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

(0) Obligation:

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

monus(S(x'), S(x)) → monus(x', x)
gcd(x, y) → gcd[Ite](equal0(x, y), x, y)
equal0(a, b) → equal0[Ite](<(a, b), a, b)

The (relative) TRS S consists of the following rules:

<(S(x), S(y)) → <(x, y)
<(0, S(y)) → True
<(x, 0) → False
gcd[Ite](False, x, y) → gcd[False][Ite](<(x, y), x, y)
gcd[Ite](True, x, y) → x
gcd[False][Ite](False, x, y) → gcd(y, monus(y, x))
gcd[False][Ite](True, x, y) → gcd(monus(x, y), y)
equal0[Ite](False, a, b) → False
equal0[Ite](True, a, b) → equal0[True][Ite](<(b, a), a, b)
equal0[True][Ite](False, a, b) → False
equal0[True][Ite](True, a, b) → True

Rewrite Strategy: INNERMOST

(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:

monus(S(x'), S(x)) → monus(x', x)
gcd(x, y) → gcd[Ite](equal0(x, y), x, y)
equal0(a, b) → equal0[Ite](<(a, b), a, b)

The (relative) TRS S consists of the following rules:

<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
gcd[Ite](False, x, y) → gcd[False][Ite](<(x, y), x, y)
gcd[Ite](True, x, y) → x
gcd[False][Ite](False, x, y) → gcd(y, monus(y, x))
gcd[False][Ite](True, x, y) → gcd(monus(x, y), y)
equal0[Ite](False, a, b) → False
equal0[Ite](True, a, b) → equal0[True][Ite](<(b, a), a, b)
equal0[True][Ite](False, a, b) → False
equal0[True][Ite](True, a, b) → True

Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
equal0[True][Ite]/1
equal0[True][Ite]/2

(4) Obligation:

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

monus(S(x'), S(x)) → monus(x', x)
gcd(x, y) → gcd[Ite](equal0(x, y), x, y)
equal0(a, b) → equal0[Ite](<(a, b), a, b)

The (relative) TRS S consists of the following rules:

<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
gcd[Ite](False, x, y) → gcd[False][Ite](<(x, y), x, y)
gcd[Ite](True, x, y) → x
gcd[False][Ite](False, x, y) → gcd(y, monus(y, x))
gcd[False][Ite](True, x, y) → gcd(monus(x, y), y)
equal0[Ite](False, a, b) → False
equal0[Ite](True, a, b) → equal0[True][Ite](<(b, a))
equal0[True][Ite](False) → False
equal0[True][Ite](True) → True

Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
monus(S(x'), S(x)) → monus(x', x)
gcd(x, y) → gcd[Ite](equal0(x, y), x, y)
equal0(a, b) → equal0[Ite](<(a, b), a, b)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
gcd[Ite](False, x, y) → gcd[False][Ite](<(x, y), x, y)
gcd[Ite](True, x, y) → x
gcd[False][Ite](False, x, y) → gcd(y, monus(y, x))
gcd[False][Ite](True, x, y) → gcd(monus(x, y), y)
equal0[Ite](False, a, b) → False
equal0[Ite](True, a, b) → equal0[True][Ite](<(b, a))
equal0[True][Ite](False) → False
equal0[True][Ite](True) → True

Types:
monus :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
gcd :: S:0' → S:0' → S:0'
gcd[Ite] :: True:False → S:0' → S:0' → S:0'
equal0 :: S:0' → S:0' → True:False
equal0[Ite] :: True:False → S:0' → S:0' → True:False
< :: S:0' → S:0' → True:False
0' :: S:0'
True :: True:False
False :: True:False
gcd[False][Ite] :: True:False → S:0' → S:0' → S:0'
equal0[True][Ite] :: True:False → True:False
hole_S:0'1_1 :: S:0'
hole_True:False2_1 :: True:False
gen_S:0'3_1 :: Nat → S:0'

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

Heuristically decided to analyse the following defined symbols:
monus, gcd, <

They will be analysed ascendingly in the following order:
monus < gcd
< < gcd

(8) Obligation:

