(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a__zeros → cons(0, zeros)
a__tail(cons(X, XS)) → mark(XS)
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__zeros → zeros
a__tail(X) → tail(X)
Rewrite Strategy: INNERMOST
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
a__tail(cons(X, tail(cons(X134_3, X235_3)))) →+ a__tail(cons(mark(X134_3), X235_3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X235_3 / tail(cons(X134_3, X235_3))].
The result substitution is [X / mark(X134_3)].
(2) BOUNDS(n^1, INF)
(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
a__zeros → cons(0', zeros)
a__tail(cons(X, XS)) → mark(XS)
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
a__zeros → zeros
a__tail(X) → tail(X)
S is empty.
Rewrite Strategy: INNERMOST
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
Innermost TRS:
Rules:
a__zeros → cons(0', zeros)
a__tail(cons(X, XS)) → mark(XS)
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
a__zeros → zeros
a__tail(X) → tail(X)
Types:
a__zeros :: 0':zeros:cons:tail
cons :: 0':zeros:cons:tail → 0':zeros:cons:tail → 0':zeros:cons:tail
0' :: 0':zeros:cons:tail
zeros :: 0':zeros:cons:tail
a__tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
mark :: 0':zeros:cons:tail → 0':zeros:cons:tail
tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
hole_0':zeros:cons:tail1_0 :: 0':zeros:cons:tail
gen_0':zeros:cons:tail2_0 :: Nat → 0':zeros:cons:tail
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
a__tail,
markThey will be analysed ascendingly in the following order:
a__tail = mark
(8) Obligation:
Innermost TRS:
Rules:
a__zeros →
cons(
0',
zeros)
a__tail(
cons(
X,
XS)) →
mark(
XS)
mark(
zeros) →
a__zerosmark(
tail(
X)) →
a__tail(
mark(
X))
mark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
0') →
0'a__zeros →
zerosa__tail(
X) →
tail(
X)
Types:
a__zeros :: 0':zeros:cons:tail
cons :: 0':zeros:cons:tail → 0':zeros:cons:tail → 0':zeros:cons:tail
0' :: 0':zeros:cons:tail
zeros :: 0':zeros:cons:tail
a__tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
mark :: 0':zeros:cons:tail → 0':zeros:cons:tail
tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
hole_0':zeros:cons:tail1_0 :: 0':zeros:cons:tail
gen_0':zeros:cons:tail2_0 :: Nat → 0':zeros:cons:tail
Generator Equations:
gen_0':zeros:cons:tail2_0(0) ⇔ 0'
gen_0':zeros:cons:tail2_0(+(x, 1)) ⇔ cons(gen_0':zeros:cons:tail2_0(x), 0')
The following defined symbols remain to be analysed:
mark, a__tail
They will be analysed ascendingly in the following order:
a__tail = mark
(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
mark(
gen_0':zeros:cons:tail2_0(
n4_0)) →
gen_0':zeros:cons:tail2_0(
n4_0), rt ∈ Ω(1 + n4
0)
Induction Base:
mark(gen_0':zeros:cons:tail2_0(0)) →RΩ(1)
0'
Induction Step:
mark(gen_0':zeros:cons:tail2_0(+(n4_0, 1))) →RΩ(1)
cons(mark(gen_0':zeros:cons:tail2_0(n4_0)), 0') →IH
cons(gen_0':zeros:cons:tail2_0(c5_0), 0')
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(10) Complex Obligation (BEST)
(11) Obligation:
Innermost TRS:
Rules:
a__zeros →
cons(
0',
zeros)
a__tail(
cons(
X,
XS)) →
mark(
XS)
mark(
zeros) →
a__zerosmark(
tail(
X)) →
a__tail(
mark(
X))
mark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
0') →
0'a__zeros →
zerosa__tail(
X) →
tail(
X)
Types:
a__zeros :: 0':zeros:cons:tail
cons :: 0':zeros:cons:tail → 0':zeros:cons:tail → 0':zeros:cons:tail
0' :: 0':zeros:cons:tail
zeros :: 0':zeros:cons:tail
a__tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
mark :: 0':zeros:cons:tail → 0':zeros:cons:tail
tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
hole_0':zeros:cons:tail1_0 :: 0':zeros:cons:tail
gen_0':zeros:cons:tail2_0 :: Nat → 0':zeros:cons:tail
Lemmas:
mark(gen_0':zeros:cons:tail2_0(n4_0)) → gen_0':zeros:cons:tail2_0(n4_0), rt ∈ Ω(1 + n40)
Generator Equations:
gen_0':zeros:cons:tail2_0(0) ⇔ 0'
gen_0':zeros:cons:tail2_0(+(x, 1)) ⇔ cons(gen_0':zeros:cons:tail2_0(x), 0')
The following defined symbols remain to be analysed:
a__tail
They will be analysed ascendingly in the following order:
a__tail = mark
(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol a__tail.
