The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0, x, y, 0)
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0) → true
isZero(s(0)) → false
isZero(s(s(x))) → isZero(s(x))
inc(0) → s(0)
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0) → 0
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0, y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1, z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
isZero(s(s(x))) →⁺ isZero(s(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x / s(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:

plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0', x, y, 0')
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0') → true
isZero(s(0')) → false
isZero(s(s(x))) → isZero(s(x))
inc(0') → s(0')
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0') → 0'
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0', y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1', z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0', x, y, 0')
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0') → true
isZero(s(0')) → false
isZero(s(s(x))) → isZero(s(x))
inc(0') → s(0')
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0') → 0'
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0', y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1', z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

Types:
plus :: 0':s:1' → 0':s:1' → 0':s:1'
ifPlus :: true:false → 0':s:1' → 0':s:1' → 0':s:1'
isZero :: 0':s:1' → true:false
inc :: 0':s:1' → 0':s:1'
true :: true:false
p :: 0':s:1' → 0':s:1'
false :: true:false
times :: 0':s:1' → 0':s:1' → 0':s:1'
timesIter :: 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
ge :: 0':s:1' → 0':s:1' → true:false
s :: 0':s:1' → 0':s:1'
f0 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f1 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f2 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat → 0':s:1'

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
plus, isZero, inc, p, timesIter, ge, f0, f1, f2

They will be analysed ascendingly in the following order:
isZero < plus
inc < plus
p < plus
plus < timesIter
inc < timesIter
ge < timesIter
f0 = f1
f0 = f2
f1 = f2

(8) Obligation:

TRS:
Rules:
plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0', x, y, 0')
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0') → true
isZero(s(0')) → false
isZero(s(s(x))) → isZero(s(x))
inc(0') → s(0')
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0') → 0'
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0', y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1', z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

Types:
plus :: 0':s:1' → 0':s:1' → 0':s:1'
ifPlus :: true:false → 0':s:1' → 0':s:1' → 0':s:1'
isZero :: 0':s:1' → true:false
inc :: 0':s:1' → 0':s:1'
true :: true:false
p :: 0':s:1' → 0':s:1'
false :: true:false
times :: 0':s:1' → 0':s:1' → 0':s:1'
timesIter :: 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
ge :: 0':s:1' → 0':s:1' → true:false
s :: 0':s:1' → 0':s:1'
f0 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f1 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f2 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat → 0':s:1'

Generator Equations:
gen_0':s:1'4_3(0) ⇔ 0'
gen_0':s:1'4_3(+(x, 1)) ⇔ s(gen_0':s:1'4_3(x))

The following defined symbols remain to be analysed:
isZero, plus, inc, p, timesIter, ge, f0, f1, f2

They will be analysed ascendingly in the following order:
isZero < plus
inc < plus
p < plus
plus < timesIter
inc < timesIter
ge < timesIter
f0 = f1
f0 = f2
f1 = f2

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)

Induction Base:
isZero(gen_0':s:1'4_3(+(1, 0))) →_R^Ω(1)
false

Induction Step:
isZero(gen_0':s:1'4_3(+(1, +(n6_3, 1)))) →_R^Ω(1)
isZero(s(gen_0':s:1'4_3(n6_3))) →_IH
false

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0', x, y, 0')
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0') → true
isZero(s(0')) → false
isZero(s(s(x))) → isZero(s(x))
inc(0') → s(0')
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0') → 0'
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0', y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1', z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

Types:
plus :: 0':s:1' → 0':s:1' → 0':s:1'
ifPlus :: true:false → 0':s:1' → 0':s:1' → 0':s:1'
isZero :: 0':s:1' → true:false
inc :: 0':s:1' → 0':s:1'
true :: true:false
p :: 0':s:1' → 0':s:1'
false :: true:false
times :: 0':s:1' → 0':s:1' → 0':s:1'
timesIter :: 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
ge :: 0':s:1' → 0':s:1' → true:false
s :: 0':s:1' → 0':s:1'
f0 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f1 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f2 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat → 0':s:1'

Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)

Generator Equations:
gen_0':s:1'4_3(0) ⇔ 0'
gen_0':s:1'4_3(+(x, 1)) ⇔ s(gen_0':s:1'4_3(x))

The following defined symbols remain to be analysed:
inc, plus, p, timesIter, ge, f0, f1, f2

They will be analysed ascendingly in the following order:
inc < plus
p < plus
plus < timesIter
inc < timesIter
ge < timesIter
f0 = f1
f0 = f2
f1 = f2

