/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 342 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), 692 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), 233 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 313 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 1477 ms] (22) BOUNDS(1, INF) ---------------------------------------- (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: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldB(t, 0) -> t foldB(t, s(n)) -> f(foldB(t, n), B) foldC(t, 0) -> t foldC(t, s(n)) -> f(foldC(t, n), C) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, s(c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b)) f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c) 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(A) -> A encArg(B) -> B encArg(C) -> C encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) 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_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2)) encArg(cons_foldC(x_1, x_2)) -> foldC(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_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_foldC(x_1, x_2) -> foldC(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 (innermost) 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 foldB(t, 0) -> t foldB(t, s(n)) -> f(foldB(t, n), B) foldC(t, 0) -> t foldC(t, s(n)) -> f(foldC(t, n), C) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, s(c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b)) f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c) The (relative) TRS S consists of the following rules: encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) 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_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2)) encArg(cons_foldC(x_1, x_2)) -> foldC(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_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_foldC(x_1, x_2) -> foldC(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: 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: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldB(t, 0) -> t foldB(t, s(n)) -> f(foldB(t, n), B) foldC(t, 0) -> t foldC(t, s(n)) -> f(foldC(t, n), C) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, s(c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b)) f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c) The (relative) TRS S consists of the following rules: encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) 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_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2)) encArg(cons_foldC(x_1, x_2)) -> foldC(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_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_foldC(x_1, x_2) -> foldC(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: INNERMOST ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) 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: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldB(t, 0') -> t foldB(t, s(n)) -> f(foldB(t, n), B) foldC(t, 0') -> t foldC(t, s(n)) -> f(foldC(t, n), C) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, s(c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0', c), b)) f''(triple(a, b, c)) -> foldC(triple(a, b, 0'), c) The (relative) TRS S consists of the following rules: encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) 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_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2)) encArg(cons_foldC(x_1, x_2)) -> foldC(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_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_foldC(x_1, x_2) -> foldC(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: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldB(t, 0') -> t foldB(t, s(n)) -> f(foldB(t, n), B) foldC(t, 0') -> t foldC(t, s(n)) -> f(foldC(t, n), C) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, s(c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0', c), b)) f''(triple(a, b, c)) -> foldC(triple(a, b, 0'), c) encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) 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_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2)) encArg(cons_foldC(x_1, x_2)) -> foldC(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_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_foldC(x_1, x_2) -> foldC(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:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' A :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' B :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' C :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' 0' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' s :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' triple :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encArg :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_g :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_g :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_A :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_B :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_C :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_0 :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_s :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_triple :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' hole_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''1_4 :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4 :: Nat -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: foldB, f, foldC, f', f'', encArg They will be analysed ascendingly in the following order: foldB = f foldB = foldC foldB = f' foldB = f'' foldB < encArg f = foldC f = f' f = f'' f < encArg foldC = f' foldC = f'' foldC < encArg f' = f'' f' < encArg f'' < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldB(t, 0') -> t foldB(t, s(n)) -> f(foldB(t, n), B) foldC(t, 0') -> t foldC(t, s(n)) -> f(foldC(t, n), C) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, s(c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0', c), b)) f''(triple(a, b, c)) -> foldC(triple(a, b, 0'), c) encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) 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_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2)) encArg(cons_foldC(x_1, x_2)) -> foldC(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_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_foldC(x_1, x_2) -> foldC(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:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' A :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' B :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' C :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' 0' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' s :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' triple :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encArg :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_g :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_g :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_A :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_B :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_C :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_0 :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_s :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_triple :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' hole_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''1_4 :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4 :: Nat -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' Generator Equations: gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(0) <=> A gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(x, 1)) <=> s(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(x)) The following defined symbols remain to be analysed: f, foldB, foldC, f', f'', encArg They will be analysed ascendingly in the following order: foldB = f foldB = foldC foldB = f' foldB = f'' foldB < encArg f = foldC f = f' f = f'' f < encArg foldC = f' foldC = f'' foldC < encArg f' = f'' f' < encArg f'' < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: foldC(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, n113_4))) -> *3_4, rt in Omega(n113_4) Induction Base: foldC(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, 0))) Induction Step: foldC(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, +(n113_4, 1)))) ->_R^Omega(1) f(foldC(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, n113_4))), C) ->_IH f(*3_4, C) 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: Innermost TRS: Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldB(t, 0') -> t foldB(t, s(n)) -> f(foldB(t, n), B) foldC(t, 0') -> t foldC(t, s(n)) -> f(foldC(t, n), C) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, s(c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0', c), b)) f''(triple(a, b, c)) -> foldC(triple(a, b, 0'), c) encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) 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_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2)) encArg(cons_foldC(x_1, x_2)) -> foldC(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_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_foldC(x_1, x_2) -> foldC(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:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' A :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' B :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' C :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' 0' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' s :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' triple :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encArg :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_g :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_g :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_A :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_B :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_C :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_0 :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_s :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_triple :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' hole_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''1_4 :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4 :: Nat -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' Generator Equations: gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(0) <=> A gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(x, 1)) <=> s(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(x)) The following defined symbols remain to be analysed: foldC, foldB, encArg They will be analysed ascendingly in the following order: foldB = f foldB = foldC foldB = f' foldB = f'' foldB < encArg f = foldC f = f' f = f'' f < encArg foldC = f' foldC = f'' foldC < encArg f' = f'' f' < encArg f'' < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldB(t, 0') -> t foldB(t, s(n)) -> f(foldB(t, n), B) foldC(t, 0') -> t foldC(t, s(n)) -> f(foldC(t, n), C) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, s(c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0', c), b)) f''(triple(a, b, c)) -> foldC(triple(a, b, 0'), c) encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) 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_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2)) encArg(cons_foldC(x_1, x_2)) -> foldC(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_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_foldC(x_1, x_2) -> foldC(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:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' A :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' B :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' C :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' 0' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' s :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' triple :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encArg :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_g :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_g :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_A :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_B :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_C :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_0 :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_s :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_triple :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' hole_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''1_4 :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4 :: Nat -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' Lemmas: foldC(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, n113_4))) -> *3_4, rt in Omega(n113_4) Generator Equations: gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(0) <=> A gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(x, 1)) <=> s(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(x)) The following defined symbols remain to be analysed: foldB, f, f', f'', encArg They will be analysed ascendingly in the following order: foldB = f foldB = foldC foldB = f' foldB = f'' foldB < encArg f = foldC f = f' f = f'' f < encArg foldC = f' foldC = f'' foldC < encArg f' = f'' f' < encArg f'' < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: foldB(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, n6192_4))) -> *3_4, rt in Omega(n6192_4) Induction Base: foldB(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, 0))) Induction Step: foldB(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, +(n6192_4, 1)))) ->_R^Omega(1) f(foldB(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, n6192_4))), B) ->_IH f(*3_4, B) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: Innermost TRS: Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldB(t, 0') -> t foldB(t, s(n)) -> f(foldB(t, n), B) foldC(t, 0') -> t foldC(t, s(n)) -> f(foldC(t, n), C) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, s(c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0', c), b)) f''(triple(a, b, c)) -> foldC(triple(a, b, 0'), c) encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) 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_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2)) encArg(cons_foldC(x_1, x_2)) -> foldC(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_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_foldC(x_1, x_2) -> foldC(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:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' A :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' B :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' C :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' 0' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' s :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' triple :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encArg :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_g :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_g :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_A :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_B :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_C :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_0 :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_s :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_triple :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' hole_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''1_4 :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4 :: Nat -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' Lemmas: foldC(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, n113_4))) -> *3_4, rt in Omega(n113_4) foldB(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, n6192_4))) -> *3_4, rt in Omega(n6192_4) Generator Equations: gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(0) <=> A gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(x, 1)) <=> s(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(x)) The following defined symbols remain to be analysed: f, foldC, f', f'', encArg They will be analysed ascendingly in the following order: foldB = f foldB = foldC foldB = f' foldB = f'' foldB < encArg f = foldC f = f' f = f'' f < encArg foldC = f' foldC = f'' foldC < encArg f' = f'' f' < encArg f'' < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: foldC(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, n10582_4))) -> *3_4, rt in Omega(n10582_4) Induction Base: foldC(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, 0))) Induction Step: foldC(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, +(n10582_4, 1)))) ->_R^Omega(1) f(foldC(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, n10582_4))), C) ->_IH f(*3_4, C) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: Innermost TRS: Rules: g(A) -> A g(B) -> A g(B) -> B g(C) -> A g(C) -> B g(C) -> C foldB(t, 0') -> t foldB(t, s(n)) -> f(foldB(t, n), B) foldC(t, 0') -> t foldC(t, s(n)) -> f(foldC(t, n), C) f(t, x) -> f'(t, g(x)) f'(triple(a, b, c), C) -> triple(a, b, s(c)) f'(triple(a, b, c), B) -> f(triple(a, b, c), A) f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0', c), b)) f''(triple(a, b, c)) -> foldC(triple(a, b, 0'), c) encArg(A) -> A encArg(B) -> B encArg(C) -> C encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) 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_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2)) encArg(cons_foldC(x_1, x_2)) -> foldC(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_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_foldC(x_1, x_2) -> foldC(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:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' A :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' B :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' C :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' 0' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' s :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' triple :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encArg :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_g :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' cons_f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_g :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_A :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_B :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_C :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_foldB :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_0 :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_s :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_foldC :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_triple :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' encode_f'' :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' hole_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''1_4 :: A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4 :: Nat -> A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f'' Lemmas: foldC(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, n10582_4))) -> *3_4, rt in Omega(n10582_4) foldB(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(a), gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(1, n6192_4))) -> *3_4, rt in Omega(n6192_4) Generator Equations: gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(0) <=> A gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(x, 1)) <=> s(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(x)) The following defined symbols remain to be analysed: encArg ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(n17445_4)) -> gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(n17445_4), rt in Omega(0) Induction Base: encArg(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(0)) ->_R^Omega(0) A Induction Step: encArg(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(+(n17445_4, 1))) ->_R^Omega(0) s(encArg(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(n17445_4))) ->_IH s(gen_A:B:C:0':s:triple:cons_g:cons_foldB:cons_foldC:cons_f:cons_f':cons_f''2_4(c17446_4)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (22) BOUNDS(1, INF)