### (0) Obligation:

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

-(0, y) → 0
-(x, 0) → x
-(x, s(y)) → if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
p(0) → 0
p(s(x)) → x

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)) →+ if(greater(x, s(y)), s(-(x, y)), 0)
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
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:

-(0', y) → 0'
-(x, 0') → x
-(x, s(y)) → if(greater(x, s(y)), s(-(x, p(s(y)))), 0')
p(0') → 0'
p(s(x)) → x

S is empty.
Rewrite Strategy: FULL

### (5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
if/0
if/2
greater/0
greater/1

### (6) Obligation:

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

-(0', y) → 0'
-(x, 0') → x
-(x, s(y)) → if(s(-(x, p(s(y)))))
p(0') → 0'
p(s(x)) → x

S is empty.
Rewrite Strategy: FULL

Infered types.

### (8) Obligation:

TRS:
Rules:
-(0', y) → 0'
-(x, 0') → x
-(x, s(y)) → if(s(-(x, p(s(y)))))
p(0') → 0'
p(s(x)) → x

Types:
- :: 0':s:if → 0':s:if → 0':s:if
0' :: 0':s:if
s :: 0':s:if → 0':s:if
if :: 0':s:if → 0':s:if
p :: 0':s:if → 0':s:if
hole_0':s:if1_0 :: 0':s:if
gen_0':s:if2_0 :: Nat → 0':s:if

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

Heuristically decided to analyse the following defined symbols:
-

### (10) Obligation:

TRS:
Rules:
-(0', y) → 0'
-(x, 0') → x
-(x, s(y)) → if(s(-(x, p(s(y)))))
p(0') → 0'
p(s(x)) → x

Types:
- :: 0':s:if → 0':s:if → 0':s:if
0' :: 0':s:if
s :: 0':s:if → 0':s:if
if :: 0':s:if → 0':s:if
p :: 0':s:if → 0':s:if
hole_0':s:if1_0 :: 0':s:if
gen_0':s:if2_0 :: Nat → 0':s:if

Generator Equations:
gen_0':s:if2_0(0) ⇔ 0'
gen_0':s:if2_0(+(x, 1)) ⇔ s(gen_0':s:if2_0(x))

The following defined symbols remain to be analysed:
-

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

Proved the following rewrite lemma:
-(gen_0':s:if2_0(a), gen_0':s:if2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Induction Base:
-(gen_0':s:if2_0(a), gen_0':s:if2_0(0))

Induction Step:
-(gen_0':s:if2_0(a), gen_0':s:if2_0(+(n4_0, 1))) →RΩ(1)
if(s(-(gen_0':s:if2_0(a), p(s(gen_0':s:if2_0(n4_0)))))) →RΩ(1)
if(s(-(gen_0':s:if2_0(a), gen_0':s:if2_0(n4_0)))) →IH
if(s(*3_0))

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

### (13) Obligation:

TRS:
Rules:
-(0', y) → 0'
-(x, 0') → x
-(x, s(y)) → if(s(-(x, p(s(y)))))
p(0') → 0'
p(s(x)) → x

Types:
- :: 0':s:if → 0':s:if → 0':s:if
0' :: 0':s:if
s :: 0':s:if → 0':s:if
if :: 0':s:if → 0':s:if
p :: 0':s:if → 0':s:if
hole_0':s:if1_0 :: 0':s:if
gen_0':s:if2_0 :: Nat → 0':s:if

Lemmas:
-(gen_0':s:if2_0(a), gen_0':s:if2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_0':s:if2_0(0) ⇔ 0'
gen_0':s:if2_0(+(x, 1)) ⇔ s(gen_0':s:if2_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':s:if2_0(a), gen_0':s:if2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

### (16) Obligation:

TRS:
Rules:
-(0', y) → 0'
-(x, 0') → x
-(x, s(y)) → if(s(-(x, p(s(y)))))
p(0') → 0'
p(s(x)) → x

Types:
- :: 0':s:if → 0':s:if → 0':s:if
0' :: 0':s:if
s :: 0':s:if → 0':s:if
if :: 0':s:if → 0':s:if
p :: 0':s:if → 0':s:if
hole_0':s:if1_0 :: 0':s:if
gen_0':s:if2_0 :: Nat → 0':s:if

Lemmas:
-(gen_0':s:if2_0(a), gen_0':s:if2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_0':s:if2_0(0) ⇔ 0'
gen_0':s:if2_0(+(x, 1)) ⇔ s(gen_0':s:if2_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':s:if2_0(a), gen_0':s:if2_0(n4_0)) → *3_0, rt ∈ Ω(n40)