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:

*(@x, @y) → #mult(@x, @y)
dyade(@l1, @l2) → dyade#1(@l1, @l2)
dyade#1(::(@x, @xs), @l2) → ::(mult(@x, @l2), dyade(@xs, @l2))
dyade#1(nil, @l2) → nil
mult(@n, @l) → mult#1(@l, @n)
mult#1(::(@x, @xs), @n) → ::(*(@n, @x), mult(@n, @xs))
mult#1(nil, @n) → nil

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

#add(#0, @y) → @y
#add(#neg(#s(#0)), @y) → #pred(@y)
#add(#neg(#s(#s(@x))), @y) → #pred(#add(#pos(#s(@x)), @y))
#add(#pos(#s(#0)), @y) → #succ(@y)
#add(#pos(#s(#s(@x))), @y) → #succ(#add(#pos(#s(@x)), @y))
#mult(#0, #0) → #0
#mult(#0, #neg(@y)) → #0
#mult(#0, #pos(@y)) → #0
#mult(#neg(@x), #0) → #0
#mult(#neg(@x), #neg(@y)) → #pos(#natmult(@x, @y))
#mult(#neg(@x), #pos(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #0) → #0
#mult(#pos(@x), #neg(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #pos(@y)) → #pos(#natmult(@x, @y))
#natmult(#0, @y) → #0
#natmult(#s(@x), @y) → #add(#pos(@y), #natmult(@x, @y))
#pred(#0) → #neg(#s(#0))
#pred(#neg(#s(@x))) → #neg(#s(#s(@x)))
#pred(#pos(#s(#0))) → #0
#pred(#pos(#s(#s(@x)))) → #pos(#s(@x))
#succ(#0) → #pos(#s(#0))
#succ(#neg(#s(#0))) → #0
#succ(#neg(#s(#s(@x)))) → #neg(#s(@x))
#succ(#pos(#s(@x))) → #pos(#s(#s(@x)))

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:

*'(@x, @y) → #mult(@x, @y)
dyade(@l1, @l2) → dyade#1(@l1, @l2)
dyade#1(::(@x, @xs), @l2) → ::(mult(@x, @l2), dyade(@xs, @l2))
dyade#1(nil, @l2) → nil
mult(@n, @l) → mult#1(@l, @n)
mult#1(::(@x, @xs), @n) → ::(*'(@n, @x), mult(@n, @xs))
mult#1(nil, @n) → nil

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

#add(#0, @y) → @y
#add(#neg(#s(#0)), @y) → #pred(@y)
#add(#neg(#s(#s(@x))), @y) → #pred(#add(#pos(#s(@x)), @y))
#add(#pos(#s(#0)), @y) → #succ(@y)
#add(#pos(#s(#s(@x))), @y) → #succ(#add(#pos(#s(@x)), @y))
#mult(#0, #0) → #0
#mult(#0, #neg(@y)) → #0
#mult(#0, #pos(@y)) → #0
#mult(#neg(@x), #0) → #0
#mult(#neg(@x), #neg(@y)) → #pos(#natmult(@x, @y))
#mult(#neg(@x), #pos(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #0) → #0
#mult(#pos(@x), #neg(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #pos(@y)) → #pos(#natmult(@x, @y))
#natmult(#0, @y) → #0
#natmult(#s(@x), @y) → #add(#pos(@y), #natmult(@x, @y))
#pred(#0) → #neg(#s(#0))
#pred(#neg(#s(@x))) → #neg(#s(#s(@x)))
#pred(#pos(#s(#0))) → #0
#pred(#pos(#s(#s(@x)))) → #pos(#s(@x))
#succ(#0) → #pos(#s(#0))
#succ(#neg(#s(#0))) → #0
#succ(#neg(#s(#s(@x)))) → #neg(#s(@x))
#succ(#pos(#s(@x))) → #pos(#s(#s(@x)))

Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
dyade(@l1, @l2) → dyade#1(@l1, @l2)
dyade#1(::(@x, @xs), @l2) → ::(mult(@x, @l2), dyade(@xs, @l2))
dyade#1(nil, @l2) → nil
mult(@n, @l) → mult#1(@l, @n)
mult#1(::(@x, @xs), @n) → ::(*'(@n, @x), mult(@n, @xs))
mult#1(nil, @n) → nil
#add(#0, @y) → @y
#add(#neg(#s(#0)), @y) → #pred(@y)
#add(#neg(#s(#s(@x))), @y) → #pred(#add(#pos(#s(@x)), @y))
#add(#pos(#s(#0)), @y) → #succ(@y)
#add(#pos(#s(#s(@x))), @y) → #succ(#add(#pos(#s(@x)), @y))
#mult(#0, #0) → #0
#mult(#0, #neg(@y)) → #0
#mult(#0, #pos(@y)) → #0
#mult(#neg(@x), #0) → #0
#mult(#neg(@x), #neg(@y)) → #pos(#natmult(@x, @y))
#mult(#neg(@x), #pos(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #0) → #0
#mult(#pos(@x), #neg(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #pos(@y)) → #pos(#natmult(@x, @y))
#natmult(#0, @y) → #0
#natmult(#s(@x), @y) → #add(#pos(@y), #natmult(@x, @y))
#pred(#0) → #neg(#s(#0))
#pred(#neg(#s(@x))) → #neg(#s(#s(@x)))
#pred(#pos(#s(#0))) → #0
#pred(#pos(#s(#s(@x)))) → #pos(#s(@x))
#succ(#0) → #pos(#s(#0))
#succ(#neg(#s(#0))) → #0
#succ(#neg(#s(#s(@x)))) → #neg(#s(@x))
#succ(#pos(#s(@x))) → #pos(#s(#s(@x)))

