(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
plus(0, x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0, y) → 0
times(s(x), y) → plus(y, times(p(s(x)), y))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0))
towerIter(0, y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))
Rewrite Strategy: FULL
(1) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.
(2) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
plus(0', x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(p(s(x)), y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0'))
towerIter(0', y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))
S is empty.
Rewrite Strategy: FULL
(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(4) Obligation:
TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(p(s(x)), y))
exp(x, 0') → s(0')
exp(x, s(y)) → times(x, exp(x, y))
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0'))
towerIter(0', y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))
Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s
(5) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
plus,
p,
times,
exp,
towerIterThey will be analysed ascendingly in the following order:
p < plus
plus < times
p < times
p < towerIter
times < exp
exp < towerIter
(6) Obligation:
TRS:
Rules:
plus(
0',
x) →
xplus(
s(
x),
y) →
s(
plus(
p(
s(
x)),
y))
times(
0',
y) →
0'times(
s(
x),
y) →
plus(
y,
times(
p(
s(
x)),
y))
exp(
x,
0') →
s(
0')
exp(
x,
s(
y)) →
times(
x,
exp(
x,
y))
p(
s(
0')) →
0'p(
s(
s(
x))) →
s(
p(
s(
x)))
tower(
x,
y) →
towerIter(
x,
y,
s(
0'))
towerIter(
0',
y,
z) →
ztowerIter(
s(
x),
y,
z) →
towerIter(
p(
s(
x)),
y,
exp(
y,
z))
Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s
Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))
The following defined symbols remain to be analysed:
p, plus, times, exp, towerIter
They will be analysed ascendingly in the following order:
p < plus
plus < times
p < times
p < towerIter
times < exp
exp < towerIter
(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
p(
gen_0':s2_0(
+(
1,
n4_0))) →
gen_0':s2_0(
n4_0), rt ∈ Ω(1 + n4
0)
Induction Base:
p(gen_0':s2_0(+(1, 0))) →RΩ(1)
0'
Induction Step:
p(gen_0':s2_0(+(1, +(n4_0, 1)))) →RΩ(1)
s(p(s(gen_0':s2_0(n4_0)))) →IH
s(gen_0':s2_0(c5_0))
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(8) Complex Obligation (BEST)
(9) Obligation:
TRS:
Rules:
plus(
0',
x) →
xplus(
s(
x),
y) →
s(
plus(
p(
s(
x)),
y))
times(
0',
y) →
0'times(
s(
x),
y) →
plus(
y,
times(
p(
s(
x)),
y))
exp(
x,
0') →
s(
0')
exp(
x,
s(
y)) →
times(
x,
exp(
x,
y))
p(
s(
0')) →
0'p(
s(
s(
x))) →
s(
p(
s(
x)))
tower(
x,
y) →
towerIter(
x,
y,
s(
0'))
towerIter(
0',
y,
z) →
ztowerIter(
s(
x),
y,
z) →
towerIter(
p(
s(
x)),
y,
exp(
y,
z))
Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s
Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)
Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))
The following defined symbols remain to be analysed:
plus, times, exp, towerIter
They will be analysed ascendingly in the following order:
plus < times
times < exp
exp < towerIter
(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
plus(
gen_0':s2_0(
n201_0),
gen_0':s2_0(
b)) →
gen_0':s2_0(
+(
n201_0,
b)), rt ∈ Ω(1 + n201
0 + n201
02)
Induction Base:
plus(gen_0':s2_0(0), gen_0':s2_0(b)) →RΩ(1)
gen_0':s2_0(b)
Induction Step:
plus(gen_0':s2_0(+(n201_0, 1)), gen_0':s2_0(b)) →RΩ(1)
s(plus(p(s(gen_0':s2_0(n201_0))), gen_0':s2_0(b))) →LΩ(1 + n2010)
s(plus(gen_0':s2_0(n201_0), gen_0':s2_0(b))) →IH
s(gen_0':s2_0(+(b, c202_0)))
We have rt ∈ Ω(n2) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n2).
