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:

double(0) → 0
double(s(x)) → s(s(double(x)))
del(x, nil) → nil
del(x, cons(y, xs)) → if(eq(x, y), x, y, xs)
if(true, x, y, xs) → xs
if(false, x, y, xs) → cons(y, del(x, xs))
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
first(nil) → 0
first(cons(x, xs)) → x
doublelist(nil) → nil
doublelist(cons(x, xs)) → cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

double(0') → 0'
double(s(x)) → s(s(double(x)))
del(x, nil) → nil
del(x, cons(y, xs)) → if(eq(x, y), x, y, xs)
if(true, x, y, xs) → xs
if(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
first(nil) → 0'
first(cons(x, xs)) → x
doublelist(nil) → nil
doublelist(cons(x, xs)) → cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
del(x, nil) → nil
del(x, cons(y, xs)) → if(eq(x, y), x, y, xs)
if(true, x, y, xs) → xs
if(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
first(nil) → 0'
first(cons(x, xs)) → x
doublelist(nil) → nil
doublelist(cons(x, xs)) → cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
del :: 0':s → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
if :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
first :: nil:cons → 0':s
doublelist :: nil:cons → nil:cons
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

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

Heuristically decided to analyse the following defined symbols:
double, del, eq, doublelist

They will be analysed ascendingly in the following order:
double < doublelist
eq < del
del < doublelist

(8) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
del(x, nil) → nil
del(x, cons(y, xs)) → if(eq(x, y), x, y, xs)
if(true, x, y, xs) → xs
if(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
first(nil) → 0'
first(cons(x, xs)) → x
doublelist(nil) → nil
doublelist(cons(x, xs)) → cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
del :: 0':s → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
if :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
first :: nil:cons → 0':s
doublelist :: nil:cons → nil:cons
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:
double, del, eq, doublelist

They will be analysed ascendingly in the following order:
double < doublelist
eq < del
del < doublelist

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

Proved the following rewrite lemma:
double(gen_0':s4_0(n7_0)) → gen_0':s4_0(*(2, n7_0)), rt ∈ Ω(1 + n7₀)

Induction Base:
double(gen_0':s4_0(0)) →_R^Ω(1)
0'

Induction Step:
double(gen_0':s4_0(+(n7_0, 1))) →_R^Ω(1)
s(s(double(gen_0':s4_0(n7_0)))) →_IH
s(s(gen_0':s4_0(*(2, c8_0))))

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
del(x, nil) → nil
del(x, cons(y, xs)) → if(eq(x, y), x, y, xs)
if(true, x, y, xs) → xs
if(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
first(nil) → 0'
first(cons(x, xs)) → x
doublelist(nil) → nil
doublelist(cons(x, xs)) → cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
del :: 0':s → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
if :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
first :: nil:cons → 0':s
doublelist :: nil:cons → nil:cons
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:
double(gen_0':s4_0(n7_0)) → gen_0':s4_0(*(2, n7_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:
eq, del, doublelist

They will be analysed ascendingly in the following order:
eq < del
del < doublelist

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

Proved the following rewrite lemma:
eq(gen_0':s4_0(n275_0), gen_0':s4_0(n275_0)) → true, rt ∈ Ω(1 + n275₀)

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

Induction Step:
eq(gen_0':s4_0(+(n275_0, 1)), gen_0':s4_0(+(n275_0, 1))) →_R^Ω(1)
eq(gen_0':s4_0(n275_0), gen_0':s4_0(n275_0)) →_IH
true

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
del(x, nil) → nil
del(x, cons(y, xs)) → if(eq(x, y), x, y, xs)
if(true, x, y, xs) → xs
if(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
first(nil) → 0'
first(cons(x, xs)) → x
doublelist(nil) → nil
doublelist(cons(x, xs)) → cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
del :: 0':s → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
if :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
first :: nil:cons → 0':s
doublelist :: nil:cons → nil:cons
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:
double(gen_0':s4_0(n7_0)) → gen_0':s4_0(*(2, n7_0)), rt ∈ Ω(1 + n7₀)
eq(gen_0':s4_0(n275_0), gen_0':s4_0(n275_0)) → true, rt ∈ Ω(1 + n275₀)

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:
del, doublelist

They will be analysed ascendingly in the following order:
del < doublelist

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

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

(16) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
del(x, nil) → nil
del(x, cons(y, xs)) → if(eq(x, y), x, y, xs)
if(true, x, y, xs) → xs
if(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
first(nil) → 0'
first(cons(x, xs)) → x
doublelist(nil) → nil
doublelist(cons(x, xs)) → cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
del :: 0':s → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
if :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
first :: nil:cons → 0':s
doublelist :: nil:cons → nil:cons
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:
double(gen_0':s4_0(n7_0)) → gen_0':s4_0(*(2, n7_0)), rt ∈ Ω(1 + n7₀)
eq(gen_0':s4_0(n275_0), gen_0':s4_0(n275_0)) → true, rt ∈ Ω(1 + n275₀)

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:
doublelist

(17) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
doublelist(gen_nil:cons5_0(n860_0)) → gen_nil:cons5_0(n860_0), rt ∈ Ω(1 + n860₀)

