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:

minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
minus(X, s(Y)) →⁺ pred(minus(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:

minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0') → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0') → false
le(0', Y) → true
gcd(0', Y) → 0'
gcd(s(X), 0') → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0') → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0') → false
le(0', Y) → true
gcd(0', Y) → 0'
gcd(s(X), 0') → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

Types:
minus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
pred :: s:0' → s:0'
0' :: s:0'
le :: s:0' → s:0' → false:true
false :: false:true
true :: false:true
gcd :: s:0' → s:0' → s:0'
if :: false:true → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
minus, le, gcd

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

(8) Obligation:

TRS:
Rules:
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0') → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0') → false
le(0', Y) → true
gcd(0', Y) → 0'
gcd(s(X), 0') → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

Types:
minus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
pred :: s:0' → s:0'
0' :: s:0'
le :: s:0' → s:0' → false:true
false :: false:true
true :: false:true
gcd :: s:0' → s:0' → s:0'
if :: false:true → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat → s:0'

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
minus, le, gcd

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

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
Rules:
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0') → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0') → false
le(0', Y) → true
gcd(0', Y) → 0'
gcd(s(X), 0') → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

Types:
minus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
pred :: s:0' → s:0'
0' :: s:0'
le :: s:0' → s:0' → false:true
false :: false:true
true :: false:true
gcd :: s:0' → s:0' → s:0'
if :: false:true → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat → s:0'

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

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

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

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

Proved the following rewrite lemma:
le(gen_s:0'3_0(+(1, n1635_0)), gen_s:0'3_0(n1635_0)) → false, rt ∈ Ω(1 + n1635₀)

Induction Base:
le(gen_s:0'3_0(+(1, 0)), gen_s:0'3_0(0)) →_R^Ω(1)
false

Induction Step:
le(gen_s:0'3_0(+(1, +(n1635_0, 1))), gen_s:0'3_0(+(n1635_0, 1))) →_R^Ω(1)
le(gen_s:0'3_0(+(1, n1635_0)), gen_s:0'3_0(n1635_0)) →_IH
false

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0') → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0') → false
le(0', Y) → true
gcd(0', Y) → 0'
gcd(s(X), 0') → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

Types:
minus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
pred :: s:0' → s:0'
0' :: s:0'
le :: s:0' → s:0' → false:true
false :: false:true
true :: false:true
gcd :: s:0' → s:0' → s:0'
if :: false:true → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
le(gen_s:0'3_0(+(1, n1635_0)), gen_s:0'3_0(n1635_0)) → false, rt ∈ Ω(1 + n1635₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(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:

TRS:
Rules:
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0') → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0') → false
le(0', Y) → true
gcd(0', Y) → 0'
gcd(s(X), 0') → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

Types:
minus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
pred :: s:0' → s:0'
0' :: s:0'
le :: s:0' → s:0' → false:true
false :: false:true
true :: false:true
gcd :: s:0' → s:0' → s:0'
if :: false:true → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
le(gen_s:0'3_0(+(1, n1635_0)), gen_s:0'3_0(n1635_0)) → false, rt ∈ Ω(1 + n1635₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_s:0'3_0(+(1, n1635_0)), gen_s:0'3_0(n1635_0)) → false, rt ∈ Ω(1 + n1635₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

TRS:
Rules:
minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0') → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0') → false
le(0', Y) → true
gcd(0', Y) → 0'
gcd(s(X), 0') → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

Types:
minus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
pred :: s:0' → s:0'
0' :: s:0'
le :: s:0' → s:0' → false:true
false :: false:true
true :: false:true
gcd :: s:0' → s:0' → s:0'
if :: false:true → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
le(gen_s:0'3_0(+(1, n1635_0)), gen_s:0'3_0(n1635_0)) → false, rt ∈ Ω(1 + n1635₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_s:0'3_0(+(1, n1635_0)), gen_s:0'3_0(n1635_0)) → false, rt ∈ Ω(1 + n1635₀)

(20) BOUNDS(n^1, INF)