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

0 CpxRelTRS
↳1 DecreasingLoopProof (⇔, 1379 ms)
↳2 BOUNDS(n^1, INF)
↳3 RenamingProof (⇔, 0 ms)
↳4 CpxRelTRS
    ↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 typed CpxTrs
    ↳7 OrderProof (LOWER BOUND(ID), 0 ms)
    ↳8 typed CpxTrs
    ↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 51 ms)
    ↳10 typed CpxTrs
    ↳11 RewriteLemmaProof (LOWER BOUND(ID), 168 ms)
    ↳12 BEST
        ↳13 typed CpxTrs
            ↳14 NoRewriteLemmaProof (LOWER BOUND(ID), 791 ms)
            ↳15 typed CpxTrs
            ↳16 LowerBoundsProof (⇔, 0 ms)
            ↳17 BOUNDS(1, INF)
        ↳18 typed CpxTrs
            ↳19 LowerBoundsProof (⇔, 0 ms)
            ↳20 BOUNDS(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) DecreasingLoopProof (EQUIVALENT transformation)

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

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

(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), a, b)
equal0[True][Ite](False, a, b) → False
equal0[True][Ite](True, a, b) → 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 → S:0' → S:0' → 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), a, b)
equal0[True][Ite](False, a, b) → False
equal0[True][Ite](True, a, b) → 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 → S:0' → S:0' → 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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) 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), a, b)
equal0[True][Ite](False, a, b) → False
equal0[True][Ite](True, a, b) → 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 → S:0' → S:0' → 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:
<, gcd

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

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
<(gen_S:0'3_1(n252_1), gen_S:0'3_1(+(1, n252_1))) → True, rt ∈ Ω(0)

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

Induction Step:
<(gen_S:0'3_1(+(n252_1, 1)), gen_S:0'3_1(+(1, +(n252_1, 1)))) →RΩ(0)
<(gen_S:0'3_1(n252_1), gen_S:0'3_1(+(1, n252_1))) →IH
True

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

(12) Complex Obligation (BEST)

(13) 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), a, b)
equal0[True][Ite](False, a, b) → False
equal0[True][Ite](True, a, b) → 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 → S:0' → S:0' → True:False
hole_S:0'1_1 :: S:0'
hole_True:False2_1 :: True:False
gen_S:0'3_1 :: Nat → S:0'

Lemmas:
<(gen_S:0'3_1(n252_1), gen_S:0'3_1(+(1, n252_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

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) 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), a, b)
equal0[True][Ite](False, a, b) → False
equal0[True][Ite](True, a, b) → 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 → S:0' → S:0' → True:False
hole_S:0'1_1 :: S:0'
hole_True:False2_1 :: True:False
gen_S:0'3_1 :: Nat → S:0'

Lemmas:
<(gen_S:0'3_1(n252_1), gen_S:0'3_1(+(1, n252_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.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(1) was proven with the following lemma:
<(gen_S:0'3_1(n252_1), gen_S:0'3_1(+(1, n252_1))) → True, rt ∈ Ω(0)

(17) BOUNDS(1, INF)

(18) 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), a, b)
equal0[True][Ite](False, a, b) → False
equal0[True][Ite](True, a, b) → 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 → S:0' → S:0' → True:False
hole_S:0'1_1 :: S:0'
hole_True:False2_1 :: True:False
gen_S:0'3_1 :: Nat → S:0'

Lemmas:
<(gen_S:0'3_1(n252_1), gen_S:0'3_1(+(1, n252_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.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(1) was proven with the following lemma:
<(gen_S:0'3_1(n252_1), gen_S:0'3_1(+(1, n252_1))) → True, rt ∈ Ω(0)

(20) BOUNDS(1, INF)