### (0) Obligation:

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

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(x, +(y, z)) → +(+(x, y), z)
f(g(f(x))) → f(h(s(0), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+(x, y), z))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
+(x, s(y)) →+ s(+(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:

+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(x, +'(y, z)) → +'(+'(x, y), z)
f(g(f(x))) → f(h(s(0'), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+'(x, y), z))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(x, +'(y, z)) → +'(+'(x, y), z)
f(g(f(x))) → f(h(s(0'), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+'(x, y), z))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: g:h → g:h
g :: g:h → g:h
h :: 0':s → g:h → g:h
hole_0':s1_0 :: 0':s
hole_g:h2_0 :: g:h
gen_0':s3_0 :: Nat → 0':s
gen_g:h4_0 :: Nat → g:h

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

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

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

### (8) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(x, +'(y, z)) → +'(+'(x, y), z)
f(g(f(x))) → f(h(s(0'), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+'(x, y), z))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: g:h → g:h
g :: g:h → g:h
h :: 0':s → g:h → g:h
hole_0':s1_0 :: 0':s
hole_g:h2_0 :: g:h
gen_0':s3_0 :: Nat → 0':s
gen_g:h4_0 :: Nat → g:h

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_g:h4_0(0) ⇔ hole_g:h2_0
gen_g:h4_0(+(x, 1)) ⇔ g(gen_g:h4_0(x))

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

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

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

Proved the following rewrite lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n6_0)) → gen_0':s3_0(+(n6_0, a)), rt ∈ Ω(1 + n60)

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(+(n6_0, 1))) →RΩ(1)
s(+'(gen_0':s3_0(a), gen_0':s3_0(n6_0))) →IH
s(gen_0':s3_0(+(a, c7_0)))

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

### (11) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(x, +'(y, z)) → +'(+'(x, y), z)
f(g(f(x))) → f(h(s(0'), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+'(x, y), z))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: g:h → g:h
g :: g:h → g:h
h :: 0':s → g:h → g:h
hole_0':s1_0 :: 0':s
hole_g:h2_0 :: g:h
gen_0':s3_0 :: Nat → 0':s
gen_g:h4_0 :: Nat → g:h

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n6_0)) → gen_0':s3_0(+(n6_0, a)), rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_g:h4_0(0) ⇔ hole_g:h2_0
gen_g:h4_0(+(x, 1)) ⇔ g(gen_g:h4_0(x))

The following defined symbols remain to be analysed:
f

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

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

### (13) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(x, +'(y, z)) → +'(+'(x, y), z)
f(g(f(x))) → f(h(s(0'), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+'(x, y), z))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: g:h → g:h
g :: g:h → g:h
h :: 0':s → g:h → g:h
hole_0':s1_0 :: 0':s
hole_g:h2_0 :: g:h
gen_0':s3_0 :: Nat → 0':s
gen_g:h4_0 :: Nat → g:h

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n6_0)) → gen_0':s3_0(+(n6_0, a)), rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_g:h4_0(0) ⇔ hole_g:h2_0
gen_g:h4_0(+(x, 1)) ⇔ g(gen_g:h4_0(x))

No more defined symbols left to analyse.

### (14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n6_0)) → gen_0':s3_0(+(n6_0, a)), rt ∈ Ω(1 + n60)

### (16) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(x, +'(y, z)) → +'(+'(x, y), z)
f(g(f(x))) → f(h(s(0'), x))
f(g(h(x, y))) → f(h(s(x), y))
f(h(x, h(y, z))) → f(h(+'(x, y), z))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: g:h → g:h
g :: g:h → g:h
h :: 0':s → g:h → g:h
hole_0':s1_0 :: 0':s
hole_g:h2_0 :: g:h
gen_0':s3_0 :: Nat → 0':s
gen_g:h4_0 :: Nat → g:h

Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n6_0)) → gen_0':s3_0(+(n6_0, a)), rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_g:h4_0(0) ⇔ hole_g:h2_0
gen_g:h4_0(+(x, 1)) ⇔ g(gen_g:h4_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n6_0)) → gen_0':s3_0(+(n6_0, a)), rt ∈ Ω(1 + n60)