(13) Obligation:
Innermost TRS:
Rules:
a__zeros →
cons(
0',
zeros)
a__tail(
cons(
X,
XS)) →
mark(
XS)
mark(
zeros) →
a__zerosmark(
tail(
X)) →
a__tail(
mark(
X))
mark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
0') →
0'a__zeros →
zerosa__tail(
X) →
tail(
X)
Types:
a__zeros :: 0':zeros:cons:tail
cons :: 0':zeros:cons:tail → 0':zeros:cons:tail → 0':zeros:cons:tail
0' :: 0':zeros:cons:tail
zeros :: 0':zeros:cons:tail
a__tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
mark :: 0':zeros:cons:tail → 0':zeros:cons:tail
tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
hole_0':zeros:cons:tail1_0 :: 0':zeros:cons:tail
gen_0':zeros:cons:tail2_0 :: Nat → 0':zeros:cons:tail
Lemmas:
mark(gen_0':zeros:cons:tail2_0(n4_0)) → gen_0':zeros:cons:tail2_0(n4_0), rt ∈ Ω(1 + n40)
Generator Equations:
gen_0':zeros:cons:tail2_0(0) ⇔ 0'
gen_0':zeros:cons:tail2_0(+(x, 1)) ⇔ cons(gen_0':zeros:cons:tail2_0(x), 0')
No more defined symbols left to analyse.
(14) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_0':zeros:cons:tail2_0(n4_0)) → gen_0':zeros:cons:tail2_0(n4_0), rt ∈ Ω(1 + n40)
(15) BOUNDS(n^1, INF)
(16) Obligation:
Innermost TRS:
Rules:
a__zeros →
cons(
0',
zeros)
a__tail(
cons(
X,
XS)) →
mark(
XS)
mark(
zeros) →
a__zerosmark(
tail(
X)) →
a__tail(
mark(
X))
mark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
0') →
0'a__zeros →
zerosa__tail(
X) →
tail(
X)
Types:
a__zeros :: 0':zeros:cons:tail
cons :: 0':zeros:cons:tail → 0':zeros:cons:tail → 0':zeros:cons:tail
0' :: 0':zeros:cons:tail
zeros :: 0':zeros:cons:tail
a__tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
mark :: 0':zeros:cons:tail → 0':zeros:cons:tail
tail :: 0':zeros:cons:tail → 0':zeros:cons:tail
hole_0':zeros:cons:tail1_0 :: 0':zeros:cons:tail
gen_0':zeros:cons:tail2_0 :: Nat → 0':zeros:cons:tail
Lemmas:
mark(gen_0':zeros:cons:tail2_0(n4_0)) → gen_0':zeros:cons:tail2_0(n4_0), rt ∈ Ω(1 + n40)
Generator Equations:
gen_0':zeros:cons:tail2_0(0) ⇔ 0'
gen_0':zeros:cons:tail2_0(+(x, 1)) ⇔ cons(gen_0':zeros:cons:tail2_0(x), 0')
No more defined symbols left to analyse.
(17) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_0':zeros:cons:tail2_0(n4_0)) → gen_0':zeros:cons:tail2_0(n4_0), rt ∈ Ω(1 + n40)
(18) BOUNDS(n^1, INF)