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:

p(0) → 0
p(s(X)) → X
leq(0, Y) → true
leq(s(X), 0) → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → X
if(false, X, Y) → Y
diff(X, Y) → if(leq(X, Y), 0, s(diff(p(X), Y)))

Rewrite Strategy: FULL

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

p(0') → 0'
p(s(X)) → X
leq(0', Y) → true
leq(s(X), 0') → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → X
if(false, X, Y) → Y
diff(X, Y) → if(leq(X, Y), 0', s(diff(p(X), Y)))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(4) Obligation:

TRS:
Rules:
p(0') → 0'
p(s(X)) → X
leq(0', Y) → true
leq(s(X), 0') → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → X
if(false, X, Y) → Y
diff(X, Y) → if(leq(X, Y), 0', s(diff(p(X), Y)))

Types:
p :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
leq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if :: true:false → 0':s → 0':s → 0':s
diff :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
leq, diff

They will be analysed ascendingly in the following order:
leq < diff

(6) Obligation:

TRS:
Rules:
p(0') → 0'
p(s(X)) → X
leq(0', Y) → true
leq(s(X), 0') → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → X
if(false, X, Y) → Y
diff(X, Y) → if(leq(X, Y), 0', s(diff(p(X), Y)))

Types:
p :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
leq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if :: true:false → 0':s → 0':s → 0':s
diff :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
leq, diff

They will be analysed ascendingly in the following order:
leq < diff

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Induction Base:
leq(gen_0':s3_0(0), gen_0':s3_0(0)) →_R^Ω(1)
true

Induction Step:
leq(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →_R^Ω(1)
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) →_IH
true

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

(8) Complex Obligation (BEST)

(9) Obligation:

TRS:
Rules:
p(0') → 0'
p(s(X)) → X
leq(0', Y) → true
leq(s(X), 0') → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → X
if(false, X, Y) → Y
diff(X, Y) → if(leq(X, Y), 0', s(diff(p(X), Y)))

Types:
p :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
leq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if :: true:false → 0':s → 0':s → 0':s
diff :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
diff

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

TRS:
Rules:
p(0') → 0'
p(s(X)) → X
leq(0', Y) → true
leq(s(X), 0') → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → X
if(false, X, Y) → Y
diff(X, Y) → if(leq(X, Y), 0', s(diff(p(X), Y)))

Types:
p :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
leq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if :: true:false → 0':s → 0':s → 0':s
diff :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

TRS:
Rules:
p(0') → 0'
p(s(X)) → X
leq(0', Y) → true
leq(s(X), 0') → false
leq(s(X), s(Y)) → leq(X, Y)
if(true, X, Y) → X
if(false, X, Y) → Y
diff(X, Y) → if(leq(X, Y), 0', s(diff(p(X), Y)))

Types:
p :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
leq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if :: true:false → 0':s → 0':s → 0':s
diff :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(16) BOUNDS(n^1, INF)