### (0) Obligation:

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

f(0, 1, x) → f(g(x), g(x), x)
f(g(x), y, z) → g(f(x, y, z))
f(x, g(y), z) → g(f(x, y, z))
f(x, y, g(z)) → g(f(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
f(g(x), y, z) →+ g(f(x, y, z))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / g(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:

f(0', 1', x) → f(g(x), g(x), x)
f(g(x), y, z) → g(f(x, y, z))
f(x, g(y), z) → g(f(x, y, z))
f(x, y, g(z)) → g(f(x, y, z))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
f(0', 1', x) → f(g(x), g(x), x)
f(g(x), y, z) → g(f(x, y, z))
f(x, g(y), z) → g(f(x, y, z))
f(x, y, g(z)) → g(f(x, y, z))

Types:
f :: 0':1':g → 0':1':g → 0':1':g → 0':1':g
0' :: 0':1':g
1' :: 0':1':g
g :: 0':1':g → 0':1':g
hole_0':1':g1_0 :: 0':1':g
gen_0':1':g2_0 :: Nat → 0':1':g

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

Heuristically decided to analyse the following defined symbols:
f

### (8) Obligation:

TRS:
Rules:
f(0', 1', x) → f(g(x), g(x), x)
f(g(x), y, z) → g(f(x, y, z))
f(x, g(y), z) → g(f(x, y, z))
f(x, y, g(z)) → g(f(x, y, z))

Types:
f :: 0':1':g → 0':1':g → 0':1':g → 0':1':g
0' :: 0':1':g
1' :: 0':1':g
g :: 0':1':g → 0':1':g
hole_0':1':g1_0 :: 0':1':g
gen_0':1':g2_0 :: Nat → 0':1':g

Generator Equations:
gen_0':1':g2_0(0) ⇔ 1'
gen_0':1':g2_0(+(x, 1)) ⇔ g(gen_0':1':g2_0(x))

The following defined symbols remain to be analysed:
f

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

Proved the following rewrite lemma:
f(gen_0':1':g2_0(+(1, n4_0)), gen_0':1':g2_0(b), gen_0':1':g2_0(c)) → *3_0, rt ∈ Ω(n40)

Induction Base:
f(gen_0':1':g2_0(+(1, 0)), gen_0':1':g2_0(b), gen_0':1':g2_0(c))

Induction Step:
f(gen_0':1':g2_0(+(1, +(n4_0, 1))), gen_0':1':g2_0(b), gen_0':1':g2_0(c)) →RΩ(1)
g(f(gen_0':1':g2_0(+(1, n4_0)), gen_0':1':g2_0(b), gen_0':1':g2_0(c))) →IH
g(*3_0)

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

### (11) Obligation:

TRS:
Rules:
f(0', 1', x) → f(g(x), g(x), x)
f(g(x), y, z) → g(f(x, y, z))
f(x, g(y), z) → g(f(x, y, z))
f(x, y, g(z)) → g(f(x, y, z))

Types:
f :: 0':1':g → 0':1':g → 0':1':g → 0':1':g
0' :: 0':1':g
1' :: 0':1':g
g :: 0':1':g → 0':1':g
hole_0':1':g1_0 :: 0':1':g
gen_0':1':g2_0 :: Nat → 0':1':g

Lemmas:
f(gen_0':1':g2_0(+(1, n4_0)), gen_0':1':g2_0(b), gen_0':1':g2_0(c)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_0':1':g2_0(0) ⇔ 1'
gen_0':1':g2_0(+(x, 1)) ⇔ g(gen_0':1':g2_0(x))

No more defined symbols left to analyse.

### (12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
f(gen_0':1':g2_0(+(1, n4_0)), gen_0':1':g2_0(b), gen_0':1':g2_0(c)) → *3_0, rt ∈ Ω(n40)

### (14) Obligation:

TRS:
Rules:
f(0', 1', x) → f(g(x), g(x), x)
f(g(x), y, z) → g(f(x, y, z))
f(x, g(y), z) → g(f(x, y, z))
f(x, y, g(z)) → g(f(x, y, z))

Types:
f :: 0':1':g → 0':1':g → 0':1':g → 0':1':g
0' :: 0':1':g
1' :: 0':1':g
g :: 0':1':g → 0':1':g
hole_0':1':g1_0 :: 0':1':g
gen_0':1':g2_0 :: Nat → 0':1':g

Lemmas:
f(gen_0':1':g2_0(+(1, n4_0)), gen_0':1':g2_0(b), gen_0':1':g2_0(c)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_0':1':g2_0(0) ⇔ 1'
gen_0':1':g2_0(+(x, 1)) ⇔ g(gen_0':1':g2_0(x))

No more defined symbols left to analyse.

### (15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
f(gen_0':1':g2_0(+(1, n4_0)), gen_0':1':g2_0(b), gen_0':1':g2_0(c)) → *3_0, rt ∈ Ω(n40)