/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). (0) CpxTRS (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (2) TRS for Loop Detection (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (4) BEST (5) proven lower bound (6) LowerBoundPropagationProof [FINISHED, 0 ms] (7) BOUNDS(n^1, INF) (8) TRS for Loop Detection (9) DecreasingLoopProof [FINISHED, 465 ms] (10) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: and(false, false) -> false and(true, false) -> false and(false, true) -> false and(true, true) -> true eq(nil, nil) -> true eq(cons(T, L), nil) -> false eq(nil, cons(T, L)) -> false eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp)) eq(var(L), var(Lp)) -> eq(L, Lp) eq(var(L), apply(T, S)) -> false eq(var(L), lambda(X, T)) -> false eq(apply(T, S), var(L)) -> false eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp)) eq(apply(T, S), lambda(X, Tp)) -> false eq(lambda(X, T), var(L)) -> false eq(lambda(X, T), apply(Tp, Sp)) -> false eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp)) if(true, var(K), var(L)) -> var(K) if(false, var(K), var(L)) -> var(L) ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp)) ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S)) ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (2) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: and(false, false) -> false and(true, false) -> false and(false, true) -> false and(true, true) -> true eq(nil, nil) -> true eq(cons(T, L), nil) -> false eq(nil, cons(T, L)) -> false eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp)) eq(var(L), var(Lp)) -> eq(L, Lp) eq(var(L), apply(T, S)) -> false eq(var(L), lambda(X, T)) -> false eq(apply(T, S), var(L)) -> false eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp)) eq(apply(T, S), lambda(X, Tp)) -> false eq(lambda(X, T), var(L)) -> false eq(lambda(X, T), apply(Tp, Sp)) -> false eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp)) if(true, var(K), var(L)) -> var(K) if(false, var(K), var(L)) -> var(L) ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp)) ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S)) ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence eq(lambda(X, T), lambda(Xp, Tp)) ->^+ and(eq(T, Tp), eq(X, Xp)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [T / lambda(X, T), Tp / lambda(Xp, Tp)]. The result substitution is [ ]. ---------------------------------------- (4) Complex Obligation (BEST) ---------------------------------------- (5) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: and(false, false) -> false and(true, false) -> false and(false, true) -> false and(true, true) -> true eq(nil, nil) -> true eq(cons(T, L), nil) -> false eq(nil, cons(T, L)) -> false eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp)) eq(var(L), var(Lp)) -> eq(L, Lp) eq(var(L), apply(T, S)) -> false eq(var(L), lambda(X, T)) -> false eq(apply(T, S), var(L)) -> false eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp)) eq(apply(T, S), lambda(X, Tp)) -> false eq(lambda(X, T), var(L)) -> false eq(lambda(X, T), apply(Tp, Sp)) -> false eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp)) if(true, var(K), var(L)) -> var(K) if(false, var(K), var(L)) -> var(L) ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp)) ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S)) ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (6) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (7) BOUNDS(n^1, INF) ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: and(false, false) -> false and(true, false) -> false and(false, true) -> false and(true, true) -> true eq(nil, nil) -> true eq(cons(T, L), nil) -> false eq(nil, cons(T, L)) -> false eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp)) eq(var(L), var(Lp)) -> eq(L, Lp) eq(var(L), apply(T, S)) -> false eq(var(L), lambda(X, T)) -> false eq(apply(T, S), var(L)) -> false eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp)) eq(apply(T, S), lambda(X, Tp)) -> false eq(lambda(X, T), var(L)) -> false eq(lambda(X, T), apply(Tp, Sp)) -> false eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp)) if(true, var(K), var(L)) -> var(K) if(false, var(K), var(L)) -> var(L) ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp)) ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S)) ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence ren(X, Y, lambda(Z, lambda(Z3_0, T4_0))) ->^+ lambda(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), lambda(var(cons(X, cons(Y, cons(lambda(var(cons(Z, cons(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), cons(lambda(Z3_0, T4_0), nil)))), ren(Z, var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), ren(Z3_0, var(cons(Z, cons(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), cons(lambda(Z3_0, T4_0), nil)))), T4_0))), nil)))), ren(X, Y, ren(var(cons(Z, cons(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), cons(lambda(Z3_0, T4_0), nil)))), var(cons(X, cons(Y, cons(lambda(var(cons(Z, cons(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), cons(lambda(Z3_0, T4_0), nil)))), ren(Z, var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), ren(Z3_0, var(cons(Z, cons(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), cons(lambda(Z3_0, T4_0), nil)))), T4_0))), nil)))), ren(Z, var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), ren(Z3_0, var(cons(Z, cons(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), cons(lambda(Z3_0, T4_0), nil)))), T4_0)))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0,1,1,0,1,2]. The pumping substitution is [T4_0 / lambda(Z, lambda(Z3_0, T4_0))]. The result substitution is [X / Z3_0, Y / var(cons(Z, cons(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), cons(lambda(Z3_0, T4_0), nil))))]. The rewrite sequence ren(X, Y, lambda(Z, lambda(Z3_0, T4_0))) ->^+ lambda(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), lambda(var(cons(X, cons(Y, cons(lambda(var(cons(Z, cons(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), cons(lambda(Z3_0, T4_0), nil)))), ren(Z, var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), ren(Z3_0, var(cons(Z, cons(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), cons(lambda(Z3_0, T4_0), nil)))), T4_0))), nil)))), ren(X, Y, ren(var(cons(Z, cons(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), cons(lambda(Z3_0, T4_0), nil)))), var(cons(X, cons(Y, cons(lambda(var(cons(Z, cons(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), cons(lambda(Z3_0, T4_0), nil)))), ren(Z, var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), ren(Z3_0, var(cons(Z, cons(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), cons(lambda(Z3_0, T4_0), nil)))), T4_0))), nil)))), ren(Z, var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), ren(Z3_0, var(cons(Z, cons(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), cons(lambda(Z3_0, T4_0), nil)))), T4_0)))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1,2,1,0,1,1,0,1,2]. The pumping substitution is [T4_0 / lambda(Z, lambda(Z3_0, T4_0))]. The result substitution is [X / Z3_0, Y / var(cons(Z, cons(var(cons(X, cons(Y, cons(lambda(Z, lambda(Z3_0, T4_0)), nil)))), cons(lambda(Z3_0, T4_0), nil))))]. ---------------------------------------- (10) BOUNDS(EXP, INF)