(11) Complex Obligation (BEST)
(12) Obligation:
TRS:
Rules:
plus(
0',
x) →
xplus(
s(
x),
y) →
s(
plus(
p(
s(
x)),
y))
times(
0',
y) →
0'times(
s(
x),
y) →
plus(
y,
times(
p(
s(
x)),
y))
exp(
x,
0') →
s(
0')
exp(
x,
s(
y)) →
times(
x,
exp(
x,
y))
p(
s(
0')) →
0'p(
s(
s(
x))) →
s(
p(
s(
x)))
tower(
x,
y) →
towerIter(
x,
y,
s(
0'))
towerIter(
0',
y,
z) →
ztowerIter(
s(
x),
y,
z) →
towerIter(
p(
s(
x)),
y,
exp(
y,
z))
Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s
Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n2010 + n20102)
Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))
The following defined symbols remain to be analysed:
times, exp, towerIter
They will be analysed ascendingly in the following order:
times < exp
exp < towerIter
(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
times(
gen_0':s2_0(
n625_0),
gen_0':s2_0(
b)) →
gen_0':s2_0(
*(
n625_0,
b)), rt ∈ Ω(1 + b·n625
0 + b
2·n625
0 + n625
0 + n625
02)
Induction Base:
times(gen_0':s2_0(0), gen_0':s2_0(b)) →RΩ(1)
0'
Induction Step:
times(gen_0':s2_0(+(n625_0, 1)), gen_0':s2_0(b)) →RΩ(1)
plus(gen_0':s2_0(b), times(p(s(gen_0':s2_0(n625_0))), gen_0':s2_0(b))) →LΩ(1 + n6250)
plus(gen_0':s2_0(b), times(gen_0':s2_0(n625_0), gen_0':s2_0(b))) →IH
plus(gen_0':s2_0(b), gen_0':s2_0(*(c626_0, b))) →LΩ(1 + b + b2)
gen_0':s2_0(+(b, *(n625_0, b)))
We have rt ∈ Ω(n3) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n3).
(14) Complex Obligation (BEST)
(15) Obligation:
TRS:
Rules:
plus(
0',
x) →
xplus(
s(
x),
y) →
s(
plus(
p(
s(
x)),
y))
times(
0',
y) →
0'times(
s(
x),
y) →
plus(
y,
times(
p(
s(
x)),
y))
exp(
x,
0') →
s(
0')
exp(
x,
s(
y)) →
times(
x,
exp(
x,
y))
p(
s(
0')) →
0'p(
s(
s(
x))) →
s(
p(
s(
x)))
tower(
x,
y) →
towerIter(
x,
y,
s(
0'))
towerIter(
0',
y,
z) →
ztowerIter(
s(
x),
y,
z) →
towerIter(
p(
s(
x)),
y,
exp(
y,
z))
Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s
Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n2010 + n20102)
times(gen_0':s2_0(n625_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n625_0, b)), rt ∈ Ω(1 + b·n6250 + b2·n6250 + n6250 + n62502)
Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))
The following defined symbols remain to be analysed:
exp, towerIter
They will be analysed ascendingly in the following order:
exp < towerIter
(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
exp(
gen_0':s2_0(
a),
gen_0':s2_0(
+(
1,
n1224_0))) →
*3_0, rt ∈ Ω(n1224
0)
Induction Base:
exp(gen_0':s2_0(a), gen_0':s2_0(+(1, 0)))
Induction Step:
exp(gen_0':s2_0(a), gen_0':s2_0(+(1, +(n1224_0, 1)))) →RΩ(1)
times(gen_0':s2_0(a), exp(gen_0':s2_0(a), gen_0':s2_0(+(1, n1224_0)))) →IH
times(gen_0':s2_0(a), *3_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(17) Complex Obligation (BEST)
(18) Obligation:
TRS:
Rules:
plus(
0',
x) →
xplus(
s(
x),
y) →
s(
plus(
p(
s(
x)),
y))
times(
0',
y) →
0'times(
s(
x),
y) →
plus(
y,
times(
p(
s(
x)),
y))
exp(
x,
0') →
s(
0')
exp(
x,
s(
y)) →
times(
x,
exp(
x,
y))
p(
s(
0')) →
0'p(
s(
s(
x))) →
s(
p(
s(
x)))
tower(
x,
y) →
towerIter(
x,
y,
s(
0'))
towerIter(
0',
y,
z) →
ztowerIter(
s(
x),
y,
z) →
towerIter(
p(
s(
x)),
y,
exp(
y,
z))
Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s
Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n2010 + n20102)
times(gen_0':s2_0(n625_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n625_0, b)), rt ∈ Ω(1 + b·n6250 + b2·n6250 + n6250 + n62502)
exp(gen_0':s2_0(a), gen_0':s2_0(+(1, n1224_0))) → *3_0, rt ∈ Ω(n12240)
Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))
The following defined symbols remain to be analysed:
towerIter
(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol towerIter.
(20) Obligation:
TRS:
Rules:
plus(
0',
x) →
xplus(
s(
x),
y) →
s(
plus(
p(
s(
x)),
y))
times(
0',
y) →
0'times(
s(
x),
y) →
plus(
y,
times(
p(
s(
x)),
y))
exp(
x,
0') →
s(
0')
exp(
x,
s(
y)) →
times(
x,
exp(
x,
y))
p(
s(
0')) →
0'p(
s(
s(
x))) →
s(
p(
s(
x)))
tower(
x,
y) →
towerIter(
x,
y,
s(
0'))
towerIter(
0',
y,
z) →
ztowerIter(
s(
x),
y,
z) →
towerIter(
p(
s(
x)),
y,
exp(
y,
z))
Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s
Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n2010 + n20102)
times(gen_0':s2_0(n625_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n625_0, b)), rt ∈ Ω(1 + b·n6250 + b2·n6250 + n6250 + n62502)
exp(gen_0':s2_0(a), gen_0':s2_0(+(1, n1224_0))) → *3_0, rt ∈ Ω(n12240)
Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))
No more defined symbols left to analyse.
