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:

ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
rev(x) → if(x, eq(0, length(x)), nil, 0, length(x))
if(x, true, z, c, l) → z
if(x, false, z, c, l) → help(s(c), l, x, z)
help(c, l, cons(x, y), z) → if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
append(nil, y) → y
append(cons(x, y), z) → cons(x, append(y, z))
length(nil) → 0
length(cons(x, y)) → s(length(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:

ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
rev(x) → if(x, eq(0', length(x)), nil, 0', length(x))
if(x, true, z, c, l) → z
if(x, false, z, c, l) → help(s(c), l, x, z)
help(c, l, cons(x, y), z) → if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
append(nil, y) → y
append(cons(x, y), z) → cons(x, append(y, z))
length(nil) → 0'
length(cons(x, y)) → s(length(y))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
rev(x) → if(x, eq(0', length(x)), nil, 0', length(x))
if(x, true, z, c, l) → z
if(x, false, z, c, l) → help(s(c), l, x, z)
help(c, l, cons(x, y), z) → if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
append(nil, y) → y
append(cons(x, y), z) → cons(x, append(y, z))
length(nil) → 0'
length(cons(x, y)) → s(length(y))

Types:
ge :: 0':s → 0':s → true:false:eq
0' :: 0':s
true :: true:false:eq
s :: 0':s → 0':s
false :: true:false:eq
rev :: nil:cons → nil:cons
if :: nil:cons → true:false:eq → nil:cons → 0':s → 0':s → nil:cons
eq :: 0':s → 0':s → true:false:eq
length :: nil:cons → 0':s
nil :: nil:cons
help :: 0':s → 0':s → nil:cons → nil:cons → nil:cons
cons :: a → nil:cons → nil:cons
append :: nil:cons → nil:cons → nil:cons
hole_true:false:eq1_0 :: true:false:eq
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
hole_a4_0 :: a
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

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

Heuristically decided to analyse the following defined symbols:
ge, length, help, append

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

(6) Obligation:

Innermost TRS:
Rules:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
rev(x) → if(x, eq(0', length(x)), nil, 0', length(x))
if(x, true, z, c, l) → z
if(x, false, z, c, l) → help(s(c), l, x, z)
help(c, l, cons(x, y), z) → if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
append(nil, y) → y
append(cons(x, y), z) → cons(x, append(y, z))
length(nil) → 0'
length(cons(x, y)) → s(length(y))

Types:
ge :: 0':s → 0':s → true:false:eq
0' :: 0':s
true :: true:false:eq
s :: 0':s → 0':s
false :: true:false:eq
rev :: nil:cons → nil:cons
if :: nil:cons → true:false:eq → nil:cons → 0':s → 0':s → nil:cons
eq :: 0':s → 0':s → true:false:eq
length :: nil:cons → 0':s
nil :: nil:cons
help :: 0':s → 0':s → nil:cons → nil:cons → nil:cons
cons :: a → nil:cons → nil:cons
append :: nil:cons → nil:cons → nil:cons
hole_true:false:eq1_0 :: true:false:eq
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
hole_a4_0 :: a
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(hole_a4_0, gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
ge, length, help, append

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

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

Proved the following rewrite lemma:
ge(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) → true, rt ∈ Ω(1 + n8₀)

Induction Base:
ge(gen_0':s5_0(0), gen_0':s5_0(0)) →_R^Ω(1)
true

Induction Step:
ge(gen_0':s5_0(+(n8_0, 1)), gen_0':s5_0(+(n8_0, 1))) →_R^Ω(1)
ge(gen_0':s5_0(n8_0), gen_0':s5_0(n8_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:

Innermost TRS:
Rules:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
rev(x) → if(x, eq(0', length(x)), nil, 0', length(x))
if(x, true, z, c, l) → z
if(x, false, z, c, l) → help(s(c), l, x, z)
help(c, l, cons(x, y), z) → if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
append(nil, y) → y
append(cons(x, y), z) → cons(x, append(y, z))
length(nil) → 0'
length(cons(x, y)) → s(length(y))

Types:
ge :: 0':s → 0':s → true:false:eq
0' :: 0':s
true :: true:false:eq
s :: 0':s → 0':s
false :: true:false:eq
rev :: nil:cons → nil:cons
if :: nil:cons → true:false:eq → nil:cons → 0':s → 0':s → nil:cons
eq :: 0':s → 0':s → true:false:eq
length :: nil:cons → 0':s
nil :: nil:cons
help :: 0':s → 0':s → nil:cons → nil:cons → nil:cons
cons :: a → nil:cons → nil:cons
append :: nil:cons → nil:cons → nil:cons
hole_true:false:eq1_0 :: true:false:eq
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
hole_a4_0 :: a
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) → true, rt ∈ Ω(1 + n8₀)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(hole_a4_0, gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
length, help, append

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

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

Proved the following rewrite lemma:
length(gen_nil:cons6_0(n301_0)) → gen_0':s5_0(n301_0), rt ∈ Ω(1 + n301₀)