Types:
*' :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
:: :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
nil :: :::nil:#0:#s:#neg:#pos
mult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#add :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#0 :: :::nil:#0:#s:#neg:#pos
#neg :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#s :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pred :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pos :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#succ :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#natmult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
hole_:::nil:#0:#s:#neg:#pos1_3 :: :::nil:#0:#s:#neg:#pos
gen_:::nil:#0:#s:#neg:#pos2_3 :: Nat → :::nil:#0:#s:#neg:#pos

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

Heuristically decided to analyse the following defined symbols:
dyade, dyade#1, mult, mult#1, #add, #natmult

They will be analysed ascendingly in the following order:
dyade = dyade#1
mult < dyade#1
mult = mult#1
#add < #natmult

(6) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
dyade(@l1, @l2) → dyade#1(@l1, @l2)
dyade#1(::(@x, @xs), @l2) → ::(mult(@x, @l2), dyade(@xs, @l2))
dyade#1(nil, @l2) → nil
mult(@n, @l) → mult#1(@l, @n)
mult#1(::(@x, @xs), @n) → ::(*'(@n, @x), mult(@n, @xs))
mult#1(nil, @n) → nil
#add(#0, @y) → @y
#add(#neg(#s(#0)), @y) → #pred(@y)
#add(#neg(#s(#s(@x))), @y) → #pred(#add(#pos(#s(@x)), @y))
#add(#pos(#s(#0)), @y) → #succ(@y)
#add(#pos(#s(#s(@x))), @y) → #succ(#add(#pos(#s(@x)), @y))
#mult(#0, #0) → #0
#mult(#0, #neg(@y)) → #0
#mult(#0, #pos(@y)) → #0
#mult(#neg(@x), #0) → #0
#mult(#neg(@x), #neg(@y)) → #pos(#natmult(@x, @y))
#mult(#neg(@x), #pos(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #0) → #0
#mult(#pos(@x), #neg(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #pos(@y)) → #pos(#natmult(@x, @y))
#natmult(#0, @y) → #0
#natmult(#s(@x), @y) → #add(#pos(@y), #natmult(@x, @y))
#pred(#0) → #neg(#s(#0))
#pred(#neg(#s(@x))) → #neg(#s(#s(@x)))
#pred(#pos(#s(#0))) → #0
#pred(#pos(#s(#s(@x)))) → #pos(#s(@x))
#succ(#0) → #pos(#s(#0))
#succ(#neg(#s(#0))) → #0
#succ(#neg(#s(#s(@x)))) → #neg(#s(@x))
#succ(#pos(#s(@x))) → #pos(#s(#s(@x)))

Types:
*' :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
:: :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
nil :: :::nil:#0:#s:#neg:#pos
mult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#add :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#0 :: :::nil:#0:#s:#neg:#pos
#neg :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#s :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pred :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pos :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#succ :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#natmult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
hole_:::nil:#0:#s:#neg:#pos1_3 :: :::nil:#0:#s:#neg:#pos
gen_:::nil:#0:#s:#neg:#pos2_3 :: Nat → :::nil:#0:#s:#neg:#pos

Generator Equations:
gen_:::nil:#0:#s:#neg:#pos2_3(0) ⇔ nil
gen_:::nil:#0:#s:#neg:#pos2_3(+(x, 1)) ⇔ ::(nil, gen_:::nil:#0:#s:#neg:#pos2_3(x))

The following defined symbols remain to be analysed:
#add, dyade, dyade#1, mult, mult#1, #natmult

They will be analysed ascendingly in the following order:
dyade = dyade#1
mult < dyade#1
mult = mult#1
#add < #natmult

(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol #add.

(8) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
dyade(@l1, @l2) → dyade#1(@l1, @l2)
dyade#1(::(@x, @xs), @l2) → ::(mult(@x, @l2), dyade(@xs, @l2))
dyade#1(nil, @l2) → nil
mult(@n, @l) → mult#1(@l, @n)
mult#1(::(@x, @xs), @n) → ::(*'(@n, @x), mult(@n, @xs))
mult#1(nil, @n) → nil
#add(#0, @y) → @y
#add(#neg(#s(#0)), @y) → #pred(@y)
#add(#neg(#s(#s(@x))), @y) → #pred(#add(#pos(#s(@x)), @y))
#add(#pos(#s(#0)), @y) → #succ(@y)
#add(#pos(#s(#s(@x))), @y) → #succ(#add(#pos(#s(@x)), @y))
#mult(#0, #0) → #0
#mult(#0, #neg(@y)) → #0
#mult(#0, #pos(@y)) → #0
#mult(#neg(@x), #0) → #0
#mult(#neg(@x), #neg(@y)) → #pos(#natmult(@x, @y))
#mult(#neg(@x), #pos(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #0) → #0
#mult(#pos(@x), #neg(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #pos(@y)) → #pos(#natmult(@x, @y))
#natmult(#0, @y) → #0
#natmult(#s(@x), @y) → #add(#pos(@y), #natmult(@x, @y))
#pred(#0) → #neg(#s(#0))
#pred(#neg(#s(@x))) → #neg(#s(#s(@x)))
#pred(#pos(#s(#0))) → #0
#pred(#pos(#s(#s(@x)))) → #pos(#s(@x))
#succ(#0) → #pos(#s(#0))
#succ(#neg(#s(#0))) → #0
#succ(#neg(#s(#s(@x)))) → #neg(#s(@x))
#succ(#pos(#s(@x))) → #pos(#s(#s(@x)))

