sub(n, s) Frozen(m, Sum_term_var(xi), n, s) -> Case(Term_sub(m, s), xi, Term_sub(n, s)) Term_sub(Term_app(m, n), s) -> Term_app(Term_sub(m, s), Term_sub(n, s)) Term_sub(Term_pair(m, n), s) -> Term_pair(Term_sub(m, s), Term_sub(n, s)) Term_sub(Term_inl(m), s) -> Term_inl(Term_sub(m, s)) Term_sub(Term_inr(m), s) -> Term_inr(Term_sub(m, s)) Term_sub(Term_var(x), Id) -> Term_var(x) Term_sub(Term_var(x), Cons_usual(y, m, s)) -> m Term_sub(Term_var(x), Cons_usual(y, m, s)) -> Term_sub(Term_var(x), s) Term_sub(Term_var(x), Cons_sum(xi, k, s)) -> Term_sub(Term_var(x), s) Term_sub(Term_sub(m, s), t) -> Term_sub(m, Concat(s, t)) Sum_sub(xi, Id) -> Sum_term_var(xi) Sum_sub(xi, Cons_sum(psi, k, s)) -> Sum_constant(k) Sum_sub(xi, Cons_sum(psi, k, s)) -> Sum_sub(xi, s) Sum_sub(xi, Cons_usual(y, m, s)) -> Sum_sub(xi, s) Concat(Concat(s, t), u) -> Concat(s, Concat(t, u)) Concat(Cons_usual(x, m, s), t) -> Cons_usual(x, Term_sub(m, t), Concat(s, t)) Concat(Cons_sum(xi, k, s), t) -> Cons_sum(xi, k, Concat(s, t)) Concat(Id, s) -> s The (relative) TRS S consists of the following rules: encArg(Case(x_1, x_2, x_3)) -> Case(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(Sum_constant(x_1)) -> Sum_constant(encArg(x_1)) encArg(Left) -> Left encArg(Right) -> Right encArg(Sum_term_var(x_1)) -> Sum_term_var(encArg(x_1)) encArg(Term_app(x_1, x_2)) -> Term_app(encArg(x_1), encArg(x_2)) encArg(Term_pair(x_1, x_2)) -> Term_pair(encArg(x_1), encArg(x_2)) encArg(Term_inl(x_1)) -> Term_inl(encArg(x_1)) encArg(Term_inr(x_1)) -> Term_inr(encArg(x_1)) encArg(Term_var(x_1)) -> Term_var(encArg(x_1)) encArg(Id) -> Id encArg(Cons_usual(x_1, x_2, x_3)) -> Cons_usual(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(Cons_sum(x_1, x_2, x_3)) -> Cons_sum(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_Term_sub(x_1, x_2)) -> Term_sub(encArg(x_1), encArg(x_2)) encArg(cons_Frozen(x_1, x_2, x_3, x_4)) -> Frozen(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_Sum_sub(x_1, x_2)) -> Sum_sub(encArg(x_1), encArg(x_2)) encArg(cons_Concat(x_1, x_2)) -> Concat(encArg(x_1), encArg(x_2)) encode_Term_sub(x_1, x_2) -> Term_sub(encArg(x_1), encArg(x_2)) encode_Case(x_1, x_2, x_3) -> Case(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Frozen(x_1, x_2, x_3, x_4) -> Frozen(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_Sum_sub(x_1, x_2) -> Sum_sub(encArg(x_1), encArg(x_2)) encode_Sum_constant(x_1) -> Sum_constant(encArg(x_1)) encode_Left -> Left encode_Right -> Right encode_Sum_term_var(x_1) -> Sum_term_var(encArg(x_1)) encode_Term_app(x_1, x_2) -> Term_app(encArg(x_1), encArg(x_2)) encode_Term_pair(x_1, x_2) -> Term_pair(encArg(x_1), encArg(x_2)) encode_Term_inl(x_1) -> Term_inl(encArg(x_1)) encode_Term_inr(x_1) -> Term_inr(encArg(x_1)) encode_Term_var(x_1) -> Term_var(encArg(x_1)) encode_Id -> Id encode_Cons_usual(x_1, x_2, x_3) -> Cons_usual(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Cons_sum(x_1, x_2, x_3) -> Cons_sum(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Concat(x_1, x_2) -> Concat(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: Term_sub(Case(m, xi, n), s) -> Frozen(m, Sum_sub(xi, s), n, s) Frozen(m, Sum_constant(Left), n, s) -> Term_sub(m, s) Frozen(m, Sum_constant(Right), n, s) -> Term_sub(n, s) Frozen(m, Sum_term_var(xi), n, s) -> Case(Term_sub(m, s), xi, Term_sub(n, s)) Term_sub(Term_app(m, n), s) -> Term_app(Term_sub(m, s), Term_sub(n, s)) Term_sub(Term_pair(m, n), s) -> Term_pair(Term_sub(m, s), Term_sub(n, s)) Term_sub(Term_inl(m), s) -> Term_inl(Term_sub(m, s)) Term_sub(Term_inr(m), s) -> Term_inr(Term_sub(m, s)) Term_sub(Term_var(x), Id) -> Term_var(x) Term_sub(Term_var(x), Cons_usual(y, m, s)) -> m Term_sub(Term_var(x), Cons_usual(y, m, s)) -> Term_sub(Term_var(x), s) Term_sub(Term_var(x), Cons_sum(xi, k, s)) -> Term_sub(Term_var(x), s) Term_sub(Term_sub(m, s), t) -> Term_sub(m, Concat(s, t)) Sum_sub(xi, Id) -> Sum_term_var(xi) Sum_sub(xi, Cons_sum(psi, k, s)) -> Sum_constant(k) Sum_sub(xi, Cons_sum(psi, k, s)) -> Sum_sub(xi, s) Sum_sub(xi, Cons_usual(y, m, s)) -> Sum_sub(xi, s) Concat(Concat(s, t), u) -> Concat(s, Concat(t, u)) Concat(Cons_usual(x, m, s), t) -> Cons_usual(x, Term_sub(m, t), Concat(s, t)) Concat(Cons_sum(xi, k, s), t) -> Cons_sum(xi, k, Concat(s, t)) Concat(Id, s) -> s The (relative) TRS S consists of the following rules: encArg(Case(x_1, x_2, x_3)) -> Case(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(Sum_constant(x_1)) -> Sum_constant(encArg(x_1)) encArg(Left) -> Left encArg(Right) -> Right encArg(Sum_term_var(x_1)) -> Sum_term_var(encArg(x_1)) encArg(Term_app(x_1, x_2)) -> Term_app(encArg(x_1), encArg(x_2)) encArg(Term_pair(x_1, x_2)) -> Term_pair(encArg(x_1), encArg(x_2)) encArg(Term_inl(x_1)) -> Term_inl(encArg(x_1)) encArg(Term_inr(x_1)) -> Term_inr(encArg(x_1)) encArg(Term_var(x_1)) -> Term_var(encArg(x_1)) encArg(Id) -> Id encArg(Cons_usual(x_1, x_2, x_3)) -> Cons_usual(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(Cons_sum(x_1, x_2, x_3)) -> Cons_sum(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_Term_sub(x_1, x_2)) -> Term_sub(encArg(x_1), encArg(x_2)) encArg(cons_Frozen(x_1, x_2, x_3, x_4)) -> Frozen(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_Sum_sub(x_1, x_2)) -> Sum_sub(encArg(x_1), encArg(x_2)) encArg(cons_Concat(x_1, x_2)) -> Concat(encArg(x_1), encArg(x_2)) encode_Term_sub(x_1, x_2) -> Term_sub(encArg(x_1), encArg(x_2)) encode_Case(x_1, x_2, x_3) -> Case(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Frozen(x_1, x_2, x_3, x_4) -> Frozen(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_Sum_sub(x_1, x_2) -> Sum_sub(encArg(x_1), encArg(x_2)) encode_Sum_constant(x_1) -> Sum_constant(encArg(x_1)) encode_Left -> Left encode_Right -> Right encode_Sum_term_var(x_1) -> Sum_term_var(encArg(x_1)) encode_Term_app(x_1, x_2) -> Term_app(encArg(x_1), encArg(x_2)) encode_Term_pair(x_1, x_2) -> Term_pair(encArg(x_1), encArg(x_2)) encode_Term_inl(x_1) -> Term_inl(encArg(x_1)) encode_Term_inr(x_1) -> Term_inr(encArg(x_1)) encode_Term_var(x_1) -> Term_var(encArg(x_1)) encode_Id -> Id encode_Cons_usual(x_1, x_2, x_3) -> Cons_usual(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Cons_sum(x_1, x_2, x_3) -> Cons_sum(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Concat(x_1, x_2) -> Concat(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 Term_sub(Term_pair(m, n), s) ->^+ Term_pair(Term_sub(m, s), Term_sub(n, s)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [m / Term_pair(m, n)]. 