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
inc(gen_0':s:1'4_3(n163_3)) → gen_0':s:1'4_3(+(1, n163_3)), rt ∈ Ω(1 + n163₃)

Induction Base:
inc(gen_0':s:1'4_3(0)) →_R^Ω(1)
s(0')

Induction Step:
inc(gen_0':s:1'4_3(+(n163_3, 1))) →_R^Ω(1)
s(inc(gen_0':s:1'4_3(n163_3))) →_IH
s(gen_0':s:1'4_3(+(1, c164_3)))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0', x, y, 0')
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0') → true
isZero(s(0')) → false
isZero(s(s(x))) → isZero(s(x))
inc(0') → s(0')
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0') → 0'
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0', y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1', z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

Types:
plus :: 0':s:1' → 0':s:1' → 0':s:1'
ifPlus :: true:false → 0':s:1' → 0':s:1' → 0':s:1'
isZero :: 0':s:1' → true:false
inc :: 0':s:1' → 0':s:1'
true :: true:false
p :: 0':s:1' → 0':s:1'
false :: true:false
times :: 0':s:1' → 0':s:1' → 0':s:1'
timesIter :: 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
ge :: 0':s:1' → 0':s:1' → true:false
s :: 0':s:1' → 0':s:1'
f0 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f1 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f2 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat → 0':s:1'

Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)
inc(gen_0':s:1'4_3(n163_3)) → gen_0':s:1'4_3(+(1, n163_3)), rt ∈ Ω(1 + n163₃)

Generator Equations:
gen_0':s:1'4_3(0) ⇔ 0'
gen_0':s:1'4_3(+(x, 1)) ⇔ s(gen_0':s:1'4_3(x))

The following defined symbols remain to be analysed:
p, plus, timesIter, ge, f0, f1, f2

They will be analysed ascendingly in the following order:
p < plus
plus < timesIter
ge < timesIter
f0 = f1
f0 = f2
f1 = f2

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol p.

(16) Obligation:

TRS:
Rules:
plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0', x, y, 0')
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0') → true
isZero(s(0')) → false
isZero(s(s(x))) → isZero(s(x))
inc(0') → s(0')
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0') → 0'
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0', y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1', z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

Types:
plus :: 0':s:1' → 0':s:1' → 0':s:1'
ifPlus :: true:false → 0':s:1' → 0':s:1' → 0':s:1'
isZero :: 0':s:1' → true:false
inc :: 0':s:1' → 0':s:1'
true :: true:false
p :: 0':s:1' → 0':s:1'
false :: true:false
times :: 0':s:1' → 0':s:1' → 0':s:1'
timesIter :: 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
ge :: 0':s:1' → 0':s:1' → true:false
s :: 0':s:1' → 0':s:1'
f0 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f1 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f2 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat → 0':s:1'

Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)
inc(gen_0':s:1'4_3(n163_3)) → gen_0':s:1'4_3(+(1, n163_3)), rt ∈ Ω(1 + n163₃)

Generator Equations:
gen_0':s:1'4_3(0) ⇔ 0'
gen_0':s:1'4_3(+(x, 1)) ⇔ s(gen_0':s:1'4_3(x))

The following defined symbols remain to be analysed:
plus, timesIter, ge, f0, f1, f2

They will be analysed ascendingly in the following order:
plus < timesIter
ge < timesIter
f0 = f1
f0 = f2
f1 = f2

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol plus.

(18) Obligation:

TRS:
Rules:
plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0', x, y, 0')
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0') → true
isZero(s(0')) → false
isZero(s(s(x))) → isZero(s(x))
inc(0') → s(0')
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0') → 0'
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0', y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1', z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

Types:
plus :: 0':s:1' → 0':s:1' → 0':s:1'
ifPlus :: true:false → 0':s:1' → 0':s:1' → 0':s:1'
isZero :: 0':s:1' → true:false
inc :: 0':s:1' → 0':s:1'
true :: true:false
p :: 0':s:1' → 0':s:1'
false :: true:false
times :: 0':s:1' → 0':s:1' → 0':s:1'
timesIter :: 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
ge :: 0':s:1' → 0':s:1' → true:false
s :: 0':s:1' → 0':s:1'
f0 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f1 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f2 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat → 0':s:1'

Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)
inc(gen_0':s:1'4_3(n163_3)) → gen_0':s:1'4_3(+(1, n163_3)), rt ∈ Ω(1 + n163₃)

Generator Equations:
gen_0':s:1'4_3(0) ⇔ 0'
gen_0':s:1'4_3(+(x, 1)) ⇔ s(gen_0':s:1'4_3(x))