Types:
*' :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
:: :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
nil :: :::nil:#0:#s:#neg:#pos
mult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#add :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#0 :: :::nil:#0:#s:#neg:#pos
#neg :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#s :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pred :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pos :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#succ :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#natmult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
hole_:::nil:#0:#s:#neg:#pos1_3 :: :::nil:#0:#s:#neg:#pos
gen_:::nil:#0:#s:#neg:#pos2_3 :: Nat → :::nil:#0:#s:#neg:#pos

Generator Equations:
gen_:::nil:#0:#s:#neg:#pos2_3(0) ⇔ nil
gen_:::nil:#0:#s:#neg:#pos2_3(+(x, 1)) ⇔ ::(nil, gen_:::nil:#0:#s:#neg:#pos2_3(x))

The following defined symbols remain to be analysed:
#natmult, dyade, dyade#1, mult, mult#1

They will be analysed ascendingly in the following order:
dyade = dyade#1
mult < dyade#1
mult = mult#1

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol #natmult.

(10) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
dyade(@l1, @l2) → dyade#1(@l1, @l2)
dyade#1(::(@x, @xs), @l2) → ::(mult(@x, @l2), dyade(@xs, @l2))
dyade#1(nil, @l2) → nil
mult(@n, @l) → mult#1(@l, @n)
mult#1(::(@x, @xs), @n) → ::(*'(@n, @x), mult(@n, @xs))
mult#1(nil, @n) → nil
#add(#0, @y) → @y
#add(#neg(#s(#0)), @y) → #pred(@y)
#add(#neg(#s(#s(@x))), @y) → #pred(#add(#pos(#s(@x)), @y))
#add(#pos(#s(#0)), @y) → #succ(@y)
#add(#pos(#s(#s(@x))), @y) → #succ(#add(#pos(#s(@x)), @y))
#mult(#0, #0) → #0
#mult(#0, #neg(@y)) → #0
#mult(#0, #pos(@y)) → #0
#mult(#neg(@x), #0) → #0
#mult(#neg(@x), #neg(@y)) → #pos(#natmult(@x, @y))
#mult(#neg(@x), #pos(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #0) → #0
#mult(#pos(@x), #neg(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #pos(@y)) → #pos(#natmult(@x, @y))
#natmult(#0, @y) → #0
#natmult(#s(@x), @y) → #add(#pos(@y), #natmult(@x, @y))
#pred(#0) → #neg(#s(#0))
#pred(#neg(#s(@x))) → #neg(#s(#s(@x)))
#pred(#pos(#s(#0))) → #0
#pred(#pos(#s(#s(@x)))) → #pos(#s(@x))
#succ(#0) → #pos(#s(#0))
#succ(#neg(#s(#0))) → #0
#succ(#neg(#s(#s(@x)))) → #neg(#s(@x))
#succ(#pos(#s(@x))) → #pos(#s(#s(@x)))

Types:
*' :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
:: :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
nil :: :::nil:#0:#s:#neg:#pos
mult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#add :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#0 :: :::nil:#0:#s:#neg:#pos
#neg :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#s :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pred :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pos :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#succ :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#natmult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
hole_:::nil:#0:#s:#neg:#pos1_3 :: :::nil:#0:#s:#neg:#pos
gen_:::nil:#0:#s:#neg:#pos2_3 :: Nat → :::nil:#0:#s:#neg:#pos

Generator Equations:
gen_:::nil:#0:#s:#neg:#pos2_3(0) ⇔ nil
gen_:::nil:#0:#s:#neg:#pos2_3(+(x, 1)) ⇔ ::(nil, gen_:::nil:#0:#s:#neg:#pos2_3(x))

The following defined symbols remain to be analysed:
mult#1, dyade, dyade#1, mult

They will be analysed ascendingly in the following order:
dyade = dyade#1
mult < dyade#1
mult = mult#1

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n112₃)

Induction Base:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, 0)), gen_:::nil:#0:#s:#neg:#pos2_3(b))

Induction Step:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, +(n112_3, 1))), gen_:::nil:#0:#s:#neg:#pos2_3(b)) →_R^Ω(1)
::(*'(gen_:::nil:#0:#s:#neg:#pos2_3(b), nil), mult(gen_:::nil:#0:#s:#neg:#pos2_3(b), gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)))) →_R^Ω(1)
::(#mult(gen_:::nil:#0:#s:#neg:#pos2_3(b), nil), mult(gen_:::nil:#0:#s:#neg:#pos2_3(b), gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)))) →_R^Ω(1)
::(#mult(gen_:::nil:#0:#s:#neg:#pos2_3(b), nil), mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b))) →_IH
::(#mult(gen_:::nil:#0:#s:#neg:#pos2_3(b), nil), *3_3)

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

(12) Complex Obligation (BEST)

