(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__p(s(0)) → 0
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
mark(f(X)) →+ a__f(mark(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / f(X)].
The result substitution is [ ].
(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__f(0') → cons(0', f(s(0')))
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(0')) → 0'
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)
S is empty.
Rewrite Strategy: FULL
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
TRS:
Rules:
a__f(0') → cons(0', f(s(0')))
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(0')) → 0'
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)
Types:
a__f :: 0':s:f:cons:p → 0':s:f:cons:p
0' :: 0':s:f:cons:p
cons :: 0':s:f:cons:p → 0':s:f:cons:p → 0':s:f:cons:p
f :: 0':s:f:cons:p → 0':s:f:cons:p
s :: 0':s:f:cons:p → 0':s:f:cons:p
a__p :: 0':s:f:cons:p → 0':s:f:cons:p
mark :: 0':s:f:cons:p → 0':s:f:cons:p
p :: 0':s:f:cons:p → 0':s:f:cons:p
hole_0':s:f:cons:p1_0 :: 0':s:f:cons:p
gen_0':s:f:cons:p2_0 :: Nat → 0':s:f:cons:p
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
a__f,
markThey will be analysed ascendingly in the following order:
a__f < mark
(8) Obligation:
TRS:
Rules:
a__f(
0') →
cons(
0',
f(
s(
0')))
a__f(
s(
0')) →
a__f(
a__p(
s(
0')))
a__p(
s(
0')) →
0'mark(
f(
X)) →
a__f(
mark(
X))
mark(
p(
X)) →
a__p(
mark(
X))
mark(
0') →
0'mark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
s(
X)) →
s(
mark(
X))
a__f(
X) →
f(
X)
a__p(
X) →
p(
X)
Types:
a__f :: 0':s:f:cons:p → 0':s:f:cons:p
0' :: 0':s:f:cons:p
cons :: 0':s:f:cons:p → 0':s:f:cons:p → 0':s:f:cons:p
f :: 0':s:f:cons:p → 0':s:f:cons:p
s :: 0':s:f:cons:p → 0':s:f:cons:p
a__p :: 0':s:f:cons:p → 0':s:f:cons:p
mark :: 0':s:f:cons:p → 0':s:f:cons:p
p :: 0':s:f:cons:p → 0':s:f:cons:p
hole_0':s:f:cons:p1_0 :: 0':s:f:cons:p
gen_0':s:f:cons:p2_0 :: Nat → 0':s:f:cons:p
Generator Equations:
gen_0':s:f:cons:p2_0(0) ⇔ 0'
gen_0':s:f:cons:p2_0(+(x, 1)) ⇔ cons(gen_0':s:f:cons:p2_0(x), 0')
The following defined symbols remain to be analysed:
a__f, mark
They will be analysed ascendingly in the following order:
a__f < mark
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol a__f.
(10) Obligation:
TRS:
Rules:
a__f(
0') →
cons(
0',
f(
s(
0')))
a__f(
s(
0')) →
a__f(
a__p(
s(
0')))
a__p(
s(
0')) →
0'mark(
f(
X)) →
a__f(
mark(
X))
mark(
p(
X)) →
a__p(
mark(
X))
mark(
0') →
0'mark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
s(
X)) →
s(
mark(
X))
a__f(
X) →
f(
X)
a__p(
X) →
p(
X)
Types:
a__f :: 0':s:f:cons:p → 0':s:f:cons:p
0' :: 0':s:f:cons:p
cons :: 0':s:f:cons:p → 0':s:f:cons:p → 0':s:f:cons:p
f :: 0':s:f:cons:p → 0':s:f:cons:p
s :: 0':s:f:cons:p → 0':s:f:cons:p
a__p :: 0':s:f:cons:p → 0':s:f:cons:p
mark :: 0':s:f:cons:p → 0':s:f:cons:p
