The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

(0) Obligation:

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

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
+(x, s(y)) →⁺ s(+(x, y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [y / s(y)].
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:

+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0'))))
iffact(x, true) → *'(x, fact(-(x, s(0'))))
iffact(x, false) → s(0')

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0'))))
iffact(x, true) → *'(x, fact(-(x, s(0'))))
iffact(x, false) → s(0')

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
- :: 0':s → 0':s → 0':s
fact :: 0':s → 0':s
iffact :: 0':s → true:false → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
+', *', ge, -, fact

They will be analysed ascendingly in the following order:
+' < *'
*' < fact
ge < fact
- < fact

(8) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0'))))
iffact(x, true) → *'(x, fact(-(x, s(0'))))
iffact(x, false) → s(0')

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
- :: 0':s → 0':s → 0':s
fact :: 0':s → 0':s
iffact :: 0':s → true:false → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
+', *', ge, -, fact

They will be analysed ascendingly in the following order:
+' < *'
*' < fact
ge < fact
- < fact

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

Proved the following rewrite lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

Induction Base:
+'(gen_0':s3_0(a), gen_0':s3_0(0)) →_R^Ω(1)
gen_0':s3_0(a)

Induction Step:
+'(gen_0':s3_0(a), gen_0':s3_0(+(n5_0, 1))) →_R^Ω(1)
s(+'(gen_0':s3_0(a), gen_0':s3_0(n5_0))) →_IH
s(gen_0':s3_0(+(a, c6_0)))

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0'))))
iffact(x, true) → *'(x, fact(-(x, s(0'))))
iffact(x, false) → s(0')

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
- :: 0':s → 0':s → 0':s
fact :: 0':s → 0':s
iffact :: 0':s → true:false → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
*', ge, -, fact

They will be analysed ascendingly in the following order:
*' < fact
ge < fact
- < fact

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

Proved the following rewrite lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) → gen_0':s3_0(*(n548_0, a)), rt ∈ Ω(1 + a·n548₀ + n548₀)

Induction Base:
*'(gen_0':s3_0(a), gen_0':s3_0(0)) →_R^Ω(1)
0'

Induction Step:
*'(gen_0':s3_0(a), gen_0':s3_0(+(n548_0, 1))) →_R^Ω(1)
+'(*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)), gen_0':s3_0(a)) →_IH
+'(gen_0':s3_0(*(c549_0, a)), gen_0':s3_0(a)) →_L^{Ω(1 + a)}
gen_0':s3_0(+(a, *(n548_0, a)))

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0'))))
iffact(x, true) → *'(x, fact(-(x, s(0'))))
iffact(x, false) → s(0')

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
- :: 0':s → 0':s → 0':s
fact :: 0':s → 0':s
iffact :: 0':s → true:false → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) → gen_0':s3_0(*(n548_0, a)), rt ∈ Ω(1 + a·n548₀ + n548₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
ge, -, fact

They will be analysed ascendingly in the following order:
ge < fact
- < fact

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
ge(gen_0':s3_0(n1208_0), gen_0':s3_0(n1208_0)) → true, rt ∈ Ω(1 + n1208₀)

Induction Base:
ge(gen_0':s3_0(0), gen_0':s3_0(0)) →_R^Ω(1)
true

Induction Step:
ge(gen_0':s3_0(+(n1208_0, 1)), gen_0':s3_0(+(n1208_0, 1))) →_R^Ω(1)
ge(gen_0':s3_0(n1208_0), gen_0':s3_0(n1208_0)) →_IH
true

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

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0'))))
iffact(x, true) → *'(x, fact(-(x, s(0'))))
iffact(x, false) → s(0')

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
- :: 0':s → 0':s → 0':s
fact :: 0':s → 0':s
iffact :: 0':s → true:false → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) → gen_0':s3_0(*(n548_0, a)), rt ∈ Ω(1 + a·n548₀ + n548₀)
ge(gen_0':s3_0(n1208_0), gen_0':s3_0(n1208_0)) → true, rt ∈ Ω(1 + n1208₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
-, fact

They will be analysed ascendingly in the following order:
- < fact

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
-(gen_0':s3_0(n1503_0), gen_0':s3_0(n1503_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n1503₀)