(13) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
dyade(@l1, @l2) → dyade#1(@l1, @l2)
dyade#1(::(@x, @xs), @l2) → ::(mult(@x, @l2), dyade(@xs, @l2))
dyade#1(nil, @l2) → nil
mult(@n, @l) → mult#1(@l, @n)
mult#1(::(@x, @xs), @n) → ::(*'(@n, @x), mult(@n, @xs))
mult#1(nil, @n) → nil
#add(#0, @y) → @y
#add(#neg(#s(#0)), @y) → #pred(@y)
#add(#neg(#s(#s(@x))), @y) → #pred(#add(#pos(#s(@x)), @y))
#add(#pos(#s(#0)), @y) → #succ(@y)
#add(#pos(#s(#s(@x))), @y) → #succ(#add(#pos(#s(@x)), @y))
#mult(#0, #0) → #0
#mult(#0, #neg(@y)) → #0
#mult(#0, #pos(@y)) → #0
#mult(#neg(@x), #0) → #0
#mult(#neg(@x), #neg(@y)) → #pos(#natmult(@x, @y))
#mult(#neg(@x), #pos(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #0) → #0
#mult(#pos(@x), #neg(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #pos(@y)) → #pos(#natmult(@x, @y))
#natmult(#0, @y) → #0
#natmult(#s(@x), @y) → #add(#pos(@y), #natmult(@x, @y))
#pred(#0) → #neg(#s(#0))
#pred(#neg(#s(@x))) → #neg(#s(#s(@x)))
#pred(#pos(#s(#0))) → #0
#pred(#pos(#s(#s(@x)))) → #pos(#s(@x))
#succ(#0) → #pos(#s(#0))
#succ(#neg(#s(#0))) → #0
#succ(#neg(#s(#s(@x)))) → #neg(#s(@x))
#succ(#pos(#s(@x))) → #pos(#s(#s(@x)))

Types:
*' :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
:: :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
nil :: :::nil:#0:#s:#neg:#pos
mult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#add :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#0 :: :::nil:#0:#s:#neg:#pos
#neg :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#s :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pred :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pos :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#succ :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#natmult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
hole_:::nil:#0:#s:#neg:#pos1_3 :: :::nil:#0:#s:#neg:#pos
gen_:::nil:#0:#s:#neg:#pos2_3 :: Nat → :::nil:#0:#s:#neg:#pos

Lemmas:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n112₃)

Generator Equations:
gen_:::nil:#0:#s:#neg:#pos2_3(0) ⇔ nil
gen_:::nil:#0:#s:#neg:#pos2_3(+(x, 1)) ⇔ ::(nil, gen_:::nil:#0:#s:#neg:#pos2_3(x))

The following defined symbols remain to be analysed:
mult, dyade, dyade#1

They will be analysed ascendingly in the following order:
dyade = dyade#1
mult < dyade#1
mult = mult#1

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n1814224_3)) → *3_3, rt ∈ Ω(n1814224₃)

Induction Base:
mult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(0))

Induction Step:
mult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(+(n1814224_3, 1))) →_R^Ω(1)
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(n1814224_3, 1)), gen_:::nil:#0:#s:#neg:#pos2_3(a)) →_R^Ω(1)
::(*'(gen_:::nil:#0:#s:#neg:#pos2_3(a), nil), mult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n1814224_3))) →_R^Ω(1)
::(#mult(gen_:::nil:#0:#s:#neg:#pos2_3(a), nil), mult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n1814224_3))) →_IH
::(#mult(gen_:::nil:#0:#s:#neg:#pos2_3(a), nil), *3_3)

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

(15) Complex Obligation (BEST)

(16) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
dyade(@l1, @l2) → dyade#1(@l1, @l2)
dyade#1(::(@x, @xs), @l2) → ::(mult(@x, @l2), dyade(@xs, @l2))
dyade#1(nil, @l2) → nil
mult(@n, @l) → mult#1(@l, @n)
mult#1(::(@x, @xs), @n) → ::(*'(@n, @x), mult(@n, @xs))
mult#1(nil, @n) → nil
#add(#0, @y) → @y
#add(#neg(#s(#0)), @y) → #pred(@y)
#add(#neg(#s(#s(@x))), @y) → #pred(#add(#pos(#s(@x)), @y))
#add(#pos(#s(#0)), @y) → #succ(@y)
#add(#pos(#s(#s(@x))), @y) → #succ(#add(#pos(#s(@x)), @y))
#mult(#0, #0) → #0
#mult(#0, #neg(@y)) → #0
#mult(#0, #pos(@y)) → #0
#mult(#neg(@x), #0) → #0
#mult(#neg(@x), #neg(@y)) → #pos(#natmult(@x, @y))
#mult(#neg(@x), #pos(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #0) → #0
#mult(#pos(@x), #neg(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #pos(@y)) → #pos(#natmult(@x, @y))
#natmult(#0, @y) → #0
#natmult(#s(@x), @y) → #add(#pos(@y), #natmult(@x, @y))
#pred(#0) → #neg(#s(#0))
#pred(#neg(#s(@x))) → #neg(#s(#s(@x)))
#pred(#pos(#s(#0))) → #0
#pred(#pos(#s(#s(@x)))) → #pos(#s(@x))
#succ(#0) → #pos(#s(#0))
#succ(#neg(#s(#0))) → #0
#succ(#neg(#s(#s(@x)))) → #neg(#s(@x))
#succ(#pos(#s(@x))) → #pos(#s(#s(@x)))

Types:
*' :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
:: :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
nil :: :::nil:#0:#s:#neg:#pos
mult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#add :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#0 :: :::nil:#0:#s:#neg:#pos
#neg :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#s :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pred :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pos :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#succ :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#natmult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
hole_:::nil:#0:#s:#neg:#pos1_3 :: :::nil:#0:#s:#neg:#pos
gen_:::nil:#0:#s:#neg:#pos2_3 :: Nat → :::nil:#0:#s:#neg:#pos

Lemmas:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n112₃)
mult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n1814224_3)) → *3_3, rt ∈ Ω(n1814224₃)