Innermost TRS:
Rules:
monus(S(x'), S(x)) → monus(x', x)
gcd(x, y) → gcd[Ite](equal0(x, y), x, y)
equal0(a, b) → equal0[Ite](<(a, b), a, b)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
gcd[Ite](False, x, y) → gcd[False][Ite](<(x, y), x, y)
gcd[Ite](True, x, y) → x
gcd[False][Ite](False, x, y) → gcd(y, monus(y, x))
gcd[False][Ite](True, x, y) → gcd(monus(x, y), y)
equal0[Ite](False, a, b) → False
equal0[Ite](True, a, b) → equal0[True][Ite](<(b, a))
equal0[True][Ite](False) → False
equal0[True][Ite](True) → True

Types:
monus :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
gcd :: S:0' → S:0' → S:0'
gcd[Ite] :: True:False → S:0' → S:0' → S:0'
equal0 :: S:0' → S:0' → True:False
equal0[Ite] :: True:False → S:0' → S:0' → True:False
< :: S:0' → S:0' → True:False
0' :: S:0'
True :: True:False
False :: True:False
gcd[False][Ite] :: True:False → S:0' → S:0' → S:0'
equal0[True][Ite] :: True:False → True:False
hole_S:0'1_1 :: S:0'
hole_True:False2_1 :: True:False
gen_S:0'3_1 :: Nat → S:0'

Generator Equations:
gen_S:0'3_1(0) ⇔ 0'
gen_S:0'3_1(+(x, 1)) ⇔ S(gen_S:0'3_1(x))

The following defined symbols remain to be analysed:
monus, gcd, <

They will be analysed ascendingly in the following order:
monus < gcd
< < gcd

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

Proved the following rewrite lemma:
monus(gen_S:0'3_1(+(1, n5_1)), gen_S:0'3_1(+(1, n5_1))) → *4_1, rt ∈ Ω(n5₁)

Induction Base:
monus(gen_S:0'3_1(+(1, 0)), gen_S:0'3_1(+(1, 0)))

Induction Step:
monus(gen_S:0'3_1(+(1, +(n5_1, 1))), gen_S:0'3_1(+(1, +(n5_1, 1)))) →_R^Ω(1)
monus(gen_S:0'3_1(+(1, n5_1)), gen_S:0'3_1(+(1, n5_1))) →_IH
*4_1

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
monus(S(x'), S(x)) → monus(x', x)
gcd(x, y) → gcd[Ite](equal0(x, y), x, y)
equal0(a, b) → equal0[Ite](<(a, b), a, b)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
gcd[Ite](False, x, y) → gcd[False][Ite](<(x, y), x, y)
gcd[Ite](True, x, y) → x
gcd[False][Ite](False, x, y) → gcd(y, monus(y, x))
gcd[False][Ite](True, x, y) → gcd(monus(x, y), y)
equal0[Ite](False, a, b) → False
equal0[Ite](True, a, b) → equal0[True][Ite](<(b, a))
equal0[True][Ite](False) → False
equal0[True][Ite](True) → True

Types:
monus :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
gcd :: S:0' → S:0' → S:0'
gcd[Ite] :: True:False → S:0' → S:0' → S:0'
equal0 :: S:0' → S:0' → True:False
equal0[Ite] :: True:False → S:0' → S:0' → True:False
< :: S:0' → S:0' → True:False
0' :: S:0'
True :: True:False
False :: True:False
gcd[False][Ite] :: True:False → S:0' → S:0' → S:0'
equal0[True][Ite] :: True:False → True:False
hole_S:0'1_1 :: S:0'
hole_True:False2_1 :: True:False
gen_S:0'3_1 :: Nat → S:0'

Lemmas:
monus(gen_S:0'3_1(+(1, n5_1)), gen_S:0'3_1(+(1, n5_1))) → *4_1, rt ∈ Ω(n5₁)

Generator Equations:
gen_S:0'3_1(0) ⇔ 0'
gen_S:0'3_1(+(x, 1)) ⇔ S(gen_S:0'3_1(x))

The following defined symbols remain to be analysed:
<, gcd

They will be analysed ascendingly in the following order:
< < gcd

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

Proved the following rewrite lemma:
<(gen_S:0'3_1(n403_1), gen_S:0'3_1(+(1, n403_1))) → True, rt ∈ Ω(0)

Induction Base:
<(gen_S:0'3_1(0), gen_S:0'3_1(+(1, 0))) →_R^Ω(0)
True

Induction Step:
<(gen_S:0'3_1(+(n403_1, 1)), gen_S:0'3_1(+(1, +(n403_1, 1)))) →_R^Ω(0)
<(gen_S:0'3_1(n403_1), gen_S:0'3_1(+(1, n403_1))) →_IH
True

We have rt ∈ Ω(1) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n⁰).