Induction Base:
-(gen_0':s3_0(0), gen_0':s3_0(0)) →_R^Ω(1)
gen_0':s3_0(0)

Induction Step:
-(gen_0':s3_0(+(n1503_0, 1)), gen_0':s3_0(+(n1503_0, 1))) →_R^Ω(1)
-(gen_0':s3_0(n1503_0), gen_0':s3_0(n1503_0)) →_IH
gen_0':s3_0(0)

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

(19) Complex Obligation (BEST)

(20) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0'))))
iffact(x, true) → *'(x, fact(-(x, s(0'))))
iffact(x, false) → s(0')

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
- :: 0':s → 0':s → 0':s
fact :: 0':s → 0':s
iffact :: 0':s → true:false → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) → gen_0':s3_0(*(n548_0, a)), rt ∈ Ω(1 + a·n548₀ + n548₀)
ge(gen_0':s3_0(n1208_0), gen_0':s3_0(n1208_0)) → true, rt ∈ Ω(1 + n1208₀)
-(gen_0':s3_0(n1503_0), gen_0':s3_0(n1503_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n1503₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
fact

(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(22) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0'))))
iffact(x, true) → *'(x, fact(-(x, s(0'))))
iffact(x, false) → s(0')

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
- :: 0':s → 0':s → 0':s
fact :: 0':s → 0':s
iffact :: 0':s → true:false → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) → gen_0':s3_0(*(n548_0, a)), rt ∈ Ω(1 + a·n548₀ + n548₀)
ge(gen_0':s3_0(n1208_0), gen_0':s3_0(n1208_0)) → true, rt ∈ Ω(1 + n1208₀)
-(gen_0':s3_0(n1503_0), gen_0':s3_0(n1503_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n1503₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) → gen_0':s3_0(*(n548_0, a)), rt ∈ Ω(1 + a·n548₀ + n548₀)

(24) BOUNDS(n^2, INF)

(25) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0'))))
iffact(x, true) → *'(x, fact(-(x, s(0'))))
iffact(x, false) → s(0')

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
- :: 0':s → 0':s → 0':s
fact :: 0':s → 0':s
iffact :: 0':s → true:false → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) → gen_0':s3_0(*(n548_0, a)), rt ∈ Ω(1 + a·n548₀ + n548₀)
ge(gen_0':s3_0(n1208_0), gen_0':s3_0(n1208_0)) → true, rt ∈ Ω(1 + n1208₀)
-(gen_0':s3_0(n1503_0), gen_0':s3_0(n1503_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n1503₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) → gen_0':s3_0(*(n548_0, a)), rt ∈ Ω(1 + a·n548₀ + n548₀)

(27) BOUNDS(n^2, INF)

(28) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0'))))
iffact(x, true) → *'(x, fact(-(x, s(0'))))
iffact(x, false) → s(0')

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
- :: 0':s → 0':s → 0':s
fact :: 0':s → 0':s
iffact :: 0':s → true:false → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) → gen_0':s3_0(*(n548_0, a)), rt ∈ Ω(1 + a·n548₀ + n548₀)
ge(gen_0':s3_0(n1208_0), gen_0':s3_0(n1208_0)) → true, rt ∈ Ω(1 + n1208₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) → gen_0':s3_0(*(n548_0, a)), rt ∈ Ω(1 + a·n548₀ + n548₀)

(30) BOUNDS(n^2, INF)

(31) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0'))))
iffact(x, true) → *'(x, fact(-(x, s(0'))))
iffact(x, false) → s(0')

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
- :: 0':s → 0':s → 0':s
fact :: 0':s → 0':s
iffact :: 0':s → true:false → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) → gen_0':s3_0(*(n548_0, a)), rt ∈ Ω(1 + a·n548₀ + n548₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(32) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) → gen_0':s3_0(*(n548_0, a)), rt ∈ Ω(1 + a·n548₀ + n548₀)

(33) BOUNDS(n^2, INF)

(34) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0'))))
iffact(x, true) → *'(x, fact(-(x, s(0'))))
iffact(x, false) → s(0')

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
- :: 0':s → 0':s → 0':s
fact :: 0':s → 0':s
iffact :: 0':s → true:false → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(35) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

(36) BOUNDS(n^1, INF)