Generator Equations:
gen_:::nil:#0:#s:#neg:#pos2_3(0) ⇔ nil
gen_:::nil:#0:#s:#neg:#pos2_3(+(x, 1)) ⇔ ::(nil, gen_:::nil:#0:#s:#neg:#pos2_3(x))

The following defined symbols remain to be analysed:
mult#1, dyade, dyade#1

They will be analysed ascendingly in the following order:
dyade = dyade#1
mult < dyade#1
mult = mult#1

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

Proved the following rewrite lemma:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n3572542_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n3572542₃)

Induction Base:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, 0)), gen_:::nil:#0:#s:#neg:#pos2_3(b))

Induction Step:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, +(n3572542_3, 1))), gen_:::nil:#0:#s:#neg:#pos2_3(b)) →_R^Ω(1)
::(*'(gen_:::nil:#0:#s:#neg:#pos2_3(b), nil), mult(gen_:::nil:#0:#s:#neg:#pos2_3(b), gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n3572542_3)))) →_R^Ω(1)
::(#mult(gen_:::nil:#0:#s:#neg:#pos2_3(b), nil), mult(gen_:::nil:#0:#s:#neg:#pos2_3(b), gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n3572542_3)))) →_R^Ω(1)
::(#mult(gen_:::nil:#0:#s:#neg:#pos2_3(b), nil), mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n3572542_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b))) →_IH
::(#mult(gen_:::nil:#0:#s:#neg:#pos2_3(b), nil), *3_3)

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

(18) Complex Obligation (BEST)

(19) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
dyade(@l1, @l2) → dyade#1(@l1, @l2)
dyade#1(::(@x, @xs), @l2) → ::(mult(@x, @l2), dyade(@xs, @l2))
dyade#1(nil, @l2) → nil
mult(@n, @l) → mult#1(@l, @n)
mult#1(::(@x, @xs), @n) → ::(*'(@n, @x), mult(@n, @xs))
mult#1(nil, @n) → nil
#add(#0, @y) → @y
#add(#neg(#s(#0)), @y) → #pred(@y)
#add(#neg(#s(#s(@x))), @y) → #pred(#add(#pos(#s(@x)), @y))
#add(#pos(#s(#0)), @y) → #succ(@y)
#add(#pos(#s(#s(@x))), @y) → #succ(#add(#pos(#s(@x)), @y))
#mult(#0, #0) → #0
#mult(#0, #neg(@y)) → #0
#mult(#0, #pos(@y)) → #0
#mult(#neg(@x), #0) → #0
#mult(#neg(@x), #neg(@y)) → #pos(#natmult(@x, @y))
#mult(#neg(@x), #pos(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #0) → #0
#mult(#pos(@x), #neg(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #pos(@y)) → #pos(#natmult(@x, @y))
#natmult(#0, @y) → #0
#natmult(#s(@x), @y) → #add(#pos(@y), #natmult(@x, @y))
#pred(#0) → #neg(#s(#0))
#pred(#neg(#s(@x))) → #neg(#s(#s(@x)))
#pred(#pos(#s(#0))) → #0
#pred(#pos(#s(#s(@x)))) → #pos(#s(@x))
#succ(#0) → #pos(#s(#0))
#succ(#neg(#s(#0))) → #0
#succ(#neg(#s(#s(@x)))) → #neg(#s(@x))
#succ(#pos(#s(@x))) → #pos(#s(#s(@x)))

Types:
*' :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
:: :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
nil :: :::nil:#0:#s:#neg:#pos
mult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#add :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#0 :: :::nil:#0:#s:#neg:#pos
#neg :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#s :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pred :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pos :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#succ :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#natmult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
hole_:::nil:#0:#s:#neg:#pos1_3 :: :::nil:#0:#s:#neg:#pos
gen_:::nil:#0:#s:#neg:#pos2_3 :: Nat → :::nil:#0:#s:#neg:#pos

Lemmas:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n3572542_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n3572542₃)
mult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n1814224_3)) → *3_3, rt ∈ Ω(n1814224₃)

Generator Equations:
gen_:::nil:#0:#s:#neg:#pos2_3(0) ⇔ nil
gen_:::nil:#0:#s:#neg:#pos2_3(+(x, 1)) ⇔ ::(nil, gen_:::nil:#0:#s:#neg:#pos2_3(x))

The following defined symbols remain to be analysed:
dyade#1, dyade

They will be analysed ascendingly in the following order:
dyade = dyade#1

(20) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
dyade#1(gen_:::nil:#0:#s:#neg:#pos2_3(n5395010_3), gen_:::nil:#0:#s:#neg:#pos2_3(0)) → gen_:::nil:#0:#s:#neg:#pos2_3(n5395010_3), rt ∈ Ω(1 + n5395010₃)

Induction Base:
dyade#1(gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(0)) →_R^Ω(1)
nil

Induction Step:
dyade#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(n5395010_3, 1)), gen_:::nil:#0:#s:#neg:#pos2_3(0)) →_R^Ω(1)
::(mult(nil, gen_:::nil:#0:#s:#neg:#pos2_3(0)), dyade(gen_:::nil:#0:#s:#neg:#pos2_3(n5395010_3), gen_:::nil:#0:#s:#neg:#pos2_3(0))) →_R^Ω(1)
::(mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(0), nil), dyade(gen_:::nil:#0:#s:#neg:#pos2_3(n5395010_3), gen_:::nil:#0:#s:#neg:#pos2_3(0))) →_R^Ω(1)
::(nil, dyade(gen_:::nil:#0:#s:#neg:#pos2_3(n5395010_3), gen_:::nil:#0:#s:#neg:#pos2_3(0))) →_R^Ω(1)
::(nil, dyade#1(gen_:::nil:#0:#s:#neg:#pos2_3(n5395010_3), gen_:::nil:#0:#s:#neg:#pos2_3(0))) →_IH
::(nil, gen_:::nil:#0:#s:#neg:#pos2_3(c5395011_3))

