/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 201 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 522 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 997 ms] (18) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(x, nil) -> x foldf(x, cons(y, z)) -> f(foldf(x, z), y) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b)) f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) S is empty. Rewrite Strategy: FULL ---------------------------------------- (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(A) -> A encArg(B) -> B encArg(C) -> C encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(triple(x_1, x_2, x_3)) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_foldf(x_1, x_2)) -> foldf(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_f'(x_1, x_2)) -> f'(encArg(x_1), encArg(x_2)) encArg(cons_f''(x_1)) -> f''(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_A -> A encode_B -> B encode_C -> C encode_foldf(x_1, x_2) -> foldf(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_f'(x_1, x_2) -> f'(encArg(x_1), encArg(x_2)) encode_triple(x_1, x_2, x_3) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f''(x_1) -> f''(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(x, nil) -> x foldf(x, cons(y, z)) -> f(foldf(x, z), y) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b)) f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) The (relative) TRS S consists of the following rules: encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(triple(x_1, x_2, x_3)) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_foldf(x_1, x_2)) -> foldf(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_f'(x_1, x_2)) -> f'(encArg(x_1), encArg(x_2)) encArg(cons_f''(x_1)) -> f''(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_A -> A encode_B -> B encode_C -> C encode_foldf(x_1, x_2) -> foldf(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_f'(x_1, x_2) -> f'(encArg(x_1), encArg(x_2)) encode_triple(x_1, x_2, x_3) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f''(x_1) -> f''(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(x, nil) -> x foldf(x, cons(y, z)) -> f(foldf(x, z), y) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b)) f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) The (relative) TRS S consists of the following rules: encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(triple(x_1, x_2, x_3)) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_foldf(x_1, x_2)) -> foldf(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_f'(x_1, x_2)) -> f'(encArg(x_1), encArg(x_2)) encArg(cons_f''(x_1)) -> f''(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_A -> A encode_B -> B encode_C -> C encode_foldf(x_1, x_2) -> foldf(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_f'(x_1, x_2) -> f'(encArg(x_1), encArg(x_2)) encode_triple(x_1, x_2, x_3) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f''(x_1) -> f''(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(x, nil) -> x foldf(x, cons(y, z)) -> f(foldf(x, z), y) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b)) f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) The (relative) TRS S consists of the following rules: encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(triple(x_1, x_2, x_3)) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_foldf(x_1, x_2)) -> foldf(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_f'(x_1, x_2)) -> f'(encArg(x_1), encArg(x_2)) encArg(cons_f''(x_1)) -> f''(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_A -> A encode_B -> B encode_C -> C encode_foldf(x_1, x_2) -> foldf(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_f'(x_1, x_2) -> f'(encArg(x_1), encArg(x_2)) encode_triple(x_1, x_2, x_3) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f''(x_1) -> f''(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: TRS: Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(x, nil) -> x foldf(x, cons(y, z)) -> f(foldf(x, z), y) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b)) f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(triple(x_1, x_2, x_3)) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_foldf(x_1, x_2)) -> foldf(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_f'(x_1, x_2)) -> f'(encArg(x_1), encArg(x_2)) encArg(cons_f''(x_1)) -> f''(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_A -> A encode_B -> B encode_C -> C encode_foldf(x_1, x_2) -> foldf(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_f'(x_1, x_2) -> f'(encArg(x_1), encArg(x_2)) encode_triple(x_1, x_2, x_3) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f''(x_1) -> f''(encArg(x_1)) Types: g :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' A :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' B :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' C :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' foldf :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' nil :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' f :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' f' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' triple :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' f'' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encArg :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_g :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_foldf :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_f :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_f' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_f'' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_g :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_A :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_B :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_C :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_foldf :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_nil :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_cons :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_f :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_f' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_triple :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_f'' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' hole_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''1_0 :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0 :: Nat -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: foldf, f, f', f'', encArg They will be analysed ascendingly in the following order: foldf = f foldf = f' foldf = f'' foldf < encArg f = f' f = f'' f < encArg f' = f'' f' < encArg f'' < encArg ---------------------------------------- (10) Obligation: TRS: Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(x, nil) -> x foldf(x, cons(y, z)) -> f(foldf(x, z), y) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b)) f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(triple(x_1, x_2, x_3)) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_foldf(x_1, x_2)) -> foldf(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_f'(x_1, x_2)) -> f'(encArg(x_1), encArg(x_2)) encArg(cons_f''(x_1)) -> f''(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_A -> A encode_B -> B encode_C -> C encode_foldf(x_1, x_2) -> foldf(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_f'(x_1, x_2) -> f'(encArg(x_1), encArg(x_2)) encode_triple(x_1, x_2, x_3) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f''(x_1) -> f''(encArg(x_1)) Types: g :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' A :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' B :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' C :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' foldf :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' nil :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' f :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' f' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' triple :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' f'' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encArg :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_g :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_foldf :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_f :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_f' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_f'' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_g :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_A :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_B :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_C :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_foldf :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_nil :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_cons :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_f :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_f' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_triple :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_f'' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' hole_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''1_0 :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0 :: Nat -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' Generator Equations: gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(0) <=> A gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(+(x, 1)) <=> cons(A, gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(x)) The following defined symbols remain to be analysed: f, foldf, f', f'', encArg They will be analysed ascendingly in the following order: foldf = f foldf = f' foldf = f'' foldf < encArg f = f' f = f'' f < encArg f' = f'' f' < encArg f'' < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: foldf(gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(a), gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(+(1, n113_0))) -> *3_0, rt in Omega(n113_0) Induction Base: foldf(gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(a), gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(+(1, 0))) Induction Step: foldf(gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(a), gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(+(1, +(n113_0, 1)))) ->_R^Omega(1) f(foldf(gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(a), gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(+(1, n113_0))), A) ->_IH f(*3_0, A) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: TRS: Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(x, nil) -> x foldf(x, cons(y, z)) -> f(foldf(x, z), y) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b)) f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(triple(x_1, x_2, x_3)) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_foldf(x_1, x_2)) -> foldf(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_f'(x_1, x_2)) -> f'(encArg(x_1), encArg(x_2)) encArg(cons_f''(x_1)) -> f''(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_A -> A encode_B -> B encode_C -> C encode_foldf(x_1, x_2) -> foldf(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_f'(x_1, x_2) -> f'(encArg(x_1), encArg(x_2)) encode_triple(x_1, x_2, x_3) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f''(x_1) -> f''(encArg(x_1)) Types: g :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' A :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' B :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' C :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' foldf :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' nil :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' f :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' f' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' triple :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' f'' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encArg :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_g :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_foldf :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_f :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_f' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_f'' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_g :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_A :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_B :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_C :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_foldf :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_nil :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_cons :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_f :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_f' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_triple :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_f'' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' hole_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''1_0 :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0 :: Nat -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' Generator Equations: gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(0) <=> A gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(+(x, 1)) <=> cons(A, gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(x)) The following defined symbols remain to be analysed: foldf, encArg They will be analysed ascendingly in the following order: foldf = f foldf = f' foldf = f'' foldf < encArg f = f' f = f'' f < encArg f' = f'' f' < encArg f'' < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: TRS: Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldf(x, nil) -> x foldf(x, cons(y, z)) -> f(foldf(x, z), y) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, cons(C, c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldf(triple(cons(A, a), nil, c), b)) f''(triple(a, b, c)) -> foldf(triple(a, b, nil), c) encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(triple(x_1, x_2, x_3)) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_foldf(x_1, x_2)) -> foldf(encArg(x_1), encArg(x_2)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_f'(x_1, x_2)) -> f'(encArg(x_1), encArg(x_2)) encArg(cons_f''(x_1)) -> f''(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_A -> A encode_B -> B encode_C -> C encode_foldf(x_1, x_2) -> foldf(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_f'(x_1, x_2) -> f'(encArg(x_1), encArg(x_2)) encode_triple(x_1, x_2, x_3) -> triple(encArg(x_1), encArg(x_2), encArg(x_3)) encode_f''(x_1) -> f''(encArg(x_1)) Types: g :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' A :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' B :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' C :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' foldf :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' nil :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' f :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' f' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' triple :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' f'' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encArg :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_g :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_foldf :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_f :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_f' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' cons_f'' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_g :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_A :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_B :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_C :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_foldf :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_nil :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_cons :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_f :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_f' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_triple :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' encode_f'' :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' hole_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''1_0 :: A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0 :: Nat -> A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f'' Lemmas: foldf(gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(a), gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(+(1, n113_0))) -> *3_0, rt in Omega(n113_0) Generator Equations: gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(0) <=> A gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(+(x, 1)) <=> cons(A, gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(x)) The following defined symbols remain to be analysed: f, f', f'', encArg They will be analysed ascendingly in the following order: foldf = f foldf = f' foldf = f'' foldf < encArg f = f' f = f'' f < encArg f' = f'' f' < encArg f'' < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(n1961_0)) -> gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(n1961_0), rt in Omega(0) Induction Base: encArg(gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(0)) ->_R^Omega(0) A Induction Step: encArg(gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(+(n1961_0, 1))) ->_R^Omega(0) cons(encArg(A), encArg(gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(n1961_0))) ->_R^Omega(0) cons(A, encArg(gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(n1961_0))) ->_IH cons(A, gen_A:B:C:nil:cons:triple:cons_g:cons_foldf:cons_f:cons_f':cons_f''2_0(c1962_0)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) BOUNDS(1, INF)