### (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 [ ].

### (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

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) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

Induction Base:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, 0)))

Induction Step:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, +(n5_0, 1)))) →RΩ(1)
pred(minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0)))) →IH
pred(*4_0)

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

### (11) 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:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

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

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

Proved the following rewrite lemma:
le(gen_s:0'3_0(+(1, n2333_0)), gen_s:0'3_0(n2333_0)) → false, rt ∈ Ω(1 + n23330)

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

Induction Step:
le(gen_s:0'3_0(+(1, +(n2333_0, 1))), gen_s:0'3_0(+(n2333_0, 1))) →RΩ(1)
le(gen_s:0'3_0(+(1, n2333_0)), gen_s:0'3_0(n2333_0)) →IH
false

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

### (14) 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:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
le(gen_s:0'3_0(+(1, n2333_0)), gen_s:0'3_0(n2333_0)) → false, rt ∈ Ω(1 + n23330)

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

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

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

### (16) 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:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
le(gen_s:0'3_0(+(1, n2333_0)), gen_s:0'3_0(n2333_0)) → false, rt ∈ Ω(1 + n23330)

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.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

### (19) 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:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
le(gen_s:0'3_0(+(1, n2333_0)), gen_s:0'3_0(n2333_0)) → false, rt ∈ Ω(1 + n23330)

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.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

### (22) 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:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

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.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)