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:

times(x, y) → help(x, y, 0)
help(x, y, c) → if(lt(c, y), x, y, c)
if(true, x, y, c) → plus(x, help(x, y, s(c)))
if(false, x, y, c) → 0
lt(0, s(x)) → true
lt(s(x), 0) → false
lt(s(x), s(y)) → lt(x, y)
plus(x, 0) → x
plus(0, x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

times(x, y) → help(x, y, 0')
help(x, y, c) → if(lt(c, y), x, y, c)
if(true, x, y, c) → plus(x, help(x, y, s(c)))
if(false, x, y, c) → 0'
lt(0', s(x)) → true
lt(s(x), 0') → false
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(0', x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
times(x, y) → help(x, y, 0')
help(x, y, c) → if(lt(c, y), x, y, c)
if(true, x, y, c) → plus(x, help(x, y, s(c)))
if(false, x, y, c) → 0'
lt(0', s(x)) → true
lt(s(x), 0') → false
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(0', x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

Types:
times :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
true :: true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
help, lt, plus

They will be analysed ascendingly in the following order:
lt < help
plus < help

(8) Obligation:

TRS:
Rules:
times(x, y) → help(x, y, 0')
help(x, y, c) → if(lt(c, y), x, y, c)
if(true, x, y, c) → plus(x, help(x, y, s(c)))
if(false, x, y, c) → 0'
lt(0', s(x)) → true
lt(s(x), 0') → false
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(0', x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

Types:
times :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
true :: true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
false :: true:false
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:
lt, help, plus

They will be analysed ascendingly in the following order:
lt < help
plus < help

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

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

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

Induction Step:
lt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, 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:
times(x, y) → help(x, y, 0')
help(x, y, c) → if(lt(c, y), x, y, c)
if(true, x, y, c) → plus(x, help(x, y, s(c)))
if(false, x, y, c) → 0'
lt(0', s(x)) → true
lt(s(x), 0') → false
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(0', x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

Types:
times :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
true :: true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, 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:
plus, help

They will be analysed ascendingly in the following order:
plus < help

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
plus(gen_0':s3_0(a), gen_0':s3_0(n306_0)) → gen_0':s3_0(+(n306_0, a)), rt ∈ Ω(1 + n306₀)

Induction Base:
plus(gen_0':s3_0(a), gen_0':s3_0(0)) →_R^Ω(1)
gen_0':s3_0(a)

Induction Step:
plus(gen_0':s3_0(a), gen_0':s3_0(+(n306_0, 1))) →_R^Ω(1)
s(plus(gen_0':s3_0(a), gen_0':s3_0(n306_0))) →_IH
s(gen_0':s3_0(+(a, c307_0)))

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
times(x, y) → help(x, y, 0')
help(x, y, c) → if(lt(c, y), x, y, c)
if(true, x, y, c) → plus(x, help(x, y, s(c)))
if(false, x, y, c) → 0'
lt(0', s(x)) → true
lt(s(x), 0') → false
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(0', x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

Types:
times :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
true :: true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
plus(gen_0':s3_0(a), gen_0':s3_0(n306_0)) → gen_0':s3_0(+(n306_0, a)), rt ∈ Ω(1 + n306₀)

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:
help

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) Obligation:

TRS:
Rules:
times(x, y) → help(x, y, 0')
help(x, y, c) → if(lt(c, y), x, y, c)
if(true, x, y, c) → plus(x, help(x, y, s(c)))
if(false, x, y, c) → 0'
lt(0', s(x)) → true
lt(s(x), 0') → false
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(0', x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

Types:
times :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
true :: true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
plus(gen_0':s3_0(a), gen_0':s3_0(n306_0)) → gen_0':s3_0(+(n306_0, a)), rt ∈ Ω(1 + n306₀)

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.

(17) LowerBoundsProof (EQUIVALENT transformation)

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

(18) BOUNDS(n^1, INF)

(19) Obligation:

TRS:
Rules:
times(x, y) → help(x, y, 0')
help(x, y, c) → if(lt(c, y), x, y, c)
if(true, x, y, c) → plus(x, help(x, y, s(c)))
if(false, x, y, c) → 0'
lt(0', s(x)) → true
lt(s(x), 0') → false
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(0', x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

Types:
times :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
true :: true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
plus(gen_0':s3_0(a), gen_0':s3_0(n306_0)) → gen_0':s3_0(+(n306_0, a)), rt ∈ Ω(1 + n306₀)

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.

(20) LowerBoundsProof (EQUIVALENT transformation)

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

(21) BOUNDS(n^1, INF)

(22) Obligation:

TRS:
Rules:
times(x, y) → help(x, y, 0')
help(x, y, c) → if(lt(c, y), x, y, c)
if(true, x, y, c) → plus(x, help(x, y, s(c)))
if(false, x, y, c) → 0'
lt(0', s(x)) → true
lt(s(x), 0') → false
lt(s(x), s(y)) → lt(x, y)
plus(x, 0') → x
plus(0', x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))

Types:
times :: 0':s → 0':s → 0':s
help :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
true :: true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, 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.

(23) LowerBoundsProof (EQUIVALENT transformation)

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

(24) BOUNDS(n^1, INF)