Induction Base:
doublelist(gen_nil:cons5_0(0)) →_R^Ω(1)
nil

Induction Step:
doublelist(gen_nil:cons5_0(+(n860_0, 1))) →_R^Ω(1)
cons(double(0'), doublelist(del(first(cons(0', gen_nil:cons5_0(n860_0))), cons(0', gen_nil:cons5_0(n860_0))))) →_L^Ω(1)
cons(gen_0':s4_0(*(2, 0)), doublelist(del(first(cons(0', gen_nil:cons5_0(n860_0))), cons(0', gen_nil:cons5_0(n860_0))))) →_R^Ω(1)
cons(gen_0':s4_0(0), doublelist(del(0', cons(0', gen_nil:cons5_0(n860_0))))) →_R^Ω(1)
cons(gen_0':s4_0(0), doublelist(if(eq(0', 0'), 0', 0', gen_nil:cons5_0(n860_0)))) →_L^Ω(1)
cons(gen_0':s4_0(0), doublelist(if(true, 0', 0', gen_nil:cons5_0(n860_0)))) →_R^Ω(1)
cons(gen_0':s4_0(0), doublelist(gen_nil:cons5_0(n860_0))) →_IH
cons(gen_0':s4_0(0), gen_nil:cons5_0(c861_0))

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

(18) Complex Obligation (BEST)

(19) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
del(x, nil) → nil
del(x, cons(y, xs)) → if(eq(x, y), x, y, xs)
if(true, x, y, xs) → xs
if(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
first(nil) → 0'
first(cons(x, xs)) → x
doublelist(nil) → nil
doublelist(cons(x, xs)) → cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
del :: 0':s → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
if :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
first :: nil:cons → 0':s
doublelist :: nil:cons → nil:cons
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:
double(gen_0':s4_0(n7_0)) → gen_0':s4_0(*(2, n7_0)), rt ∈ Ω(1 + n7₀)
eq(gen_0':s4_0(n275_0), gen_0':s4_0(n275_0)) → true, rt ∈ Ω(1 + n275₀)
doublelist(gen_nil:cons5_0(n860_0)) → gen_nil:cons5_0(n860_0), rt ∈ Ω(1 + n860₀)

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.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
double(gen_0':s4_0(n7_0)) → gen_0':s4_0(*(2, n7_0)), rt ∈ Ω(1 + n7₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
del(x, nil) → nil
del(x, cons(y, xs)) → if(eq(x, y), x, y, xs)
if(true, x, y, xs) → xs
if(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
first(nil) → 0'
first(cons(x, xs)) → x
doublelist(nil) → nil
doublelist(cons(x, xs)) → cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
del :: 0':s → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
if :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
first :: nil:cons → 0':s
doublelist :: nil:cons → nil:cons
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:
double(gen_0':s4_0(n7_0)) → gen_0':s4_0(*(2, n7_0)), rt ∈ Ω(1 + n7₀)
eq(gen_0':s4_0(n275_0), gen_0':s4_0(n275_0)) → true, rt ∈ Ω(1 + n275₀)
doublelist(gen_nil:cons5_0(n860_0)) → gen_nil:cons5_0(n860_0), rt ∈ Ω(1 + n860₀)

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.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
double(gen_0':s4_0(n7_0)) → gen_0':s4_0(*(2, n7_0)), rt ∈ Ω(1 + n7₀)

(24) BOUNDS(n^1, INF)

(25) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
del(x, nil) → nil
del(x, cons(y, xs)) → if(eq(x, y), x, y, xs)
if(true, x, y, xs) → xs
if(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
first(nil) → 0'
first(cons(x, xs)) → x
doublelist(nil) → nil
doublelist(cons(x, xs)) → cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
del :: 0':s → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
if :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
first :: nil:cons → 0':s
doublelist :: nil:cons → nil:cons
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:
double(gen_0':s4_0(n7_0)) → gen_0':s4_0(*(2, n7_0)), rt ∈ Ω(1 + n7₀)
eq(gen_0':s4_0(n275_0), gen_0':s4_0(n275_0)) → true, rt ∈ Ω(1 + n275₀)

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.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
double(gen_0':s4_0(n7_0)) → gen_0':s4_0(*(2, n7_0)), rt ∈ Ω(1 + n7₀)

(27) BOUNDS(n^1, INF)

(28) Obligation:

TRS:
Rules:
double(0') → 0'
double(s(x)) → s(s(double(x)))
del(x, nil) → nil
del(x, cons(y, xs)) → if(eq(x, y), x, y, xs)
if(true, x, y, xs) → xs
if(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
first(nil) → 0'
first(cons(x, xs)) → x
doublelist(nil) → nil
doublelist(cons(x, xs)) → cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

Types:
double :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
del :: 0':s → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
if :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
first :: nil:cons → 0':s
doublelist :: nil:cons → nil:cons
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:
double(gen_0':s4_0(n7_0)) → gen_0':s4_0(*(2, n7_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.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
double(gen_0':s4_0(n7_0)) → gen_0':s4_0(*(2, n7_0)), rt ∈ Ω(1 + n7₀)

(30) BOUNDS(n^1, INF)