### (0) Obligation:

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

bin(x, 0) → s(0)
bin(0, s(y)) → 0
bin(s(x), s(y)) → +(bin(x, s(y)), bin(x, y))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
bin(s(x), s(y)) →+ +(bin(x, s(y)), bin(x, y))
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 [ ].

The rewrite sequence
bin(s(x), s(y)) →+ +(bin(x, s(y)), bin(x, y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [x / s(x), 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:

bin(x, 0') → s(0')
bin(0', s(y)) → 0'
bin(s(x), s(y)) → +'(bin(x, s(y)), bin(x, y))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
bin(x, 0') → s(0')
bin(0', s(y)) → 0'
bin(s(x), s(y)) → +'(bin(x, s(y)), bin(x, y))

Types:
bin :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

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

Heuristically decided to analyse the following defined symbols:
bin

### (8) Obligation:

TRS:
Rules:
bin(x, 0') → s(0')
bin(0', s(y)) → 0'
bin(s(x), s(y)) → +'(bin(x, s(y)), bin(x, y))

Types:
bin :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

The following defined symbols remain to be analysed:
bin

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

Proved the following rewrite lemma:
bin(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(1)) → *3_0, rt ∈ Ω(n40)

Induction Base:
bin(gen_0':s:+'2_0(+(1, 0)), gen_0':s:+'2_0(1))

Induction Step:
bin(gen_0':s:+'2_0(+(1, +(n4_0, 1))), gen_0':s:+'2_0(1)) →RΩ(1)
+'(bin(gen_0':s:+'2_0(+(1, n4_0)), s(gen_0':s:+'2_0(0))), bin(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(0))) →IH
+'(*3_0, bin(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(0))) →RΩ(1)
+'(*3_0, s(0'))

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

### (11) Obligation:

TRS:
Rules:
bin(x, 0') → s(0')
bin(0', s(y)) → 0'
bin(s(x), s(y)) → +'(bin(x, s(y)), bin(x, y))

Types:
bin :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
bin(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(1)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

No more defined symbols left to analyse.

### (12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
bin(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(1)) → *3_0, rt ∈ Ω(n40)

### (14) Obligation:

TRS:
Rules:
bin(x, 0') → s(0')
bin(0', s(y)) → 0'
bin(s(x), s(y)) → +'(bin(x, s(y)), bin(x, y))

Types:
bin :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
bin(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(1)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

No more defined symbols left to analyse.

### (15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
bin(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(1)) → *3_0, rt ∈ Ω(n40)