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

(21) Complex Obligation (BEST)

(22) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
dyade(@l1, @l2) → dyade#1(@l1, @l2)
dyade#1(::(@x, @xs), @l2) → ::(mult(@x, @l2), dyade(@xs, @l2))
dyade#1(nil, @l2) → nil
mult(@n, @l) → mult#1(@l, @n)
mult#1(::(@x, @xs), @n) → ::(*'(@n, @x), mult(@n, @xs))
mult#1(nil, @n) → nil
#add(#0, @y) → @y
#add(#neg(#s(#0)), @y) → #pred(@y)
#add(#neg(#s(#s(@x))), @y) → #pred(#add(#pos(#s(@x)), @y))
#add(#pos(#s(#0)), @y) → #succ(@y)
#add(#pos(#s(#s(@x))), @y) → #succ(#add(#pos(#s(@x)), @y))
#mult(#0, #0) → #0
#mult(#0, #neg(@y)) → #0
#mult(#0, #pos(@y)) → #0
#mult(#neg(@x), #0) → #0
#mult(#neg(@x), #neg(@y)) → #pos(#natmult(@x, @y))
#mult(#neg(@x), #pos(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #0) → #0
#mult(#pos(@x), #neg(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #pos(@y)) → #pos(#natmult(@x, @y))
#natmult(#0, @y) → #0
#natmult(#s(@x), @y) → #add(#pos(@y), #natmult(@x, @y))
#pred(#0) → #neg(#s(#0))
#pred(#neg(#s(@x))) → #neg(#s(#s(@x)))
#pred(#pos(#s(#0))) → #0
#pred(#pos(#s(#s(@x)))) → #pos(#s(@x))
#succ(#0) → #pos(#s(#0))
#succ(#neg(#s(#0))) → #0
#succ(#neg(#s(#s(@x)))) → #neg(#s(@x))
#succ(#pos(#s(@x))) → #pos(#s(#s(@x)))

Types:
*' :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
:: :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
nil :: :::nil:#0:#s:#neg:#pos
mult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#add :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#0 :: :::nil:#0:#s:#neg:#pos
#neg :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#s :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pred :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pos :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#succ :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#natmult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
hole_:::nil:#0:#s:#neg:#pos1_3 :: :::nil:#0:#s:#neg:#pos
gen_:::nil:#0:#s:#neg:#pos2_3 :: Nat → :::nil:#0:#s:#neg:#pos

Lemmas:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n3572542_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n3572542₃)
mult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n1814224_3)) → *3_3, rt ∈ Ω(n1814224₃)
dyade#1(gen_:::nil:#0:#s:#neg:#pos2_3(n5395010_3), gen_:::nil:#0:#s:#neg:#pos2_3(0)) → gen_:::nil:#0:#s:#neg:#pos2_3(n5395010_3), rt ∈ Ω(1 + n5395010₃)

Generator Equations:
gen_:::nil:#0:#s:#neg:#pos2_3(0) ⇔ nil
gen_:::nil:#0:#s:#neg:#pos2_3(+(x, 1)) ⇔ ::(nil, gen_:::nil:#0:#s:#neg:#pos2_3(x))

The following defined symbols remain to be analysed:
dyade

They will be analysed ascendingly in the following order:
dyade = dyade#1

(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(24) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
dyade(@l1, @l2) → dyade#1(@l1, @l2)
dyade#1(::(@x, @xs), @l2) → ::(mult(@x, @l2), dyade(@xs, @l2))
dyade#1(nil, @l2) → nil
mult(@n, @l) → mult#1(@l, @n)
mult#1(::(@x, @xs), @n) → ::(*'(@n, @x), mult(@n, @xs))
mult#1(nil, @n) → nil
#add(#0, @y) → @y
#add(#neg(#s(#0)), @y) → #pred(@y)
#add(#neg(#s(#s(@x))), @y) → #pred(#add(#pos(#s(@x)), @y))
#add(#pos(#s(#0)), @y) → #succ(@y)
#add(#pos(#s(#s(@x))), @y) → #succ(#add(#pos(#s(@x)), @y))
#mult(#0, #0) → #0
#mult(#0, #neg(@y)) → #0
#mult(#0, #pos(@y)) → #0
#mult(#neg(@x), #0) → #0
#mult(#neg(@x), #neg(@y)) → #pos(#natmult(@x, @y))
#mult(#neg(@x), #pos(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #0) → #0
#mult(#pos(@x), #neg(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #pos(@y)) → #pos(#natmult(@x, @y))
#natmult(#0, @y) → #0
#natmult(#s(@x), @y) → #add(#pos(@y), #natmult(@x, @y))
#pred(#0) → #neg(#s(#0))
#pred(#neg(#s(@x))) → #neg(#s(#s(@x)))
#pred(#pos(#s(#0))) → #0
#pred(#pos(#s(#s(@x)))) → #pos(#s(@x))
#succ(#0) → #pos(#s(#0))
#succ(#neg(#s(#0))) → #0
#succ(#neg(#s(#s(@x)))) → #neg(#s(@x))
#succ(#pos(#s(@x))) → #pos(#s(#s(@x)))

