### (0) Obligation:

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

f(true, x, y) → f(and(gt(x, y), gt(y, s(s(0)))), plus(s(0), x), double(y))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)
and(x, true) → x
and(x, false) → false
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
double(0) → 0
double(s(x)) → s(s(double(x)))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

f(true, x, y) → f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)
and(x, true) → x
and(x, false) → false
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))
double(0') → 0'
double(s(x)) → s(s(double(x)))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
f(true, x, y) → f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)
and(x, true) → x
and(x, false) → false
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))
double(0') → 0'
double(s(x)) → s(s(double(x)))

Types:
f :: true:false → 0':s → 0':s → f
true :: true:false
and :: true:false → true:false → true:false
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
plus :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
f, gt, plus, double

They will be analysed ascendingly in the following order:
gt < f
plus < f
double < f

### (8) Obligation:

TRS:
Rules:
f(true, x, y) → f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)
and(x, true) → x
and(x, false) → false
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))
double(0') → 0'
double(s(x)) → s(s(double(x)))

Types:
f :: true:false → 0':s → 0':s → f
true :: true:false
and :: true:false → true:false → true:false
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
plus :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
gt, f, plus, double

They will be analysed ascendingly in the following order:
gt < f
plus < f
double < f

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

Proved the following rewrite lemma:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)

Induction Base:
gt(gen_0':s4_0(0), gen_0':s4_0(0)) →RΩ(1)
false

Induction Step:
gt(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) →RΩ(1)
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) →IH
false

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

### (11) Obligation:

TRS:
Rules:
f(true, x, y) → f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)
and(x, true) → x
and(x, false) → false
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))
double(0') → 0'
double(s(x)) → s(s(double(x)))

Types:
f :: true:false → 0':s → 0':s → f
true :: true:false
and :: true:false → true:false → true:false
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
plus :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
plus, f, double

They will be analysed ascendingly in the following order:
plus < f
double < f

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

Proved the following rewrite lemma:
plus(gen_0':s4_0(a), gen_0':s4_0(n277_0)) → gen_0':s4_0(+(n277_0, a)), rt ∈ Ω(1 + n2770)

Induction Base:
plus(gen_0':s4_0(a), gen_0':s4_0(0)) →RΩ(1)
gen_0':s4_0(a)

Induction Step:
plus(gen_0':s4_0(a), gen_0':s4_0(+(n277_0, 1))) →RΩ(1)
s(plus(gen_0':s4_0(a), gen_0':s4_0(n277_0))) →IH
s(gen_0':s4_0(+(a, c278_0)))

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

### (14) Obligation:

TRS:
Rules:
f(true, x, y) → f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)
and(x, true) → x
and(x, false) → false
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))
double(0') → 0'
double(s(x)) → s(s(double(x)))

Types:
f :: true:false → 0':s → 0':s → f
true :: true:false
and :: true:false → true:false → true:false
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
plus :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)
plus(gen_0':s4_0(a), gen_0':s4_0(n277_0)) → gen_0':s4_0(+(n277_0, a)), rt ∈ Ω(1 + n2770)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
double, f

They will be analysed ascendingly in the following order:
double < f

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

Proved the following rewrite lemma:
double(gen_0':s4_0(n802_0)) → gen_0':s4_0(*(2, n802_0)), rt ∈ Ω(1 + n8020)

Induction Base:
double(gen_0':s4_0(0)) →RΩ(1)
0'

Induction Step:
double(gen_0':s4_0(+(n802_0, 1))) →RΩ(1)
s(s(double(gen_0':s4_0(n802_0)))) →IH
s(s(gen_0':s4_0(*(2, c803_0))))

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

### (17) Obligation:

TRS:
Rules:
f(true, x, y) → f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)
and(x, true) → x
and(x, false) → false
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))
double(0') → 0'
double(s(x)) → s(s(double(x)))

Types:
f :: true:false → 0':s → 0':s → f
true :: true:false
and :: true:false → true:false → true:false
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
plus :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)
plus(gen_0':s4_0(a), gen_0':s4_0(n277_0)) → gen_0':s4_0(+(n277_0, a)), rt ∈ Ω(1 + n2770)
double(gen_0':s4_0(n802_0)) → gen_0':s4_0(*(2, n802_0)), rt ∈ Ω(1 + n8020)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
f

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

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

### (19) Obligation:

TRS:
Rules:
f(true, x, y) → f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)
and(x, true) → x
and(x, false) → false
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))
double(0') → 0'
double(s(x)) → s(s(double(x)))

Types:
f :: true:false → 0':s → 0':s → f
true :: true:false
and :: true:false → true:false → true:false
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
plus :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)
plus(gen_0':s4_0(a), gen_0':s4_0(n277_0)) → gen_0':s4_0(+(n277_0, a)), rt ∈ Ω(1 + n2770)
double(gen_0':s4_0(n802_0)) → gen_0':s4_0(*(2, n802_0)), rt ∈ Ω(1 + n8020)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)

### (22) Obligation:

TRS:
Rules:
f(true, x, y) → f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)
and(x, true) → x
and(x, false) → false
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))
double(0') → 0'
double(s(x)) → s(s(double(x)))

Types:
f :: true:false → 0':s → 0':s → f
true :: true:false
and :: true:false → true:false → true:false
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
plus :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)
plus(gen_0':s4_0(a), gen_0':s4_0(n277_0)) → gen_0':s4_0(+(n277_0, a)), rt ∈ Ω(1 + n2770)
double(gen_0':s4_0(n802_0)) → gen_0':s4_0(*(2, n802_0)), rt ∈ Ω(1 + n8020)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)

### (25) Obligation:

TRS:
Rules:
f(true, x, y) → f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)
and(x, true) → x
and(x, false) → false
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))
double(0') → 0'
double(s(x)) → s(s(double(x)))

Types:
f :: true:false → 0':s → 0':s → f
true :: true:false
and :: true:false → true:false → true:false
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
plus :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)
plus(gen_0':s4_0(a), gen_0':s4_0(n277_0)) → gen_0':s4_0(+(n277_0, a)), rt ∈ Ω(1 + n2770)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)

### (28) Obligation:

TRS:
Rules:
f(true, x, y) → f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) → false
gt(s(u), 0') → true
gt(s(u), s(v)) → gt(u, v)
and(x, true) → x
and(x, false) → false
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))
double(0') → 0'
double(s(x)) → s(s(double(x)))

Types:
f :: true:false → 0':s → 0':s → f
true :: true:false
and :: true:false → true:false → true:false
gt :: 0':s → 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
plus :: 0':s → 0':s → 0':s
double :: 0':s → 0':s
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)