### (0) Obligation:

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

minus(minus(x)) → x
minus(+(x, y)) → *(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*(x, y)) → +(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) → minus(minus(minus(f(x))))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
minus(+(x, y)) →+ *(minus(minus(minus(x))), minus(minus(minus(y))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
The pumping substitution is [x / +(x, 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:

minus(minus(x)) → x
minus(+'(x, y)) → *'(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*'(x, y)) → +'(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) → minus(minus(minus(f(x))))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
minus(minus(x)) → x
minus(+'(x, y)) → *'(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*'(x, y)) → +'(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) → minus(minus(minus(f(x))))

Types:
minus :: +':*' → +':*'
+' :: +':*' → +':*' → +':*'
*' :: +':*' → +':*' → +':*'
f :: +':*' → +':*'
hole_+':*'1_0 :: +':*'
gen_+':*'2_0 :: Nat → +':*'

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

Heuristically decided to analyse the following defined symbols:
minus, f

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

### (8) Obligation:

TRS:
Rules:
minus(minus(x)) → x
minus(+'(x, y)) → *'(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*'(x, y)) → +'(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) → minus(minus(minus(f(x))))

Types:
minus :: +':*' → +':*'
+' :: +':*' → +':*' → +':*'
*' :: +':*' → +':*' → +':*'
f :: +':*' → +':*'
hole_+':*'1_0 :: +':*'
gen_+':*'2_0 :: Nat → +':*'

Generator Equations:
gen_+':*'2_0(0) ⇔ hole_+':*'1_0
gen_+':*'2_0(+(x, 1)) ⇔ +'(hole_+':*'1_0, gen_+':*'2_0(x))

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

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

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

Proved the following rewrite lemma:
minus(gen_+':*'2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Induction Base:
minus(gen_+':*'2_0(0))

Induction Step:
minus(gen_+':*'2_0(+(n4_0, 1))) →RΩ(1)
*'(minus(minus(minus(hole_+':*'1_0))), minus(minus(minus(gen_+':*'2_0(n4_0))))) →RΩ(1)
*'(minus(hole_+':*'1_0), minus(minus(minus(gen_+':*'2_0(n4_0))))) →IH
*'(minus(hole_+':*'1_0), minus(minus(*3_0)))

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

### (11) Obligation:

TRS:
Rules:
minus(minus(x)) → x
minus(+'(x, y)) → *'(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*'(x, y)) → +'(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) → minus(minus(minus(f(x))))

Types:
minus :: +':*' → +':*'
+' :: +':*' → +':*' → +':*'
*' :: +':*' → +':*' → +':*'
f :: +':*' → +':*'
hole_+':*'1_0 :: +':*'
gen_+':*'2_0 :: Nat → +':*'

Lemmas:
minus(gen_+':*'2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_+':*'2_0(0) ⇔ hole_+':*'1_0
gen_+':*'2_0(+(x, 1)) ⇔ +'(hole_+':*'1_0, gen_+':*'2_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:
minus(minus(x)) → x
minus(+'(x, y)) → *'(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*'(x, y)) → +'(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) → minus(minus(minus(f(x))))

Types:
minus :: +':*' → +':*'
+' :: +':*' → +':*' → +':*'
*' :: +':*' → +':*' → +':*'
f :: +':*' → +':*'
hole_+':*'1_0 :: +':*'
gen_+':*'2_0 :: Nat → +':*'

Lemmas:
minus(gen_+':*'2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_+':*'2_0(0) ⇔ hole_+':*'1_0
gen_+':*'2_0(+(x, 1)) ⇔ +'(hole_+':*'1_0, gen_+':*'2_0(x))

No more defined symbols left to analyse.

### (14) LowerBoundsProof (EQUIVALENT transformation)

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

### (16) Obligation:

TRS:
Rules:
minus(minus(x)) → x
minus(+'(x, y)) → *'(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*'(x, y)) → +'(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) → minus(minus(minus(f(x))))

Types:
minus :: +':*' → +':*'
+' :: +':*' → +':*' → +':*'
*' :: +':*' → +':*' → +':*'
f :: +':*' → +':*'
hole_+':*'1_0 :: +':*'
gen_+':*'2_0 :: Nat → +':*'

Lemmas:
minus(gen_+':*'2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_+':*'2_0(0) ⇔ hole_+':*'1_0
gen_+':*'2_0(+(x, 1)) ⇔ +'(hole_+':*'1_0, gen_+':*'2_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

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