The following defined symbols remain to be analysed:
ge, timesIter, f0, f1, f2

They will be analysed ascendingly in the following order:
ge < timesIter
f0 = f1
f0 = f2
f1 = f2

(19) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
ge(gen_0':s:1'4_3(n1329_3), gen_0':s:1'4_3(n1329_3)) → true, rt ∈ Ω(1 + n1329₃)

Induction Base:
ge(gen_0':s:1'4_3(0), gen_0':s:1'4_3(0)) →_R^Ω(1)
true

Induction Step:
ge(gen_0':s:1'4_3(+(n1329_3, 1)), gen_0':s:1'4_3(+(n1329_3, 1))) →_R^Ω(1)
ge(gen_0':s:1'4_3(n1329_3), gen_0':s:1'4_3(n1329_3)) →_IH
true

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(20) Complex Obligation (BEST)

(21) Obligation:

TRS:
Rules:
plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0', x, y, 0')
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0') → true
isZero(s(0')) → false
isZero(s(s(x))) → isZero(s(x))
inc(0') → s(0')
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0') → 0'
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0', y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1', z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

Types:
plus :: 0':s:1' → 0':s:1' → 0':s:1'
ifPlus :: true:false → 0':s:1' → 0':s:1' → 0':s:1'
isZero :: 0':s:1' → true:false
inc :: 0':s:1' → 0':s:1'
true :: true:false
p :: 0':s:1' → 0':s:1'
false :: true:false
times :: 0':s:1' → 0':s:1' → 0':s:1'
timesIter :: 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
ge :: 0':s:1' → 0':s:1' → true:false
s :: 0':s:1' → 0':s:1'
f0 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f1 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f2 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat → 0':s:1'

Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)
inc(gen_0':s:1'4_3(n163_3)) → gen_0':s:1'4_3(+(1, n163_3)), rt ∈ Ω(1 + n163₃)
ge(gen_0':s:1'4_3(n1329_3), gen_0':s:1'4_3(n1329_3)) → true, rt ∈ Ω(1 + n1329₃)

Generator Equations:
gen_0':s:1'4_3(0) ⇔ 0'
gen_0':s:1'4_3(+(x, 1)) ⇔ s(gen_0':s:1'4_3(x))

The following defined symbols remain to be analysed:
timesIter, f0, f1, f2

They will be analysed ascendingly in the following order:
f0 = f1
f0 = f2
f1 = f2

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol timesIter.

(23) Obligation:

TRS:
Rules:
plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0', x, y, 0')
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0') → true
isZero(s(0')) → false
isZero(s(s(x))) → isZero(s(x))
inc(0') → s(0')
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0') → 0'
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0', y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1', z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

Types:
plus :: 0':s:1' → 0':s:1' → 0':s:1'
ifPlus :: true:false → 0':s:1' → 0':s:1' → 0':s:1'
isZero :: 0':s:1' → true:false
inc :: 0':s:1' → 0':s:1'
true :: true:false
p :: 0':s:1' → 0':s:1'
false :: true:false
times :: 0':s:1' → 0':s:1' → 0':s:1'
timesIter :: 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
ge :: 0':s:1' → 0':s:1' → true:false
s :: 0':s:1' → 0':s:1'
f0 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f1 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f2 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat → 0':s:1'

Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)
inc(gen_0':s:1'4_3(n163_3)) → gen_0':s:1'4_3(+(1, n163_3)), rt ∈ Ω(1 + n163₃)
ge(gen_0':s:1'4_3(n1329_3), gen_0':s:1'4_3(n1329_3)) → true, rt ∈ Ω(1 + n1329₃)

Generator Equations:
gen_0':s:1'4_3(0) ⇔ 0'
gen_0':s:1'4_3(+(x, 1)) ⇔ s(gen_0':s:1'4_3(x))

The following defined symbols remain to be analysed:
f1, f0, f2

They will be analysed ascendingly in the following order:
f0 = f1
f0 = f2
f1 = f2

(24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f1.