Types:
*' :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
:: :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
nil :: :::nil:#0:#s:#neg:#pos
mult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#add :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#0 :: :::nil:#0:#s:#neg:#pos
#neg :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#s :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pred :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pos :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#succ :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#natmult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
hole_:::nil:#0:#s:#neg:#pos1_3 :: :::nil:#0:#s:#neg:#pos
gen_:::nil:#0:#s:#neg:#pos2_3 :: Nat → :::nil:#0:#s:#neg:#pos

Lemmas:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n3572542_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n3572542₃)
mult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n1814224_3)) → *3_3, rt ∈ Ω(n1814224₃)
dyade#1(gen_:::nil:#0:#s:#neg:#pos2_3(n5395010_3), gen_:::nil:#0:#s:#neg:#pos2_3(0)) → gen_:::nil:#0:#s:#neg:#pos2_3(n5395010_3), rt ∈ Ω(1 + n5395010₃)

Generator Equations:
gen_:::nil:#0:#s:#neg:#pos2_3(0) ⇔ nil
gen_:::nil:#0:#s:#neg:#pos2_3(+(x, 1)) ⇔ ::(nil, gen_:::nil:#0:#s:#neg:#pos2_3(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n3572542_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n3572542₃)

(26) BOUNDS(n^1, INF)

(27) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
dyade(@l1, @l2) → dyade#1(@l1, @l2)
dyade#1(::(@x, @xs), @l2) → ::(mult(@x, @l2), dyade(@xs, @l2))
dyade#1(nil, @l2) → nil
mult(@n, @l) → mult#1(@l, @n)
mult#1(::(@x, @xs), @n) → ::(*'(@n, @x), mult(@n, @xs))
mult#1(nil, @n) → nil
#add(#0, @y) → @y
#add(#neg(#s(#0)), @y) → #pred(@y)
#add(#neg(#s(#s(@x))), @y) → #pred(#add(#pos(#s(@x)), @y))
#add(#pos(#s(#0)), @y) → #succ(@y)
#add(#pos(#s(#s(@x))), @y) → #succ(#add(#pos(#s(@x)), @y))
#mult(#0, #0) → #0
#mult(#0, #neg(@y)) → #0
#mult(#0, #pos(@y)) → #0
#mult(#neg(@x), #0) → #0
#mult(#neg(@x), #neg(@y)) → #pos(#natmult(@x, @y))
#mult(#neg(@x), #pos(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #0) → #0
#mult(#pos(@x), #neg(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #pos(@y)) → #pos(#natmult(@x, @y))
#natmult(#0, @y) → #0
#natmult(#s(@x), @y) → #add(#pos(@y), #natmult(@x, @y))
#pred(#0) → #neg(#s(#0))
#pred(#neg(#s(@x))) → #neg(#s(#s(@x)))
#pred(#pos(#s(#0))) → #0
#pred(#pos(#s(#s(@x)))) → #pos(#s(@x))
#succ(#0) → #pos(#s(#0))
#succ(#neg(#s(#0))) → #0
#succ(#neg(#s(#s(@x)))) → #neg(#s(@x))
#succ(#pos(#s(@x))) → #pos(#s(#s(@x)))

Types:
*' :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
:: :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
nil :: :::nil:#0:#s:#neg:#pos
mult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#add :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#0 :: :::nil:#0:#s:#neg:#pos
#neg :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#s :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pred :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pos :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#succ :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#natmult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
hole_:::nil:#0:#s:#neg:#pos1_3 :: :::nil:#0:#s:#neg:#pos
gen_:::nil:#0:#s:#neg:#pos2_3 :: Nat → :::nil:#0:#s:#neg:#pos

Lemmas:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n3572542_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n3572542₃)
mult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n1814224_3)) → *3_3, rt ∈ Ω(n1814224₃)
dyade#1(gen_:::nil:#0:#s:#neg:#pos2_3(n5395010_3), gen_:::nil:#0:#s:#neg:#pos2_3(0)) → gen_:::nil:#0:#s:#neg:#pos2_3(n5395010_3), rt ∈ Ω(1 + n5395010₃)

Generator Equations:
gen_:::nil:#0:#s:#neg:#pos2_3(0) ⇔ nil
gen_:::nil:#0:#s:#neg:#pos2_3(+(x, 1)) ⇔ ::(nil, gen_:::nil:#0:#s:#neg:#pos2_3(x))

No more defined symbols left to analyse.

(28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n3572542_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n3572542₃)

(29) BOUNDS(n^1, INF)

(30) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
dyade(@l1, @l2) → dyade#1(@l1, @l2)
dyade#1(::(@x, @xs), @l2) → ::(mult(@x, @l2), dyade(@xs, @l2))
dyade#1(nil, @l2) → nil
mult(@n, @l) → mult#1(@l, @n)
mult#1(::(@x, @xs), @n) → ::(*'(@n, @x), mult(@n, @xs))
mult#1(nil, @n) → nil
#add(#0, @y) → @y
#add(#neg(#s(#0)), @y) → #pred(@y)
#add(#neg(#s(#s(@x))), @y) → #pred(#add(#pos(#s(@x)), @y))
#add(#pos(#s(#0)), @y) → #succ(@y)
#add(#pos(#s(#s(@x))), @y) → #succ(#add(#pos(#s(@x)), @y))
#mult(#0, #0) → #0
#mult(#0, #neg(@y)) → #0
#mult(#0, #pos(@y)) → #0
#mult(#neg(@x), #0) → #0
#mult(#neg(@x), #neg(@y)) → #pos(#natmult(@x, @y))
#mult(#neg(@x), #pos(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #0) → #0
#mult(#pos(@x), #neg(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #pos(@y)) → #pos(#natmult(@x, @y))
#natmult(#0, @y) → #0
#natmult(#s(@x), @y) → #add(#pos(@y), #natmult(@x, @y))
#pred(#0) → #neg(#s(#0))
#pred(#neg(#s(@x))) → #neg(#s(#s(@x)))
#pred(#pos(#s(#0))) → #0
#pred(#pos(#s(#s(@x)))) → #pos(#s(@x))
#succ(#0) → #pos(#s(#0))
#succ(#neg(#s(#0))) → #0
#succ(#neg(#s(#s(@x)))) → #neg(#s(@x))
#succ(#pos(#s(@x))) → #pos(#s(#s(@x)))