Induction Base:
length(gen_nil:cons6_0(0)) →_R^Ω(1)
0'

Induction Step:
length(gen_nil:cons6_0(+(n301_0, 1))) →_R^Ω(1)
s(length(gen_nil:cons6_0(n301_0))) →_IH
s(gen_0':s5_0(c302_0))

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:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
rev(x) → if(x, eq(0', length(x)), nil, 0', length(x))
if(x, true, z, c, l) → z
if(x, false, z, c, l) → help(s(c), l, x, z)
help(c, l, cons(x, y), z) → if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
append(nil, y) → y
append(cons(x, y), z) → cons(x, append(y, z))
length(nil) → 0'
length(cons(x, y)) → s(length(y))

Types:
ge :: 0':s → 0':s → true:false:eq
0' :: 0':s
true :: true:false:eq
s :: 0':s → 0':s
false :: true:false:eq
rev :: nil:cons → nil:cons
if :: nil:cons → true:false:eq → nil:cons → 0':s → 0':s → nil:cons
eq :: 0':s → 0':s → true:false:eq
length :: nil:cons → 0':s
nil :: nil:cons
help :: 0':s → 0':s → nil:cons → nil:cons → nil:cons
cons :: a → nil:cons → nil:cons
append :: nil:cons → nil:cons → nil:cons
hole_true:false:eq1_0 :: true:false:eq
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
hole_a4_0 :: a
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) → true, rt ∈ Ω(1 + n8₀)
length(gen_nil:cons6_0(n301_0)) → gen_0':s5_0(n301_0), rt ∈ Ω(1 + n301₀)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(hole_a4_0, gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
append, help

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

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
append(gen_nil:cons6_0(n541_0), gen_nil:cons6_0(b)) → gen_nil:cons6_0(+(n541_0, b)), rt ∈ Ω(1 + n541₀)

Induction Base:
append(gen_nil:cons6_0(0), gen_nil:cons6_0(b)) →_R^Ω(1)
gen_nil:cons6_0(b)

Induction Step:
append(gen_nil:cons6_0(+(n541_0, 1)), gen_nil:cons6_0(b)) →_R^Ω(1)
cons(hole_a4_0, append(gen_nil:cons6_0(n541_0), gen_nil:cons6_0(b))) →_IH
cons(hole_a4_0, gen_nil:cons6_0(+(b, c542_0)))

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

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost TRS:
Rules:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
rev(x) → if(x, eq(0', length(x)), nil, 0', length(x))
if(x, true, z, c, l) → z
if(x, false, z, c, l) → help(s(c), l, x, z)
help(c, l, cons(x, y), z) → if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
append(nil, y) → y
append(cons(x, y), z) → cons(x, append(y, z))
length(nil) → 0'
length(cons(x, y)) → s(length(y))

Types:
ge :: 0':s → 0':s → true:false:eq
0' :: 0':s
true :: true:false:eq
s :: 0':s → 0':s
false :: true:false:eq
rev :: nil:cons → nil:cons
if :: nil:cons → true:false:eq → nil:cons → 0':s → 0':s → nil:cons
eq :: 0':s → 0':s → true:false:eq
length :: nil:cons → 0':s
nil :: nil:cons
help :: 0':s → 0':s → nil:cons → nil:cons → nil:cons
cons :: a → nil:cons → nil:cons
append :: nil:cons → nil:cons → nil:cons
hole_true:false:eq1_0 :: true:false:eq
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
hole_a4_0 :: a
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) → true, rt ∈ Ω(1 + n8₀)
length(gen_nil:cons6_0(n301_0)) → gen_0':s5_0(n301_0), rt ∈ Ω(1 + n301₀)
append(gen_nil:cons6_0(n541_0), gen_nil:cons6_0(b)) → gen_nil:cons6_0(+(n541_0, b)), rt ∈ Ω(1 + n541₀)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(hole_a4_0, gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
help

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(17) Obligation:

Innermost TRS:
Rules:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
rev(x) → if(x, eq(0', length(x)), nil, 0', length(x))
if(x, true, z, c, l) → z
if(x, false, z, c, l) → help(s(c), l, x, z)
help(c, l, cons(x, y), z) → if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
append(nil, y) → y
append(cons(x, y), z) → cons(x, append(y, z))
length(nil) → 0'
length(cons(x, y)) → s(length(y))

Types:
ge :: 0':s → 0':s → true:false:eq
0' :: 0':s
true :: true:false:eq
s :: 0':s → 0':s
false :: true:false:eq
rev :: nil:cons → nil:cons
if :: nil:cons → true:false:eq → nil:cons → 0':s → 0':s → nil:cons
eq :: 0':s → 0':s → true:false:eq
length :: nil:cons → 0':s
nil :: nil:cons
help :: 0':s → 0':s → nil:cons → nil:cons → nil:cons
cons :: a → nil:cons → nil:cons
append :: nil:cons → nil:cons → nil:cons
hole_true:false:eq1_0 :: true:false:eq
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
hole_a4_0 :: a
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) → true, rt ∈ Ω(1 + n8₀)
length(gen_nil:cons6_0(n301_0)) → gen_0':s5_0(n301_0), rt ∈ Ω(1 + n301₀)
append(gen_nil:cons6_0(n541_0), gen_nil:cons6_0(b)) → gen_nil:cons6_0(+(n541_0, b)), rt ∈ Ω(1 + n541₀)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(hole_a4_0, gen_nil:cons6_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ge(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) → true, rt ∈ Ω(1 + n8₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
rev(x) → if(x, eq(0', length(x)), nil, 0', length(x))
if(x, true, z, c, l) → z
if(x, false, z, c, l) → help(s(c), l, x, z)
help(c, l, cons(x, y), z) → if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
append(nil, y) → y
append(cons(x, y), z) → cons(x, append(y, z))
length(nil) → 0'
length(cons(x, y)) → s(length(y))

Types:
ge :: 0':s → 0':s → true:false:eq
0' :: 0':s
true :: true:false:eq
s :: 0':s → 0':s
false :: true:false:eq
rev :: nil:cons → nil:cons
if :: nil:cons → true:false:eq → nil:cons → 0':s → 0':s → nil:cons
eq :: 0':s → 0':s → true:false:eq
length :: nil:cons → 0':s
nil :: nil:cons
help :: 0':s → 0':s → nil:cons → nil:cons → nil:cons
cons :: a → nil:cons → nil:cons
append :: nil:cons → nil:cons → nil:cons
hole_true:false:eq1_0 :: true:false:eq
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
hole_a4_0 :: a
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) → true, rt ∈ Ω(1 + n8₀)
length(gen_nil:cons6_0(n301_0)) → gen_0':s5_0(n301_0), rt ∈ Ω(1 + n301₀)
append(gen_nil:cons6_0(n541_0), gen_nil:cons6_0(b)) → gen_nil:cons6_0(+(n541_0, b)), rt ∈ Ω(1 + n541₀)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(hole_a4_0, gen_nil:cons6_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ge(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) → true, rt ∈ Ω(1 + n8₀)

(22) BOUNDS(n^1, INF)

(23) Obligation:

Innermost TRS:
Rules:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
rev(x) → if(x, eq(0', length(x)), nil, 0', length(x))
if(x, true, z, c, l) → z
if(x, false, z, c, l) → help(s(c), l, x, z)
help(c, l, cons(x, y), z) → if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
append(nil, y) → y
append(cons(x, y), z) → cons(x, append(y, z))
length(nil) → 0'
length(cons(x, y)) → s(length(y))

Types:
ge :: 0':s → 0':s → true:false:eq
0' :: 0':s
true :: true:false:eq
s :: 0':s → 0':s
false :: true:false:eq
rev :: nil:cons → nil:cons
if :: nil:cons → true:false:eq → nil:cons → 0':s → 0':s → nil:cons
eq :: 0':s → 0':s → true:false:eq
length :: nil:cons → 0':s
nil :: nil:cons
help :: 0':s → 0':s → nil:cons → nil:cons → nil:cons
cons :: a → nil:cons → nil:cons
append :: nil:cons → nil:cons → nil:cons
hole_true:false:eq1_0 :: true:false:eq
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
hole_a4_0 :: a
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) → true, rt ∈ Ω(1 + n8₀)
length(gen_nil:cons6_0(n301_0)) → gen_0':s5_0(n301_0), rt ∈ Ω(1 + n301₀)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(hole_a4_0, gen_nil:cons6_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ge(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) → true, rt ∈ Ω(1 + n8₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

Innermost TRS:
Rules:
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
rev(x) → if(x, eq(0', length(x)), nil, 0', length(x))
if(x, true, z, c, l) → z
if(x, false, z, c, l) → help(s(c), l, x, z)
help(c, l, cons(x, y), z) → if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
append(nil, y) → y
append(cons(x, y), z) → cons(x, append(y, z))
length(nil) → 0'
length(cons(x, y)) → s(length(y))

Types:
ge :: 0':s → 0':s → true:false:eq
0' :: 0':s
true :: true:false:eq
s :: 0':s → 0':s
false :: true:false:eq
rev :: nil:cons → nil:cons
if :: nil:cons → true:false:eq → nil:cons → 0':s → 0':s → nil:cons
eq :: 0':s → 0':s → true:false:eq
length :: nil:cons → 0':s
nil :: nil:cons
help :: 0':s → 0':s → nil:cons → nil:cons → nil:cons
cons :: a → nil:cons → nil:cons
append :: nil:cons → nil:cons → nil:cons
hole_true:false:eq1_0 :: true:false:eq
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
hole_a4_0 :: a
gen_0':s5_0 :: Nat → 0':s
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) → true, rt ∈ Ω(1 + n8₀)

Generator Equations:
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(hole_a4_0, gen_nil:cons6_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ge(gen_0':s5_0(n8_0), gen_0':s5_0(n8_0)) → true, rt ∈ Ω(1 + n8₀)

(28) BOUNDS(n^1, INF)