(25) Obligation:

TRS:
Rules:
plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0', x, y, 0')
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0') → true
isZero(s(0')) → false
isZero(s(s(x))) → isZero(s(x))
inc(0') → s(0')
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0') → 0'
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0', y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1', z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

Types:
plus :: 0':s:1' → 0':s:1' → 0':s:1'
ifPlus :: true:false → 0':s:1' → 0':s:1' → 0':s:1'
isZero :: 0':s:1' → true:false
inc :: 0':s:1' → 0':s:1'
true :: true:false
p :: 0':s:1' → 0':s:1'
false :: true:false
times :: 0':s:1' → 0':s:1' → 0':s:1'
timesIter :: 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
ge :: 0':s:1' → 0':s:1' → true:false
s :: 0':s:1' → 0':s:1'
f0 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f1 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f2 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat → 0':s:1'

Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)
inc(gen_0':s:1'4_3(n163_3)) → gen_0':s:1'4_3(+(1, n163_3)), rt ∈ Ω(1 + n163₃)
ge(gen_0':s:1'4_3(n1329_3), gen_0':s:1'4_3(n1329_3)) → true, rt ∈ Ω(1 + n1329₃)

Generator Equations:
gen_0':s:1'4_3(0) ⇔ 0'
gen_0':s:1'4_3(+(x, 1)) ⇔ s(gen_0':s:1'4_3(x))

The following defined symbols remain to be analysed:
f2, f0

They will be analysed ascendingly in the following order:
f0 = f1
f0 = f2
f1 = f2

(26) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f2.

(27) Obligation:

TRS:
Rules:
plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0', x, y, 0')
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0') → true
isZero(s(0')) → false
isZero(s(s(x))) → isZero(s(x))
inc(0') → s(0')
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0') → 0'
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0', y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1', z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

Types:
plus :: 0':s:1' → 0':s:1' → 0':s:1'
ifPlus :: true:false → 0':s:1' → 0':s:1' → 0':s:1'
isZero :: 0':s:1' → true:false
inc :: 0':s:1' → 0':s:1'
true :: true:false
p :: 0':s:1' → 0':s:1'
false :: true:false
times :: 0':s:1' → 0':s:1' → 0':s:1'
timesIter :: 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
ge :: 0':s:1' → 0':s:1' → true:false
s :: 0':s:1' → 0':s:1'
f0 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f1 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f2 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat → 0':s:1'

Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)
inc(gen_0':s:1'4_3(n163_3)) → gen_0':s:1'4_3(+(1, n163_3)), rt ∈ Ω(1 + n163₃)
ge(gen_0':s:1'4_3(n1329_3), gen_0':s:1'4_3(n1329_3)) → true, rt ∈ Ω(1 + n1329₃)

Generator Equations:
gen_0':s:1'4_3(0) ⇔ 0'
gen_0':s:1'4_3(+(x, 1)) ⇔ s(gen_0':s:1'4_3(x))

The following defined symbols remain to be analysed:
f0

They will be analysed ascendingly in the following order:
f0 = f1
f0 = f2
f1 = f2

(28) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f0.

(29) Obligation:

TRS:
Rules:
plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0', x, y, 0')
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0') → true
isZero(s(0')) → false
isZero(s(s(x))) → isZero(s(x))
inc(0') → s(0')
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0') → 0'
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0', y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1', z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

Types:
plus :: 0':s:1' → 0':s:1' → 0':s:1'
ifPlus :: true:false → 0':s:1' → 0':s:1' → 0':s:1'
isZero :: 0':s:1' → true:false
inc :: 0':s:1' → 0':s:1'
true :: true:false
p :: 0':s:1' → 0':s:1'
false :: true:false
times :: 0':s:1' → 0':s:1' → 0':s:1'
timesIter :: 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
ge :: 0':s:1' → 0':s:1' → true:false
s :: 0':s:1' → 0':s:1'
f0 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f1 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f2 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat → 0':s:1'

Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)
inc(gen_0':s:1'4_3(n163_3)) → gen_0':s:1'4_3(+(1, n163_3)), rt ∈ Ω(1 + n163₃)
ge(gen_0':s:1'4_3(n1329_3), gen_0':s:1'4_3(n1329_3)) → true, rt ∈ Ω(1 + n1329₃)

Generator Equations:
gen_0':s:1'4_3(0) ⇔ 0'
gen_0':s:1'4_3(+(x, 1)) ⇔ s(gen_0':s:1'4_3(x))

No more defined symbols left to analyse.

(30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)

(31) BOUNDS(n^1, INF)

(32) Obligation:

TRS:
Rules:
plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0', x, y, 0')
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0') → true
isZero(s(0')) → false
isZero(s(s(x))) → isZero(s(x))
inc(0') → s(0')
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0') → 0'
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0', y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1', z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

Types:
plus :: 0':s:1' → 0':s:1' → 0':s:1'
ifPlus :: true:false → 0':s:1' → 0':s:1' → 0':s:1'
isZero :: 0':s:1' → true:false
inc :: 0':s:1' → 0':s:1'
true :: true:false
p :: 0':s:1' → 0':s:1'
false :: true:false
times :: 0':s:1' → 0':s:1' → 0':s:1'
timesIter :: 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
ge :: 0':s:1' → 0':s:1' → true:false
s :: 0':s:1' → 0':s:1'
f0 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f1 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f2 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat → 0':s:1'

Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)
inc(gen_0':s:1'4_3(n163_3)) → gen_0':s:1'4_3(+(1, n163_3)), rt ∈ Ω(1 + n163₃)
ge(gen_0':s:1'4_3(n1329_3), gen_0':s:1'4_3(n1329_3)) → true, rt ∈ Ω(1 + n1329₃)

Generator Equations:
gen_0':s:1'4_3(0) ⇔ 0'
gen_0':s:1'4_3(+(x, 1)) ⇔ s(gen_0':s:1'4_3(x))

No more defined symbols left to analyse.

(33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)

(34) BOUNDS(n^1, INF)

(35) Obligation:

TRS:
Rules:
plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0', x, y, 0')
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0') → true
isZero(s(0')) → false
isZero(s(s(x))) → isZero(s(x))
inc(0') → s(0')
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0') → 0'
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0', y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1', z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

Types:
plus :: 0':s:1' → 0':s:1' → 0':s:1'
ifPlus :: true:false → 0':s:1' → 0':s:1' → 0':s:1'
isZero :: 0':s:1' → true:false
inc :: 0':s:1' → 0':s:1'
true :: true:false
p :: 0':s:1' → 0':s:1'
false :: true:false
times :: 0':s:1' → 0':s:1' → 0':s:1'
timesIter :: 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
ge :: 0':s:1' → 0':s:1' → true:false
s :: 0':s:1' → 0':s:1'
f0 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f1 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f2 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat → 0':s:1'

Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)
inc(gen_0':s:1'4_3(n163_3)) → gen_0':s:1'4_3(+(1, n163_3)), rt ∈ Ω(1 + n163₃)

Generator Equations:
gen_0':s:1'4_3(0) ⇔ 0'
gen_0':s:1'4_3(+(x, 1)) ⇔ s(gen_0':s:1'4_3(x))

No more defined symbols left to analyse.

(36) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)

(37) BOUNDS(n^1, INF)

(38) Obligation:

TRS:
Rules:
plus(x, y) → ifPlus(isZero(x), x, inc(y))
ifPlus(true, x, y) → p(y)
ifPlus(false, x, y) → plus(p(x), y)
times(x, y) → timesIter(0', x, y, 0')
timesIter(i, x, y, z) → ifTimes(ge(i, x), i, x, y, z)
ifTimes(true, i, x, y, z) → z
ifTimes(false, i, x, y, z) → timesIter(inc(i), x, y, plus(z, y))
isZero(0') → true
isZero(s(0')) → false
isZero(s(s(x))) → isZero(s(x))
inc(0') → s(0')
inc(s(x)) → s(inc(x))
inc(x) → s(x)
p(0') → 0'
p(s(x)) → x
p(s(s(x))) → s(p(s(x)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
f0(0', y, x) → f1(x, y, x)
f1(x, y, z) → f2(x, y, z)
f2(x, 1', z) → f0(x, z, z)
f0(x, y, z) → d
f1(x, y, z) → c

Types:
plus :: 0':s:1' → 0':s:1' → 0':s:1'
ifPlus :: true:false → 0':s:1' → 0':s:1' → 0':s:1'
isZero :: 0':s:1' → true:false
inc :: 0':s:1' → 0':s:1'
true :: true:false
p :: 0':s:1' → 0':s:1'
false :: true:false
times :: 0':s:1' → 0':s:1' → 0':s:1'
timesIter :: 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
0' :: 0':s:1'
ifTimes :: true:false → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1' → 0':s:1'
ge :: 0':s:1' → 0':s:1' → true:false
s :: 0':s:1' → 0':s:1'
f0 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f1 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
f2 :: 0':s:1' → 0':s:1' → 0':s:1' → d:c
1' :: 0':s:1'
d :: d:c
c :: d:c
hole_0':s:1'1_3 :: 0':s:1'
hole_true:false2_3 :: true:false
hole_d:c3_3 :: d:c
gen_0':s:1'4_3 :: Nat → 0':s:1'

Lemmas:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)

Generator Equations:
gen_0':s:1'4_3(0) ⇔ 0'
gen_0':s:1'4_3(+(x, 1)) ⇔ s(gen_0':s:1'4_3(x))

No more defined symbols left to analyse.

(39) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
isZero(gen_0':s:1'4_3(+(1, n6_3))) → false, rt ∈ Ω(1 + n6₃)

(40) BOUNDS(n^1, INF)