p :: 0':s:f:cons:p → 0':s:f:cons:p
hole_0':s:f:cons:p1_0 :: 0':s:f:cons:p
gen_0':s:f:cons:p2_0 :: Nat → 0':s:f:cons:p
Generator Equations:
gen_0':s:f:cons:p2_0(0) ⇔ 0'
gen_0':s:f:cons:p2_0(+(x, 1)) ⇔ cons(gen_0':s:f:cons:p2_0(x), 0')
The following defined symbols remain to be analysed:
mark
(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
mark(
gen_0':s:f:cons:p2_0(
n11_0)) →
gen_0':s:f:cons:p2_0(
n11_0), rt ∈ Ω(1 + n11
0)
Induction Base:
mark(gen_0':s:f:cons:p2_0(0)) →RΩ(1)
0'
Induction Step:
mark(gen_0':s:f:cons:p2_0(+(n11_0, 1))) →RΩ(1)
cons(mark(gen_0':s:f:cons:p2_0(n11_0)), 0') →IH
cons(gen_0':s:f:cons:p2_0(c12_0), 0')
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(12) Complex Obligation (BEST)
(13) Obligation:
TRS:
Rules:
a__f(
0') →
cons(
0',
f(
s(
0')))
a__f(
s(
0')) →
a__f(
a__p(
s(
0')))
a__p(
s(
0')) →
0'mark(
f(
X)) →
a__f(
mark(
X))
mark(
p(
X)) →
a__p(
mark(
X))
mark(
0') →
0'mark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
s(
X)) →
s(
mark(
X))
a__f(
X) →
f(
X)
a__p(
X) →
p(
X)
Types:
a__f :: 0':s:f:cons:p → 0':s:f:cons:p
0' :: 0':s:f:cons:p
cons :: 0':s:f:cons:p → 0':s:f:cons:p → 0':s:f:cons:p
f :: 0':s:f:cons:p → 0':s:f:cons:p
s :: 0':s:f:cons:p → 0':s:f:cons:p
a__p :: 0':s:f:cons:p → 0':s:f:cons:p
mark :: 0':s:f:cons:p → 0':s:f:cons:p
p :: 0':s:f:cons:p → 0':s:f:cons:p
hole_0':s:f:cons:p1_0 :: 0':s:f:cons:p
gen_0':s:f:cons:p2_0 :: Nat → 0':s:f:cons:p
Lemmas:
mark(gen_0':s:f:cons:p2_0(n11_0)) → gen_0':s:f:cons:p2_0(n11_0), rt ∈ Ω(1 + n110)
Generator Equations:
gen_0':s:f:cons:p2_0(0) ⇔ 0'
gen_0':s:f:cons:p2_0(+(x, 1)) ⇔ cons(gen_0':s:f:cons:p2_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':s:f:cons:p2_0(n11_0)) → gen_0':s:f:cons:p2_0(n11_0), rt ∈ Ω(1 + n110)
(15) BOUNDS(n^1, INF)
(16) Obligation:
TRS:
Rules:
a__f(
0') →
cons(
0',
f(
s(
0')))
a__f(
s(
0')) →
a__f(
a__p(
s(
0')))
a__p(
s(
0')) →
0'mark(
f(
X)) →
a__f(
mark(
X))
mark(
p(
X)) →
a__p(
mark(
X))
mark(
0') →
0'mark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
s(
X)) →
s(
mark(
X))
a__f(
X) →
f(
X)
a__p(
X) →
p(
X)
Types:
a__f :: 0':s:f:cons:p → 0':s:f:cons:p
0' :: 0':s:f:cons:p
cons :: 0':s:f:cons:p → 0':s:f:cons:p → 0':s:f:cons:p
f :: 0':s:f:cons:p → 0':s:f:cons:p
s :: 0':s:f:cons:p → 0':s:f:cons:p
a__p :: 0':s:f:cons:p → 0':s:f:cons:p
mark :: 0':s:f:cons:p → 0':s:f:cons:p
p :: 0':s:f:cons:p → 0':s:f:cons:p
hole_0':s:f:cons:p1_0 :: 0':s:f:cons:p
gen_0':s:f:cons:p2_0 :: Nat → 0':s:f:cons:p
Lemmas:
mark(gen_0':s:f:cons:p2_0(n11_0)) → gen_0':s:f:cons:p2_0(n11_0), rt ∈ Ω(1 + n110)
Generator Equations:
gen_0':s:f:cons:p2_0(0) ⇔ 0'
gen_0':s:f:cons:p2_0(+(x, 1)) ⇔ cons(gen_0':s:f:cons:p2_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':s:f:cons:p2_0(n11_0)) → gen_0':s:f:cons:p2_0(n11_0), rt ∈ Ω(1 + n110)
(18) BOUNDS(n^1, INF)