The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

loop(Cons(x, xs), Nil, pp, ss) → False
loop(Cons(x', xs'), Cons(x, xs), pp, ss) → loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss)
loop(Nil, s, pp, ss) → True
match1(p, s) → loop(p, s, p, s)

The (relative) TRS S consists of the following rules:

!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0, S(y)) → False
!EQ(S(x), 0) → False
!EQ(0, 0) → True
loop[Ite](False, p, s, pp, Cons(x, xs)) → loop(pp, xs, pp, xs)
loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) → loop(xs', xs, pp, ss)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

loop(Cons(x, xs), Nil, pp, ss) → False
loop(Cons(x', xs'), Cons(x, xs), pp, ss) → loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss)
loop(Nil, s, pp, ss) → True
match1(p, s) → loop(p, s, p, s)

The (relative) TRS S consists of the following rules:

!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
loop[Ite](False, p, s, pp, Cons(x, xs)) → loop(pp, xs, pp, xs)
loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) → loop(xs', xs, pp, ss)

Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
loop(Cons(x, xs), Nil, pp, ss) → False
loop(Cons(x', xs'), Cons(x, xs), pp, ss) → loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss)
loop(Nil, s, pp, ss) → True
match1(p, s) → loop(p, s, p, s)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
loop[Ite](False, p, s, pp, Cons(x, xs)) → loop(pp, xs, pp, xs)
loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) → loop(xs', xs, pp, ss)

Types:
loop :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil → False:True
Cons :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
False :: False:True
loop[Ite] :: False:True → Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil → False:True
!EQ :: S:0' → S:0' → False:True
True :: False:True
match1 :: Cons:Nil → Cons:Nil → False:True
S :: S:0' → S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

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

Heuristically decided to analyse the following defined symbols:
loop, !EQ

They will be analysed ascendingly in the following order:
!EQ < loop

(8) Obligation:

Innermost TRS:
Rules:
loop(Cons(x, xs), Nil, pp, ss) → False
loop(Cons(x', xs'), Cons(x, xs), pp, ss) → loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss)
loop(Nil, s, pp, ss) → True
match1(p, s) → loop(p, s, p, s)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
loop[Ite](False, p, s, pp, Cons(x, xs)) → loop(pp, xs, pp, xs)
loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) → loop(xs', xs, pp, ss)

Types:
loop :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil → False:True
Cons :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
False :: False:True
loop[Ite] :: False:True → Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil → False:True
!EQ :: S:0' → S:0' → False:True
True :: False:True
match1 :: Cons:Nil → Cons:Nil → False:True
S :: S:0' → S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

The following defined symbols remain to be analysed:
!EQ, loop

They will be analysed ascendingly in the following order:
!EQ < loop

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

Proved the following rewrite lemma:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)

Induction Base:
!EQ(gen_S:0'5_0(0), gen_S:0'5_0(+(1, 0))) →_R^Ω(0)
False

Induction Step:
!EQ(gen_S:0'5_0(+(n7_0, 1)), gen_S:0'5_0(+(1, +(n7_0, 1)))) →_R^Ω(0)
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) →_IH
False

We have rt ∈ Ω(1) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n⁰).

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
loop(Cons(x, xs), Nil, pp, ss) → False
loop(Cons(x', xs'), Cons(x, xs), pp, ss) → loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss)
loop(Nil, s, pp, ss) → True
match1(p, s) → loop(p, s, p, s)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
loop[Ite](False, p, s, pp, Cons(x, xs)) → loop(pp, xs, pp, xs)
loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) → loop(xs', xs, pp, ss)

Types:
loop :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil → False:True
Cons :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
False :: False:True
loop[Ite] :: False:True → Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil → False:True
!EQ :: S:0' → S:0' → False:True
True :: False:True
match1 :: Cons:Nil → Cons:Nil → False:True
S :: S:0' → S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

The following defined symbols remain to be analysed:
loop

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

Proved the following rewrite lemma:
loop(gen_Cons:Nil4_0(+(1, n306_0)), gen_Cons:Nil4_0(n306_0), gen_Cons:Nil4_0(c), gen_Cons:Nil4_0(d)) → False, rt ∈ Ω(1 + n306₀)

Induction Base:
loop(gen_Cons:Nil4_0(+(1, 0)), gen_Cons:Nil4_0(0), gen_Cons:Nil4_0(c), gen_Cons:Nil4_0(d)) →_R^Ω(1)
False