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
monus(S(x'), S(x)) → monus(x', x)
gcd(x, y) → gcd[Ite](equal0(x, y), x, y)
equal0(a, b) → equal0[Ite](<(a, b), a, b)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
gcd[Ite](False, x, y) → gcd[False][Ite](<(x, y), x, y)
gcd[Ite](True, x, y) → x
gcd[False][Ite](False, x, y) → gcd(y, monus(y, x))
gcd[False][Ite](True, x, y) → gcd(monus(x, y), y)
equal0[Ite](False, a, b) → False
equal0[Ite](True, a, b) → equal0[True][Ite](<(b, a))
equal0[True][Ite](False) → False
equal0[True][Ite](True) → True

Types:
monus :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
gcd :: S:0' → S:0' → S:0'
gcd[Ite] :: True:False → S:0' → S:0' → S:0'
equal0 :: S:0' → S:0' → True:False
equal0[Ite] :: True:False → S:0' → S:0' → True:False
< :: S:0' → S:0' → True:False
0' :: S:0'
True :: True:False
False :: True:False
gcd[False][Ite] :: True:False → S:0' → S:0' → S:0'
equal0[True][Ite] :: True:False → True:False
hole_S:0'1_1 :: S:0'
hole_True:False2_1 :: True:False
gen_S:0'3_1 :: Nat → S:0'

Lemmas:
monus(gen_S:0'3_1(+(1, n5_1)), gen_S:0'3_1(+(1, n5_1))) → *4_1, rt ∈ Ω(n5₁)
<(gen_S:0'3_1(n403_1), gen_S:0'3_1(+(1, n403_1))) → True, rt ∈ Ω(0)

Generator Equations:
gen_S:0'3_1(0) ⇔ 0'
gen_S:0'3_1(+(x, 1)) ⇔ S(gen_S:0'3_1(x))

The following defined symbols remain to be analysed:
gcd

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

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

(16) Obligation:

Innermost TRS:
Rules:
monus(S(x'), S(x)) → monus(x', x)
gcd(x, y) → gcd[Ite](equal0(x, y), x, y)
equal0(a, b) → equal0[Ite](<(a, b), a, b)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
gcd[Ite](False, x, y) → gcd[False][Ite](<(x, y), x, y)
gcd[Ite](True, x, y) → x
gcd[False][Ite](False, x, y) → gcd(y, monus(y, x))
gcd[False][Ite](True, x, y) → gcd(monus(x, y), y)
equal0[Ite](False, a, b) → False
equal0[Ite](True, a, b) → equal0[True][Ite](<(b, a))
equal0[True][Ite](False) → False
equal0[True][Ite](True) → True

Types:
monus :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
gcd :: S:0' → S:0' → S:0'
gcd[Ite] :: True:False → S:0' → S:0' → S:0'
equal0 :: S:0' → S:0' → True:False
equal0[Ite] :: True:False → S:0' → S:0' → True:False
< :: S:0' → S:0' → True:False
0' :: S:0'
True :: True:False
False :: True:False
gcd[False][Ite] :: True:False → S:0' → S:0' → S:0'
equal0[True][Ite] :: True:False → True:False
hole_S:0'1_1 :: S:0'
hole_True:False2_1 :: True:False
gen_S:0'3_1 :: Nat → S:0'

Lemmas:
monus(gen_S:0'3_1(+(1, n5_1)), gen_S:0'3_1(+(1, n5_1))) → *4_1, rt ∈ Ω(n5₁)
<(gen_S:0'3_1(n403_1), gen_S:0'3_1(+(1, n403_1))) → True, rt ∈ Ω(0)

Generator Equations:
gen_S:0'3_1(0) ⇔ 0'
gen_S:0'3_1(+(x, 1)) ⇔ S(gen_S:0'3_1(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
monus(gen_S:0'3_1(+(1, n5_1)), gen_S:0'3_1(+(1, n5_1))) → *4_1, rt ∈ Ω(n5₁)

