### (0) Obligation:

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

not(true) → false
not(false) → true
odd(0) → false
odd(s(x)) → not(odd(x))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(s(x), y) → s(+(x, y))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
odd(s(x)) →+ not(odd(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / s(x)].
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:

not(true) → false
not(false) → true
odd(0') → false
odd(s(x)) → not(odd(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(s(x), y) → s(+'(x, y))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
not(true) → false
not(false) → true
odd(0') → false
odd(s(x)) → not(odd(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(s(x), y) → s(+'(x, y))

Types:
not :: true:false → true:false
true :: true:false
false :: true:false
odd :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
odd, +'

### (8) Obligation:

TRS:
Rules:
not(true) → false
not(false) → true
odd(0') → false
odd(s(x)) → not(odd(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(s(x), y) → s(+'(x, y))

Types:
not :: true:false → true:false
true :: true:false
false :: true:false
odd :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
odd, +'

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

Proved the following rewrite lemma:
odd(gen_0':s3_0(n5_0)) → *4_0, rt ∈ Ω(n50)

Induction Base:
odd(gen_0':s3_0(0))

Induction Step:
odd(gen_0':s3_0(+(n5_0, 1))) →RΩ(1)
not(odd(gen_0':s3_0(n5_0))) →IH
not(*4_0)

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

### (11) Obligation:

TRS:
Rules:
not(true) → false
not(false) → true
odd(0') → false
odd(s(x)) → not(odd(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(s(x), y) → s(+'(x, y))

Types:
not :: true:false → true:false
true :: true:false
false :: true:false
odd :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
odd(gen_0':s3_0(n5_0)) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
+'

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

Proved the following rewrite lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n341_0)) → gen_0':s3_0(+(n341_0, a)), rt ∈ Ω(1 + n3410)

Induction Base:
+'(gen_0':s3_0(a), gen_0':s3_0(0)) →RΩ(1)
gen_0':s3_0(a)

Induction Step:
+'(gen_0':s3_0(a), gen_0':s3_0(+(n341_0, 1))) →RΩ(1)
s(+'(gen_0':s3_0(a), gen_0':s3_0(n341_0))) →IH
s(gen_0':s3_0(+(a, c342_0)))

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

### (14) Obligation:

TRS:
Rules:
not(true) → false
not(false) → true
odd(0') → false
odd(s(x)) → not(odd(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(s(x), y) → s(+'(x, y))

Types:
not :: true:false → true:false
true :: true:false
false :: true:false
odd :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
odd(gen_0':s3_0(n5_0)) → *4_0, rt ∈ Ω(n50)
+'(gen_0':s3_0(a), gen_0':s3_0(n341_0)) → gen_0':s3_0(+(n341_0, a)), rt ∈ Ω(1 + n3410)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

### (15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
odd(gen_0':s3_0(n5_0)) → *4_0, rt ∈ Ω(n50)

### (17) Obligation:

TRS:
Rules:
not(true) → false
not(false) → true
odd(0') → false
odd(s(x)) → not(odd(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(s(x), y) → s(+'(x, y))

Types:
not :: true:false → true:false
true :: true:false
false :: true:false
odd :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
odd(gen_0':s3_0(n5_0)) → *4_0, rt ∈ Ω(n50)
+'(gen_0':s3_0(a), gen_0':s3_0(n341_0)) → gen_0':s3_0(+(n341_0, a)), rt ∈ Ω(1 + n3410)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
odd(gen_0':s3_0(n5_0)) → *4_0, rt ∈ Ω(n50)

### (20) Obligation:

TRS:
Rules:
not(true) → false
not(false) → true
odd(0') → false
odd(s(x)) → not(odd(x))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(s(x), y) → s(+'(x, y))

Types:
not :: true:false → true:false
true :: true:false
false :: true:false
odd :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
odd(gen_0':s3_0(n5_0)) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
odd(gen_0':s3_0(n5_0)) → *4_0, rt ∈ Ω(n50)