The Runtime Complexity (innermost) 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) → sum(generate(x, y))
generate(x, y) → gen(x, y, 0)
gen(x, y, z) → if(ge(z, x), x, y, z)
if(true, x, y, z) → nil
if(false, x, y, z) → cons(y, gen(x, y, s(z)))
sum(nil) → 0
sum(cons(0, xs)) → sum(xs)
sum(cons(s(x), xs)) → s(sum(cons(x, xs)))
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)

Rewrite Strategy: INNERMOST

(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:

times(x, y) → sum(generate(x, y))
generate(x, y) → gen(x, y, 0')
gen(x, y, z) → if(ge(z, x), x, y, z)
if(true, x, y, z) → nil
if(false, x, y, z) → cons(y, gen(x, y, s(z)))
sum(nil) → 0'
sum(cons(0', xs)) → sum(xs)
sum(cons(s(x), xs)) → s(sum(cons(x, xs)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
times(x, y) → sum(generate(x, y))
generate(x, y) → gen(x, y, 0')
gen(x, y, z) → if(ge(z, x), x, y, z)
if(true, x, y, z) → nil
if(false, x, y, z) → cons(y, gen(x, y, s(z)))
sum(nil) → 0'
sum(cons(0', xs)) → sum(xs)
sum(cons(s(x), xs)) → s(sum(cons(x, xs)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
times :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
generate :: 0':s → 0':s → nil:cons
gen :: 0':s → 0':s → 0':s → nil:cons
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → nil:cons
ge :: 0':s → 0':s → true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s → nil:cons → nil:cons
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

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

Heuristically decided to analyse the following defined symbols:
sum, gen, ge

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

(6) Obligation:

Innermost TRS:
Rules:
times(x, y) → sum(generate(x, y))
generate(x, y) → gen(x, y, 0')
gen(x, y, z) → if(ge(z, x), x, y, z)
if(true, x, y, z) → nil
if(false, x, y, z) → cons(y, gen(x, y, s(z)))
sum(nil) → 0'
sum(cons(0', xs)) → sum(xs)
sum(cons(s(x), xs)) → s(sum(cons(x, xs)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
times :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
generate :: 0':s → 0':s → nil:cons
gen :: 0':s → 0':s → 0':s → nil:cons
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → nil:cons
ge :: 0':s → 0':s → true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s → nil:cons → nil:cons
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

The following defined symbols remain to be analysed:
sum, gen, ge

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

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

Proved the following rewrite lemma:
sum(gen_nil:cons5_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)

Induction Base:
sum(gen_nil:cons5_0(0)) →_R^Ω(1)
0'

Induction Step:
sum(gen_nil:cons5_0(+(n7_0, 1))) →_R^Ω(1)
sum(gen_nil:cons5_0(n7_0)) →_IH
gen_0':s4_0(0)

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
times(x, y) → sum(generate(x, y))
generate(x, y) → gen(x, y, 0')
gen(x, y, z) → if(ge(z, x), x, y, z)
if(true, x, y, z) → nil
if(false, x, y, z) → cons(y, gen(x, y, s(z)))
sum(nil) → 0'
sum(cons(0', xs)) → sum(xs)
sum(cons(s(x), xs)) → s(sum(cons(x, xs)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
times :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
generate :: 0':s → 0':s → nil:cons
gen :: 0':s → 0':s → 0':s → nil:cons
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → nil:cons
ge :: 0':s → 0':s → true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s → nil:cons → nil:cons
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
sum(gen_nil:cons5_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

The following defined symbols remain to be analysed:
ge, gen

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

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
ge(gen_0':s4_0(n300_0), gen_0':s4_0(n300_0)) → true, rt ∈ Ω(1 + n300₀)

Induction Base:
ge(gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
true

Induction Step:
ge(gen_0':s4_0(+(n300_0, 1)), gen_0':s4_0(+(n300_0, 1))) →_R^Ω(1)
ge(gen_0':s4_0(n300_0), gen_0':s4_0(n300_0)) →_IH
true

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
times(x, y) → sum(generate(x, y))
generate(x, y) → gen(x, y, 0')
gen(x, y, z) → if(ge(z, x), x, y, z)
if(true, x, y, z) → nil
if(false, x, y, z) → cons(y, gen(x, y, s(z)))
sum(nil) → 0'
sum(cons(0', xs)) → sum(xs)
sum(cons(s(x), xs)) → s(sum(cons(x, xs)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
times :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
generate :: 0':s → 0':s → nil:cons
gen :: 0':s → 0':s → 0':s → nil:cons
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → nil:cons
ge :: 0':s → 0':s → true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s → nil:cons → nil:cons
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
sum(gen_nil:cons5_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)
ge(gen_0':s4_0(n300_0), gen_0':s4_0(n300_0)) → true, rt ∈ Ω(1 + n300₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

The following defined symbols remain to be analysed:
gen

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

Innermost TRS:
Rules:
times(x, y) → sum(generate(x, y))
generate(x, y) → gen(x, y, 0')
gen(x, y, z) → if(ge(z, x), x, y, z)
if(true, x, y, z) → nil
if(false, x, y, z) → cons(y, gen(x, y, s(z)))
sum(nil) → 0'
sum(cons(0', xs)) → sum(xs)
sum(cons(s(x), xs)) → s(sum(cons(x, xs)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
times :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
generate :: 0':s → 0':s → nil:cons
gen :: 0':s → 0':s → 0':s → nil:cons
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → nil:cons
ge :: 0':s → 0':s → true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s → nil:cons → nil:cons
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
sum(gen_nil:cons5_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)
ge(gen_0':s4_0(n300_0), gen_0':s4_0(n300_0)) → true, rt ∈ Ω(1 + n300₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sum(gen_nil:cons5_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
times(x, y) → sum(generate(x, y))
generate(x, y) → gen(x, y, 0')
gen(x, y, z) → if(ge(z, x), x, y, z)
if(true, x, y, z) → nil
if(false, x, y, z) → cons(y, gen(x, y, s(z)))
sum(nil) → 0'
sum(cons(0', xs)) → sum(xs)
sum(cons(s(x), xs)) → s(sum(cons(x, xs)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
times :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
generate :: 0':s → 0':s → nil:cons
gen :: 0':s → 0':s → 0':s → nil:cons
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → nil:cons
ge :: 0':s → 0':s → true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s → nil:cons → nil:cons
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
sum(gen_nil:cons5_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)
ge(gen_0':s4_0(n300_0), gen_0':s4_0(n300_0)) → true, rt ∈ Ω(1 + n300₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sum(gen_nil:cons5_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
times(x, y) → sum(generate(x, y))
generate(x, y) → gen(x, y, 0')
gen(x, y, z) → if(ge(z, x), x, y, z)
if(true, x, y, z) → nil
if(false, x, y, z) → cons(y, gen(x, y, s(z)))
sum(nil) → 0'
sum(cons(0', xs)) → sum(xs)
sum(cons(s(x), xs)) → s(sum(cons(x, xs)))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
times :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
generate :: 0':s → 0':s → nil:cons
gen :: 0':s → 0':s → 0':s → nil:cons
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → nil:cons
ge :: 0':s → 0':s → true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s → nil:cons → nil:cons
s :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
sum(gen_nil:cons5_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sum(gen_nil:cons5_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)

(22) BOUNDS(n^1, INF)