Induction Step:
loop(gen_Cons:Nil4_0(+(1, +(n306_0, 1))), gen_Cons:Nil4_0(+(n306_0, 1)), gen_Cons:Nil4_0(c), gen_Cons:Nil4_0(d)) →_R^Ω(1)
loop[Ite](!EQ(0', 0'), Cons(0', gen_Cons:Nil4_0(+(1, n306_0))), Cons(0', gen_Cons:Nil4_0(n306_0)), gen_Cons:Nil4_0(c), gen_Cons:Nil4_0(d)) →_R^Ω(0)
loop[Ite](True, Cons(0', gen_Cons:Nil4_0(+(1, n306_0))), Cons(0', gen_Cons:Nil4_0(n306_0)), gen_Cons:Nil4_0(c), gen_Cons:Nil4_0(d)) →_R^Ω(0)
loop(gen_Cons:Nil4_0(+(1, n306_0)), gen_Cons:Nil4_0(n306_0), gen_Cons:Nil4_0(c), gen_Cons:Nil4_0(d)) →_IH
False

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
loop(Cons(x, xs), Nil, pp, ss) → False
loop(Cons(x', xs'), Cons(x, xs), pp, ss) → loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss)
loop(Nil, s, pp, ss) → True
match1(p, s) → loop(p, s, p, s)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
loop[Ite](False, p, s, pp, Cons(x, xs)) → loop(pp, xs, pp, xs)
loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) → loop(xs', xs, pp, ss)

Types:
loop :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil → False:True
Cons :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
False :: False:True
loop[Ite] :: False:True → Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil → False:True
!EQ :: S:0' → S:0' → False:True
True :: False:True
match1 :: Cons:Nil → Cons:Nil → False:True
S :: S:0' → S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)
loop(gen_Cons:Nil4_0(+(1, n306_0)), gen_Cons:Nil4_0(n306_0), gen_Cons:Nil4_0(c), gen_Cons:Nil4_0(d)) → False, rt ∈ Ω(1 + n306₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
loop(gen_Cons:Nil4_0(+(1, n306_0)), gen_Cons:Nil4_0(n306_0), gen_Cons:Nil4_0(c), gen_Cons:Nil4_0(d)) → False, rt ∈ Ω(1 + n306₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
loop(Cons(x, xs), Nil, pp, ss) → False
loop(Cons(x', xs'), Cons(x, xs), pp, ss) → loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss)
loop(Nil, s, pp, ss) → True
match1(p, s) → loop(p, s, p, s)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
loop[Ite](False, p, s, pp, Cons(x, xs)) → loop(pp, xs, pp, xs)
loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) → loop(xs', xs, pp, ss)

Types:
loop :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil → False:True
Cons :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
False :: False:True
loop[Ite] :: False:True → Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil → False:True
!EQ :: S:0' → S:0' → False:True
True :: False:True
match1 :: Cons:Nil → Cons:Nil → False:True
S :: S:0' → S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)
loop(gen_Cons:Nil4_0(+(1, n306_0)), gen_Cons:Nil4_0(n306_0), gen_Cons:Nil4_0(c), gen_Cons:Nil4_0(d)) → False, rt ∈ Ω(1 + n306₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
loop(gen_Cons:Nil4_0(+(1, n306_0)), gen_Cons:Nil4_0(n306_0), gen_Cons:Nil4_0(c), gen_Cons:Nil4_0(d)) → False, rt ∈ Ω(1 + n306₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
loop(Cons(x, xs), Nil, pp, ss) → False
loop(Cons(x', xs'), Cons(x, xs), pp, ss) → loop[Ite](!EQ(x', x), Cons(x', xs'), Cons(x, xs), pp, ss)
loop(Nil, s, pp, ss) → True
match1(p, s) → loop(p, s, p, s)
!EQ(S(x), S(y)) → !EQ(x, y)
!EQ(0', S(y)) → False
!EQ(S(x), 0') → False
!EQ(0', 0') → True
loop[Ite](False, p, s, pp, Cons(x, xs)) → loop(pp, xs, pp, xs)
loop[Ite](True, Cons(x', xs'), Cons(x, xs), pp, ss) → loop(xs', xs, pp, ss)

Types:
loop :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil → False:True
Cons :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
False :: False:True
loop[Ite] :: False:True → Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil → False:True
!EQ :: S:0' → S:0' → False:True
True :: False:True
match1 :: Cons:Nil → Cons:Nil → False:True
S :: S:0' → S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(1) was proven with the following lemma:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → False, rt ∈ Ω(0)

(22) BOUNDS(1, INF)