The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(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 [ ].

(2) BOUNDS(n^1, INF)

(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) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
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

Types:
- :: 0':s:if → 0':s:if → 0':s:if
0' :: 0':s:if
s :: 0':s:if → 0':s:if
if :: greater → 0':s:if → 0':s:if → 0':s:if
greater :: 0':s:if → 0':s:if → greater
p :: 0':s:if → 0':s:if
hole_0':s:if1_0 :: 0':s:if
hole_greater2_0 :: greater
gen_0':s:if3_0 :: Nat → 0':s:if

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

Heuristically decided to analyse the following defined symbols:
-

(8) Obligation:

TRS:
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

Types:
- :: 0':s:if → 0':s:if → 0':s:if
0' :: 0':s:if
s :: 0':s:if → 0':s:if
if :: greater → 0':s:if → 0':s:if → 0':s:if
greater :: 0':s:if → 0':s:if → greater
p :: 0':s:if → 0':s:if
hole_0':s:if1_0 :: 0':s:if
hole_greater2_0 :: greater
gen_0':s:if3_0 :: Nat → 0':s:if

Generator Equations:
gen_0':s:if3_0(0) ⇔ 0'
gen_0':s:if3_0(+(x, 1)) ⇔ s(gen_0':s:if3_0(x))

The following defined symbols remain to be analysed:
-

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
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

Types:
- :: 0':s:if → 0':s:if → 0':s:if
0' :: 0':s:if
s :: 0':s:if → 0':s:if
if :: greater → 0':s:if → 0':s:if → 0':s:if
greater :: 0':s:if → 0':s:if → greater
p :: 0':s:if → 0':s:if
hole_0':s:if1_0 :: 0':s:if
hole_greater2_0 :: greater
gen_0':s:if3_0 :: Nat → 0':s:if

Generator Equations:
gen_0':s:if3_0(0) ⇔ 0'
gen_0':s:if3_0(+(x, 1)) ⇔ s(gen_0':s:if3_0(x))

No more defined symbols left to analyse.