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: Term_sub(Case(m, xi, n), s) -> Frozen(m, Sum_sub(xi, s), n, s) Frozen(m, Sum_constant(Left), n, s) -> Term_sub(m, s) Frozen(m, Sum_constant(Right), n, s) -> Term_sub(n, s) Frozen(m, Sum_term_var(xi), n, s) -> Case(Term_sub(m, s), xi, Term_sub(n, s)) Term_sub(Term_app(m, n), s) -> Term_app(Term_sub(m, s), Term_sub(n, s)) Term_sub(Term_pair(m, n), s) -> Term_pair(Term_sub(m, s), Term_sub(n, s)) Term_sub(Term_inl(m), s) -> Term_inl(Term_sub(m, s)) Term_sub(Term_inr(m), s) -> Term_inr(Term_sub(m, s)) Term_sub(Term_var(x), Id) -> Term_var(x) Term_sub(Term_var(x), Cons_usual(y, m, s)) -> m Term_sub(Term_var(x), Cons_usual(y, m, s)) -> Term_sub(Term_var(x), s) Term_sub(Term_var(x), Cons_sum(xi, k, s)) -> Term_sub(Term_var(x), s) Term_sub(Term_sub(m, s), t) -> Term_sub(m, Concat(s, t)) Sum_sub(xi, Id) -> Sum_term_var(xi) Sum_sub(xi, Cons_sum(psi, k, s)) -> Sum_constant(k) Sum_sub(xi, Cons_sum(psi, k, s)) -> Sum_sub(xi, s) Sum_sub(xi, Cons_usual(y, m, s)) -> Sum_sub(xi, s) Concat(Concat(s, t), u) -> Concat(s, Concat(t, u)) Concat(Cons_usual(x, m, s), t) -> Cons_usual(x, Term_sub(m, t), Concat(s, t)) Concat(Cons_sum(xi, k, s), t) -> Cons_sum(xi, k, Concat(s, t)) Concat(Id, s) -> s The (relative) TRS S consists of the following rules: encArg(Case(x_1, x_2, x_3)) -> Case(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(Sum_constant(x_1)) -> Sum_constant(encArg(x_1)) encArg(Left) -> Left encArg(Right) -> Right encArg(Sum_term_var(x_1)) -> Sum_term_var(encArg(x_1)) encArg(Term_app(x_1, x_2)) -> Term_app(encArg(x_1), encArg(x_2)) encArg(Term_pair(x_1, x_2)) -> Term_pair(encArg(x_1), encArg(x_2)) encArg(Term_inl(x_1)) -> Term_inl(encArg(x_1)) encArg(Term_inr(x_1)) -> Term_inr(encArg(x_1)) encArg(Term_var(x_1)) -> Term_var(encArg(x_1)) encArg(Id) -> Id encArg(Cons_usual(x_1, x_2, x_3)) -> Cons_usual(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(Cons_sum(x_1, x_2, x_3)) -> Cons_sum(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_Term_sub(x_1, x_2)) -> Term_sub(encArg(x_1), encArg(x_2)) encArg(cons_Frozen(x_1, x_2, x_3, x_4)) -> Frozen(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_Sum_sub(x_1, x_2)) -> Sum_sub(encArg(x_1), encArg(x_2)) encArg(cons_Concat(x_1, x_2)) -> Concat(encArg(x_1), encArg(x_2)) encode_Term_sub(x_1, x_2) -> Term_sub(encArg(x_1), encArg(x_2)) encode_Case(x_1, x_2, x_3) -> Case(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Frozen(x_1, x_2, x_3, x_4) -> Frozen(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_Sum_sub(x_1, x_2) -> Sum_sub(encArg(x_1), encArg(x_2)) encode_Sum_constant(x_1) -> Sum_constant(encArg(x_1)) encode_Left -> Left encode_Right -> Right encode_Sum_term_var(x_1) -> Sum_term_var(encArg(x_1)) encode_Term_app(x_1, x_2) -> Term_app(encArg(x_1), encArg(x_2)) encode_Term_pair(x_1, x_2) -> Term_pair(encArg(x_1), encArg(x_2)) encode_Term_inl(x_1) -> Term_inl(encArg(x_1)) encode_Term_inr(x_1) -> Term_inr(encArg(x_1)) encode_Term_var(x_1) -> Term_var(encArg(x_1)) encode_Id -> Id encode_Cons_usual(x_1, x_2, x_3) -> Cons_usual(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Cons_sum(x_1, x_2, x_3) -> Cons_sum(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Concat(x_1, x_2) -> Concat(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: Term_sub(Case(m, xi, n), s) -> Frozen(m, Sum_sub(xi, s), n, s) Frozen(m, Sum_constant(Left), n, s) -> Term_sub(m, s) Frozen(m, Sum_constant(Right), n, s) -> Term_sub(n, s) Frozen(m, Sum_term_var(xi), n, s) -> Case(Term_sub(m, s), xi, Term_sub(n, s)) Term_sub(Term_app(m, n), s) -> Term_app(Term_sub(m, s), Term_sub(n, s)) Term_sub(Term_pair(m, n), s) -> Term_pair(Term_sub(m, s), Term_sub(n, s)) Term_sub(Term_inl(m), s) -> Term_inl(Term_sub(m, s)) Term_sub(Term_inr(m), s) -> Term_inr(Term_sub(m, s)) Term_sub(Term_var(x), Id) -> Term_var(x) Term_sub(Term_var(x), Cons_usual(y, m, s)) -> m Term_sub(Term_var(x), Cons_usual(y, m, s)) -> Term_sub(Term_var(x), s) Term_sub(Term_var(x), Cons_sum(xi, k, s)) -> Term_sub(Term_var(x), s) Term_sub(Term_sub(m, s), t) -> Term_sub(m, Concat(s, t)) Sum_sub(xi, Id) -> Sum_term_var(xi) Sum_sub(xi, Cons_sum(psi, k, s)) -> Sum_constant(k) Sum_sub(xi, Cons_sum(psi, k, s)) -> Sum_sub(xi, s) Sum_sub(xi, Cons_usual(y, m, s)) -> Sum_sub(xi, s) Concat(Concat(s, t), u) -> Concat(s, Concat(t, u)) Concat(Cons_usual(x, m, s), t) -> Cons_usual(x, Term_sub(m, t), Concat(s, t)) Concat(Cons_sum(xi, k, s), t) -> Cons_sum(xi, k, Concat(s, t)) Concat(Id, s) -> s The (relative) TRS S consists of the following rules: encArg(Case(x_1, x_2, x_3)) -> Case(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(Sum_constant(x_1)) -> Sum_constant(encArg(x_1)) encArg(Left) -> Left encArg(Right) -> Right encArg(Sum_term_var(x_1)) -> Sum_term_var(encArg(x_1)) encArg(Term_app(x_1, x_2)) -> Term_app(encArg(x_1), encArg(x_2)) encArg(Term_pair(x_1, x_2)) -> Term_pair(encArg(x_1), encArg(x_2)) encArg(Term_inl(x_1)) -> Term_inl(encArg(x_1)) encArg(Term_inr(x_1)) -> Term_inr(encArg(x_1)) encArg(Term_var(x_1)) -> Term_var(encArg(x_1)) encArg(Id) -> Id encArg(Cons_usual(x_1, x_2, x_3)) -> Cons_usual(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(Cons_sum(x_1, x_2, x_3)) -> Cons_sum(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_Term_sub(x_1, x_2)) -> Term_sub(encArg(x_1), encArg(x_2)) encArg(cons_Frozen(x_1, x_2, x_3, x_4)) -> Frozen(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_Sum_sub(x_1, x_2)) -> Sum_sub(encArg(x_1), encArg(x_2)) encArg(cons_Concat(x_1, x_2)) -> Concat(encArg(x_1), encArg(x_2)) encode_Term_sub(x_1, x_2) -> Term_sub(encArg(x_1), encArg(x_2)) encode_Case(x_1, x_2, x_3) -> Case(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Frozen(x_1, x_2, x_3, x_4) -> Frozen(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_Sum_sub(x_1, x_2) -> Sum_sub(encArg(x_1), encArg(x_2)) encode_Sum_constant(x_1) -> Sum_constant(encArg(x_1)) encode_Left -> Left encode_Right -> Right encode_Sum_term_var(x_1) -> Sum_term_var(encArg(x_1)) encode_Term_app(x_1, x_2) -> Term_app(encArg(x_1), encArg(x_2)) encode_Term_pair(x_1, x_2) -> Term_pair(encArg(x_1), encArg(x_2)) encode_Term_inl(x_1) -> Term_inl(encArg(x_1)) encode_Term_inr(x_1) -> Term_inr(encArg(x_1)) encode_Term_var(x_1) -> Term_var(encArg(x_1)) encode_Id -> Id encode_Cons_usual(x_1, x_2, x_3) -> Cons_usual(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Cons_sum(x_1, x_2, x_3) -> Cons_sum(encArg(x_1), encArg(x_2), encArg(x_3)) encode_Concat(x_1, x_2) -> Concat(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST