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:

bsort(nil) → nil
bsort(.(x, y)) → last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
bubble(nil) → nil
bubble(.(x, nil)) → .(x, nil)
bubble(.(x, .(y, z))) → if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
last(nil) → 0
last(.(x, nil)) → x
last(.(x, .(y, z))) → last(.(y, z))
butlast(nil) → nil
butlast(.(x, nil)) → nil
butlast(.(x, .(y, z))) → .(x, butlast(.(y, z)))

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:

bsort(nil) → nil
bsort(.(x, y)) → last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
bubble(nil) → nil
bubble(.(x, nil)) → .(x, nil)
bubble(.(x, .(y, z))) → if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
last(nil) → 0'
last(.(x, nil)) → x
last(.(x, .(y, z))) → last(.(y, z))
butlast(nil) → nil
butlast(.(x, nil)) → nil
butlast(.(x, .(y, z))) → .(x, butlast(.(y, z)))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
bsort(nil) → nil
bsort(.(x, y)) → last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
bubble(nil) → nil
bubble(.(x, nil)) → .(x, nil)
bubble(.(x, .(y, z))) → if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
last(nil) → 0'
last(.(x, nil)) → x
last(.(x, .(y, z))) → last(.(y, z))
butlast(nil) → nil
butlast(.(x, nil)) → nil
butlast(.(x, .(y, z))) → .(x, butlast(.(y, z)))

Types:
bsort :: nil:.:if:0' → nil:.:if:0'
nil :: nil:.:if:0'
. :: nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
last :: nil:.:if:0' → nil:.:if:0'
bubble :: nil:.:if:0' → nil:.:if:0'
butlast :: nil:.:if:0' → nil:.:if:0'
if :: <= → nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
<= :: nil:.:if:0' → nil:.:if:0' → <=
0' :: nil:.:if:0'
hole_nil:.:if:0'1_0 :: nil:.:if:0'
hole_<=2_0 :: <=
gen_nil:.:if:0'3_0 :: Nat → nil:.:if:0'

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

Heuristically decided to analyse the following defined symbols:
bsort, last, bubble, butlast

They will be analysed ascendingly in the following order:
last < bsort
bubble < bsort
butlast < bsort

(6) Obligation:

Innermost TRS:
Rules:
bsort(nil) → nil
bsort(.(x, y)) → last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
bubble(nil) → nil
bubble(.(x, nil)) → .(x, nil)
bubble(.(x, .(y, z))) → if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
last(nil) → 0'
last(.(x, nil)) → x
last(.(x, .(y, z))) → last(.(y, z))
butlast(nil) → nil
butlast(.(x, nil)) → nil
butlast(.(x, .(y, z))) → .(x, butlast(.(y, z)))

Types:
bsort :: nil:.:if:0' → nil:.:if:0'
nil :: nil:.:if:0'
. :: nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
last :: nil:.:if:0' → nil:.:if:0'
bubble :: nil:.:if:0' → nil:.:if:0'
butlast :: nil:.:if:0' → nil:.:if:0'
if :: <= → nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
<= :: nil:.:if:0' → nil:.:if:0' → <=
0' :: nil:.:if:0'
hole_nil:.:if:0'1_0 :: nil:.:if:0'
hole_<=2_0 :: <=
gen_nil:.:if:0'3_0 :: Nat → nil:.:if:0'

Generator Equations:
gen_nil:.:if:0'3_0(0) ⇔ nil
gen_nil:.:if:0'3_0(+(x, 1)) ⇔ .(nil, gen_nil:.:if:0'3_0(x))

The following defined symbols remain to be analysed:
last, bsort, bubble, butlast

They will be analysed ascendingly in the following order:
last < bsort
bubble < bsort
butlast < bsort

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

Proved the following rewrite lemma:
last(gen_nil:.:if:0'3_0(+(1, n5_0))) → gen_nil:.:if:0'3_0(0), rt ∈ Ω(1 + n5₀)

Induction Base:
last(gen_nil:.:if:0'3_0(+(1, 0))) →_R^Ω(1)
nil

Induction Step:
last(gen_nil:.:if:0'3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
last(.(nil, gen_nil:.:if:0'3_0(n5_0))) →_IH
gen_nil:.:if:0'3_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:
bsort(nil) → nil
bsort(.(x, y)) → last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
bubble(nil) → nil
bubble(.(x, nil)) → .(x, nil)
bubble(.(x, .(y, z))) → if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
last(nil) → 0'
last(.(x, nil)) → x
last(.(x, .(y, z))) → last(.(y, z))
butlast(nil) → nil
butlast(.(x, nil)) → nil
butlast(.(x, .(y, z))) → .(x, butlast(.(y, z)))

Types:
bsort :: nil:.:if:0' → nil:.:if:0'
nil :: nil:.:if:0'
. :: nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
last :: nil:.:if:0' → nil:.:if:0'
bubble :: nil:.:if:0' → nil:.:if:0'
butlast :: nil:.:if:0' → nil:.:if:0'
if :: <= → nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
<= :: nil:.:if:0' → nil:.:if:0' → <=
0' :: nil:.:if:0'
hole_nil:.:if:0'1_0 :: nil:.:if:0'
hole_<=2_0 :: <=
gen_nil:.:if:0'3_0 :: Nat → nil:.:if:0'

Lemmas:
last(gen_nil:.:if:0'3_0(+(1, n5_0))) → gen_nil:.:if:0'3_0(0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_nil:.:if:0'3_0(0) ⇔ nil
gen_nil:.:if:0'3_0(+(x, 1)) ⇔ .(nil, gen_nil:.:if:0'3_0(x))

The following defined symbols remain to be analysed:
bubble, bsort, butlast

They will be analysed ascendingly in the following order:
bubble < bsort
butlast < bsort

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
bsort(nil) → nil
bsort(.(x, y)) → last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
bubble(nil) → nil
bubble(.(x, nil)) → .(x, nil)
bubble(.(x, .(y, z))) → if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
last(nil) → 0'
last(.(x, nil)) → x
last(.(x, .(y, z))) → last(.(y, z))
butlast(nil) → nil
butlast(.(x, nil)) → nil
butlast(.(x, .(y, z))) → .(x, butlast(.(y, z)))

Types:
bsort :: nil:.:if:0' → nil:.:if:0'
nil :: nil:.:if:0'
. :: nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
last :: nil:.:if:0' → nil:.:if:0'
bubble :: nil:.:if:0' → nil:.:if:0'
butlast :: nil:.:if:0' → nil:.:if:0'
if :: <= → nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
<= :: nil:.:if:0' → nil:.:if:0' → <=
0' :: nil:.:if:0'
hole_nil:.:if:0'1_0 :: nil:.:if:0'
hole_<=2_0 :: <=
gen_nil:.:if:0'3_0 :: Nat → nil:.:if:0'

Lemmas:
last(gen_nil:.:if:0'3_0(+(1, n5_0))) → gen_nil:.:if:0'3_0(0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_nil:.:if:0'3_0(0) ⇔ nil
gen_nil:.:if:0'3_0(+(x, 1)) ⇔ .(nil, gen_nil:.:if:0'3_0(x))

The following defined symbols remain to be analysed:
butlast, bsort

They will be analysed ascendingly in the following order:
butlast < bsort

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

Proved the following rewrite lemma:
butlast(gen_nil:.:if:0'3_0(+(1, n15924_0))) → gen_nil:.:if:0'3_0(n15924_0), rt ∈ Ω(1 + n15924₀)

Induction Base:
butlast(gen_nil:.:if:0'3_0(+(1, 0))) →_R^Ω(1)
nil

Induction Step:
butlast(gen_nil:.:if:0'3_0(+(1, +(n15924_0, 1)))) →_R^Ω(1)
.(nil, butlast(.(nil, gen_nil:.:if:0'3_0(n15924_0)))) →_IH
.(nil, gen_nil:.:if:0'3_0(c15925_0))

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:
bsort(nil) → nil
bsort(.(x, y)) → last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
bubble(nil) → nil
bubble(.(x, nil)) → .(x, nil)
bubble(.(x, .(y, z))) → if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
last(nil) → 0'
last(.(x, nil)) → x
last(.(x, .(y, z))) → last(.(y, z))
butlast(nil) → nil
butlast(.(x, nil)) → nil
butlast(.(x, .(y, z))) → .(x, butlast(.(y, z)))

Types:
bsort :: nil:.:if:0' → nil:.:if:0'
nil :: nil:.:if:0'
. :: nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
last :: nil:.:if:0' → nil:.:if:0'
bubble :: nil:.:if:0' → nil:.:if:0'
butlast :: nil:.:if:0' → nil:.:if:0'
if :: <= → nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
<= :: nil:.:if:0' → nil:.:if:0' → <=
0' :: nil:.:if:0'
hole_nil:.:if:0'1_0 :: nil:.:if:0'
hole_<=2_0 :: <=
gen_nil:.:if:0'3_0 :: Nat → nil:.:if:0'

Lemmas:
last(gen_nil:.:if:0'3_0(+(1, n5_0))) → gen_nil:.:if:0'3_0(0), rt ∈ Ω(1 + n5₀)
butlast(gen_nil:.:if:0'3_0(+(1, n15924_0))) → gen_nil:.:if:0'3_0(n15924_0), rt ∈ Ω(1 + n15924₀)

Generator Equations:
gen_nil:.:if:0'3_0(0) ⇔ nil
gen_nil:.:if:0'3_0(+(x, 1)) ⇔ .(nil, gen_nil:.:if:0'3_0(x))

The following defined symbols remain to be analysed:
bsort

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

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

(16) Obligation:

Innermost TRS:
Rules:
bsort(nil) → nil
bsort(.(x, y)) → last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
bubble(nil) → nil
bubble(.(x, nil)) → .(x, nil)
bubble(.(x, .(y, z))) → if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
last(nil) → 0'
last(.(x, nil)) → x
last(.(x, .(y, z))) → last(.(y, z))
butlast(nil) → nil
butlast(.(x, nil)) → nil
butlast(.(x, .(y, z))) → .(x, butlast(.(y, z)))

Types:
bsort :: nil:.:if:0' → nil:.:if:0'
nil :: nil:.:if:0'
. :: nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
last :: nil:.:if:0' → nil:.:if:0'
bubble :: nil:.:if:0' → nil:.:if:0'
butlast :: nil:.:if:0' → nil:.:if:0'
if :: <= → nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
<= :: nil:.:if:0' → nil:.:if:0' → <=
0' :: nil:.:if:0'
hole_nil:.:if:0'1_0 :: nil:.:if:0'
hole_<=2_0 :: <=
gen_nil:.:if:0'3_0 :: Nat → nil:.:if:0'

Lemmas:
last(gen_nil:.:if:0'3_0(+(1, n5_0))) → gen_nil:.:if:0'3_0(0), rt ∈ Ω(1 + n5₀)
butlast(gen_nil:.:if:0'3_0(+(1, n15924_0))) → gen_nil:.:if:0'3_0(n15924_0), rt ∈ Ω(1 + n15924₀)

Generator Equations:
gen_nil:.:if:0'3_0(0) ⇔ nil
gen_nil:.:if:0'3_0(+(x, 1)) ⇔ .(nil, gen_nil:.:if:0'3_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
last(gen_nil:.:if:0'3_0(+(1, n5_0))) → gen_nil:.:if:0'3_0(0), rt ∈ Ω(1 + n5₀)

(18) BOUNDS(n^1, INF)

(19) Obligation:

Innermost TRS:
Rules:
bsort(nil) → nil
bsort(.(x, y)) → last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
bubble(nil) → nil
bubble(.(x, nil)) → .(x, nil)
bubble(.(x, .(y, z))) → if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
last(nil) → 0'
last(.(x, nil)) → x
last(.(x, .(y, z))) → last(.(y, z))
butlast(nil) → nil
butlast(.(x, nil)) → nil
butlast(.(x, .(y, z))) → .(x, butlast(.(y, z)))

Types:
bsort :: nil:.:if:0' → nil:.:if:0'
nil :: nil:.:if:0'
. :: nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
last :: nil:.:if:0' → nil:.:if:0'
bubble :: nil:.:if:0' → nil:.:if:0'
butlast :: nil:.:if:0' → nil:.:if:0'
if :: <= → nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
<= :: nil:.:if:0' → nil:.:if:0' → <=
0' :: nil:.:if:0'
hole_nil:.:if:0'1_0 :: nil:.:if:0'
hole_<=2_0 :: <=
gen_nil:.:if:0'3_0 :: Nat → nil:.:if:0'

Lemmas:
last(gen_nil:.:if:0'3_0(+(1, n5_0))) → gen_nil:.:if:0'3_0(0), rt ∈ Ω(1 + n5₀)
butlast(gen_nil:.:if:0'3_0(+(1, n15924_0))) → gen_nil:.:if:0'3_0(n15924_0), rt ∈ Ω(1 + n15924₀)

Generator Equations:
gen_nil:.:if:0'3_0(0) ⇔ nil
gen_nil:.:if:0'3_0(+(x, 1)) ⇔ .(nil, gen_nil:.:if:0'3_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
last(gen_nil:.:if:0'3_0(+(1, n5_0))) → gen_nil:.:if:0'3_0(0), rt ∈ Ω(1 + n5₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

Innermost TRS:
Rules:
bsort(nil) → nil
bsort(.(x, y)) → last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
bubble(nil) → nil
bubble(.(x, nil)) → .(x, nil)
bubble(.(x, .(y, z))) → if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
last(nil) → 0'
last(.(x, nil)) → x
last(.(x, .(y, z))) → last(.(y, z))
butlast(nil) → nil
butlast(.(x, nil)) → nil
butlast(.(x, .(y, z))) → .(x, butlast(.(y, z)))

Types:
bsort :: nil:.:if:0' → nil:.:if:0'
nil :: nil:.:if:0'
. :: nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
last :: nil:.:if:0' → nil:.:if:0'
bubble :: nil:.:if:0' → nil:.:if:0'
butlast :: nil:.:if:0' → nil:.:if:0'
if :: <= → nil:.:if:0' → nil:.:if:0' → nil:.:if:0'
<= :: nil:.:if:0' → nil:.:if:0' → <=
0' :: nil:.:if:0'
hole_nil:.:if:0'1_0 :: nil:.:if:0'
hole_<=2_0 :: <=
gen_nil:.:if:0'3_0 :: Nat → nil:.:if:0'

Lemmas:
last(gen_nil:.:if:0'3_0(+(1, n5_0))) → gen_nil:.:if:0'3_0(0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_nil:.:if:0'3_0(0) ⇔ nil
gen_nil:.:if:0'3_0(+(x, 1)) ⇔ .(nil, gen_nil:.:if:0'3_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
last(gen_nil:.:if:0'3_0(+(1, n5_0))) → gen_nil:.:if:0'3_0(0), rt ∈ Ω(1 + n5₀)

(24) BOUNDS(n^1, INF)