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:

half(x) → if(ge(x, s(s(0))), x)
if(false, x) → 0
if(true, x) → s(half(p(p(x))))
p(0) → 0
p(s(x)) → x
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
log(0) → 0
log(s(x)) → s(log(half(s(x))))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
half(s(s(x97_1))) →⁺ s(half(x97_1))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x97_1 / s(s(x97_1))].
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:

half(x) → if(ge(x, s(s(0'))), x)
if(false, x) → 0'
if(true, x) → s(half(p(p(x))))
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
log(0') → 0'
log(s(x)) → s(log(half(s(x))))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
half(x) → if(ge(x, s(s(0'))), x)
if(false, x) → 0'
if(true, x) → s(half(p(p(x))))
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
log(0') → 0'
log(s(x)) → s(log(half(s(x))))

Types:
half :: 0':s → 0':s
if :: false:true → 0':s → 0':s
ge :: 0':s → 0':s → false:true
s :: 0':s → 0':s
0' :: 0':s
false :: false:true
true :: false:true
p :: 0':s → 0':s
log :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
half, ge, log

They will be analysed ascendingly in the following order:
ge < half
half < log

(8) Obligation:

TRS:
Rules:
half(x) → if(ge(x, s(s(0'))), x)
if(false, x) → 0'
if(true, x) → s(half(p(p(x))))
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
log(0') → 0'
log(s(x)) → s(log(half(s(x))))

Types:
half :: 0':s → 0':s
if :: false:true → 0':s → 0':s
ge :: 0':s → 0':s → false:true
s :: 0':s → 0':s
0' :: 0':s
false :: false:true
true :: false:true
p :: 0':s → 0':s
log :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
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:
ge, half, log

They will be analysed ascendingly in the following order:
ge < half
half < log

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

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

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

Induction Step:
ge(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →_R^Ω(1)
ge(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).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
half(x) → if(ge(x, s(s(0'))), x)
if(false, x) → 0'
if(true, x) → s(half(p(p(x))))
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
log(0') → 0'
log(s(x)) → s(log(half(s(x))))

Types:
half :: 0':s → 0':s
if :: false:true → 0':s → 0':s
ge :: 0':s → 0':s → false:true
s :: 0':s → 0':s
0' :: 0':s
false :: false:true
true :: false:true
p :: 0':s → 0':s
log :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

Lemmas:
ge(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:
half, log

They will be analysed ascendingly in the following order:
half < log

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

TRS:
Rules:
half(x) → if(ge(x, s(s(0'))), x)
if(false, x) → 0'
if(true, x) → s(half(p(p(x))))
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
log(0') → 0'
log(s(x)) → s(log(half(s(x))))

Types:
half :: 0':s → 0':s
if :: false:true → 0':s → 0':s
ge :: 0':s → 0':s → false:true
s :: 0':s → 0':s
0' :: 0':s
false :: false:true
true :: false:true
p :: 0':s → 0':s
log :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

Lemmas:
ge(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:
log

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) Obligation:

TRS:
Rules:
half(x) → if(ge(x, s(s(0'))), x)
if(false, x) → 0'
if(true, x) → s(half(p(p(x))))
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
log(0') → 0'
log(s(x)) → s(log(half(s(x))))

Types:
half :: 0':s → 0':s
if :: false:true → 0':s → 0':s
ge :: 0':s → 0':s → false:true
s :: 0':s → 0':s
0' :: 0':s
false :: false:true
true :: false:true
p :: 0':s → 0':s
log :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

Lemmas:
ge(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.

(16) LowerBoundsProof (EQUIVALENT transformation)

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

(17) BOUNDS(n^1, INF)

(18) Obligation:

TRS:
Rules:
half(x) → if(ge(x, s(s(0'))), x)
if(false, x) → 0'
if(true, x) → s(half(p(p(x))))
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
log(0') → 0'
log(s(x)) → s(log(half(s(x))))

Types:
half :: 0':s → 0':s
if :: false:true → 0':s → 0':s
ge :: 0':s → 0':s → false:true
s :: 0':s → 0':s
0' :: 0':s
false :: false:true
true :: false:true
p :: 0':s → 0':s
log :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

Lemmas:
ge(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.

(19) LowerBoundsProof (EQUIVALENT transformation)

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

(20) BOUNDS(n^1, INF)