(21) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n3) was proven with the following lemma:
times(gen_0':s2_0(n625_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n625_0, b)), rt ∈ Ω(1 + b·n6250 + b2·n6250 + n6250 + n62502)
(22) BOUNDS(n^3, INF)
(23) Obligation:
TRS:
Rules:
plus(
0',
x) →
xplus(
s(
x),
y) →
s(
plus(
p(
s(
x)),
y))
times(
0',
y) →
0'times(
s(
x),
y) →
plus(
y,
times(
p(
s(
x)),
y))
exp(
x,
0') →
s(
0')
exp(
x,
s(
y)) →
times(
x,
exp(
x,
y))
p(
s(
0')) →
0'p(
s(
s(
x))) →
s(
p(
s(
x)))
tower(
x,
y) →
towerIter(
x,
y,
s(
0'))
towerIter(
0',
y,
z) →
ztowerIter(
s(
x),
y,
z) →
towerIter(
p(
s(
x)),
y,
exp(
y,
z))
Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s
Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n2010 + n20102)
times(gen_0':s2_0(n625_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n625_0, b)), rt ∈ Ω(1 + b·n6250 + b2·n6250 + n6250 + n62502)
exp(gen_0':s2_0(a), gen_0':s2_0(+(1, n1224_0))) → *3_0, rt ∈ Ω(n12240)
Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))
No more defined symbols left to analyse.
(24) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n3) was proven with the following lemma:
times(gen_0':s2_0(n625_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n625_0, b)), rt ∈ Ω(1 + b·n6250 + b2·n6250 + n6250 + n62502)
(25) BOUNDS(n^3, INF)
(26) Obligation:
TRS:
Rules:
plus(
0',
x) →
xplus(
s(
x),
y) →
s(
plus(
p(
s(
x)),
y))
times(
0',
y) →
0'times(
s(
x),
y) →
plus(
y,
times(
p(
s(
x)),
y))
exp(
x,
0') →
s(
0')
exp(
x,
s(
y)) →
times(
x,
exp(
x,
y))
p(
s(
0')) →
0'p(
s(
s(
x))) →
s(
p(
s(
x)))
tower(
x,
y) →
towerIter(
x,
y,
s(
0'))
towerIter(
0',
y,
z) →
ztowerIter(
s(
x),
y,
z) →
towerIter(
p(
s(
x)),
y,
exp(
y,
z))
Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s
Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n2010 + n20102)
times(gen_0':s2_0(n625_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n625_0, b)), rt ∈ Ω(1 + b·n6250 + b2·n6250 + n6250 + n62502)
Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))
No more defined symbols left to analyse.
(27) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n3) was proven with the following lemma:
times(gen_0':s2_0(n625_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n625_0, b)), rt ∈ Ω(1 + b·n6250 + b2·n6250 + n6250 + n62502)
(28) BOUNDS(n^3, INF)
(29) Obligation:
TRS:
Rules:
plus(
0',
x) →
xplus(
s(
x),
y) →
s(
plus(
p(
s(
x)),
y))
times(
0',
y) →
0'times(
s(
x),
y) →
plus(
y,
times(
p(
s(
x)),
y))
exp(
x,
0') →
s(
0')
exp(
x,
s(
y)) →
times(
x,
exp(
x,
y))
p(
s(
0')) →
0'p(
s(
s(
x))) →
s(
p(
s(
x)))
tower(
x,
y) →
towerIter(
x,
y,
s(
0'))
towerIter(
0',
y,
z) →
ztowerIter(
s(
x),
y,
z) →
towerIter(
p(
s(
x)),
y,
exp(
y,
z))
Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s
Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n2010 + n20102)
Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))
No more defined symbols left to analyse.
(30) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n2) was proven with the following lemma:
plus(gen_0':s2_0(n201_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n201_0, b)), rt ∈ Ω(1 + n2010 + n20102)
(31) BOUNDS(n^2, INF)
(32) Obligation:
TRS:
Rules:
plus(
0',
x) →
xplus(
s(
x),
y) →
s(
plus(
p(
s(
x)),
y))
times(
0',
y) →
0'times(
s(
x),
y) →
plus(
y,
times(
p(
s(
x)),
y))
exp(
x,
0') →
s(
0')
exp(
x,
s(
y)) →
times(
x,
exp(
x,
y))
p(
s(
0')) →
0'p(
s(
s(
x))) →
s(
p(
s(
x)))
tower(
x,
y) →
towerIter(
x,
y,
s(
0'))
towerIter(
0',
y,
z) →
ztowerIter(
s(
x),
y,
z) →
towerIter(
p(
s(
x)),
y,
exp(
y,
z))
Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
exp :: 0':s → 0':s → 0':s
tower :: 0':s → 0':s → 0':s
towerIter :: 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s
Lemmas:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)
Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))
No more defined symbols left to analyse.
(33) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
p(gen_0':s2_0(+(1, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n40)
(34) BOUNDS(n^1, INF)