(18) BOUNDS(n^1, INF)

(19) Obligation:

Innermost TRS:
Rules:
monus(S(x'), S(x)) → monus(x', x)
gcd(x, y) → gcd[Ite](equal0(x, y), x, y)
equal0(a, b) → equal0[Ite](<(a, b), a, b)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
gcd[Ite](False, x, y) → gcd[False][Ite](<(x, y), x, y)
gcd[Ite](True, x, y) → x
gcd[False][Ite](False, x, y) → gcd(y, monus(y, x))
gcd[False][Ite](True, x, y) → gcd(monus(x, y), y)
equal0[Ite](False, a, b) → False
equal0[Ite](True, a, b) → equal0[True][Ite](<(b, a))
equal0[True][Ite](False) → False
equal0[True][Ite](True) → True

Types:
monus :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
gcd :: S:0' → S:0' → S:0'
gcd[Ite] :: True:False → S:0' → S:0' → S:0'
equal0 :: S:0' → S:0' → True:False
equal0[Ite] :: True:False → S:0' → S:0' → True:False
< :: S:0' → S:0' → True:False
0' :: S:0'
True :: True:False
False :: True:False
gcd[False][Ite] :: True:False → S:0' → S:0' → S:0'
equal0[True][Ite] :: True:False → True:False
hole_S:0'1_1 :: S:0'
hole_True:False2_1 :: True:False
gen_S:0'3_1 :: Nat → S:0'

Lemmas:
monus(gen_S:0'3_1(+(1, n5_1)), gen_S:0'3_1(+(1, n5_1))) → *4_1, rt ∈ Ω(n5₁)
<(gen_S:0'3_1(n403_1), gen_S:0'3_1(+(1, n403_1))) → True, rt ∈ Ω(0)

Generator Equations:
gen_S:0'3_1(0) ⇔ 0'
gen_S:0'3_1(+(x, 1)) ⇔ S(gen_S:0'3_1(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
monus(gen_S:0'3_1(+(1, n5_1)), gen_S:0'3_1(+(1, n5_1))) → *4_1, rt ∈ Ω(n5₁)

(21) BOUNDS(n^1, INF)

(22) Obligation:

Innermost TRS:
Rules:
monus(S(x'), S(x)) → monus(x', x)
gcd(x, y) → gcd[Ite](equal0(x, y), x, y)
equal0(a, b) → equal0[Ite](<(a, b), a, b)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
gcd[Ite](False, x, y) → gcd[False][Ite](<(x, y), x, y)
gcd[Ite](True, x, y) → x
gcd[False][Ite](False, x, y) → gcd(y, monus(y, x))
gcd[False][Ite](True, x, y) → gcd(monus(x, y), y)
equal0[Ite](False, a, b) → False
equal0[Ite](True, a, b) → equal0[True][Ite](<(b, a))
equal0[True][Ite](False) → False
equal0[True][Ite](True) → True

Types:
monus :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
gcd :: S:0' → S:0' → S:0'
gcd[Ite] :: True:False → S:0' → S:0' → S:0'
equal0 :: S:0' → S:0' → True:False
equal0[Ite] :: True:False → S:0' → S:0' → True:False
< :: S:0' → S:0' → True:False
0' :: S:0'
True :: True:False
False :: True:False
gcd[False][Ite] :: True:False → S:0' → S:0' → S:0'
equal0[True][Ite] :: True:False → True:False
hole_S:0'1_1 :: S:0'
hole_True:False2_1 :: True:False
gen_S:0'3_1 :: Nat → S:0'

Lemmas:
monus(gen_S:0'3_1(+(1, n5_1)), gen_S:0'3_1(+(1, n5_1))) → *4_1, rt ∈ Ω(n5₁)

Generator Equations:
gen_S:0'3_1(0) ⇔ 0'
gen_S:0'3_1(+(x, 1)) ⇔ S(gen_S:0'3_1(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
monus(gen_S:0'3_1(+(1, n5_1)), gen_S:0'3_1(+(1, n5_1))) → *4_1, rt ∈ Ω(n5₁)

(24) BOUNDS(n^1, INF)