Types:
*' :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
:: :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
nil :: :::nil:#0:#s:#neg:#pos
mult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#add :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#0 :: :::nil:#0:#s:#neg:#pos
#neg :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#s :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pred :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pos :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#succ :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#natmult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
hole_:::nil:#0:#s:#neg:#pos1_3 :: :::nil:#0:#s:#neg:#pos
gen_:::nil:#0:#s:#neg:#pos2_3 :: Nat → :::nil:#0:#s:#neg:#pos

Lemmas:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n3572542_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n3572542₃)
mult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n1814224_3)) → *3_3, rt ∈ Ω(n1814224₃)

Generator Equations:
gen_:::nil:#0:#s:#neg:#pos2_3(0) ⇔ nil
gen_:::nil:#0:#s:#neg:#pos2_3(+(x, 1)) ⇔ ::(nil, gen_:::nil:#0:#s:#neg:#pos2_3(x))

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n3572542_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n3572542₃)

(32) BOUNDS(n^1, INF)

(33) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
dyade(@l1, @l2) → dyade#1(@l1, @l2)
dyade#1(::(@x, @xs), @l2) → ::(mult(@x, @l2), dyade(@xs, @l2))
dyade#1(nil, @l2) → nil
mult(@n, @l) → mult#1(@l, @n)
mult#1(::(@x, @xs), @n) → ::(*'(@n, @x), mult(@n, @xs))
mult#1(nil, @n) → nil
#add(#0, @y) → @y
#add(#neg(#s(#0)), @y) → #pred(@y)
#add(#neg(#s(#s(@x))), @y) → #pred(#add(#pos(#s(@x)), @y))
#add(#pos(#s(#0)), @y) → #succ(@y)
#add(#pos(#s(#s(@x))), @y) → #succ(#add(#pos(#s(@x)), @y))
#mult(#0, #0) → #0
#mult(#0, #neg(@y)) → #0
#mult(#0, #pos(@y)) → #0
#mult(#neg(@x), #0) → #0
#mult(#neg(@x), #neg(@y)) → #pos(#natmult(@x, @y))
#mult(#neg(@x), #pos(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #0) → #0
#mult(#pos(@x), #neg(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #pos(@y)) → #pos(#natmult(@x, @y))
#natmult(#0, @y) → #0
#natmult(#s(@x), @y) → #add(#pos(@y), #natmult(@x, @y))
#pred(#0) → #neg(#s(#0))
#pred(#neg(#s(@x))) → #neg(#s(#s(@x)))
#pred(#pos(#s(#0))) → #0
#pred(#pos(#s(#s(@x)))) → #pos(#s(@x))
#succ(#0) → #pos(#s(#0))
#succ(#neg(#s(#0))) → #0
#succ(#neg(#s(#s(@x)))) → #neg(#s(@x))
#succ(#pos(#s(@x))) → #pos(#s(#s(@x)))

Types:
*' :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
:: :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
nil :: :::nil:#0:#s:#neg:#pos
mult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#add :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#0 :: :::nil:#0:#s:#neg:#pos
#neg :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#s :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pred :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pos :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#succ :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#natmult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
hole_:::nil:#0:#s:#neg:#pos1_3 :: :::nil:#0:#s:#neg:#pos
gen_:::nil:#0:#s:#neg:#pos2_3 :: Nat → :::nil:#0:#s:#neg:#pos

Lemmas:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n112₃)
mult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n1814224_3)) → *3_3, rt ∈ Ω(n1814224₃)

Generator Equations:
gen_:::nil:#0:#s:#neg:#pos2_3(0) ⇔ nil
gen_:::nil:#0:#s:#neg:#pos2_3(+(x, 1)) ⇔ ::(nil, gen_:::nil:#0:#s:#neg:#pos2_3(x))

No more defined symbols left to analyse.

(34) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n112₃)

(35) BOUNDS(n^1, INF)

(36) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
dyade(@l1, @l2) → dyade#1(@l1, @l2)
dyade#1(::(@x, @xs), @l2) → ::(mult(@x, @l2), dyade(@xs, @l2))
dyade#1(nil, @l2) → nil
mult(@n, @l) → mult#1(@l, @n)
mult#1(::(@x, @xs), @n) → ::(*'(@n, @x), mult(@n, @xs))
mult#1(nil, @n) → nil
#add(#0, @y) → @y
#add(#neg(#s(#0)), @y) → #pred(@y)
#add(#neg(#s(#s(@x))), @y) → #pred(#add(#pos(#s(@x)), @y))
#add(#pos(#s(#0)), @y) → #succ(@y)
#add(#pos(#s(#s(@x))), @y) → #succ(#add(#pos(#s(@x)), @y))
#mult(#0, #0) → #0
#mult(#0, #neg(@y)) → #0
#mult(#0, #pos(@y)) → #0
#mult(#neg(@x), #0) → #0
#mult(#neg(@x), #neg(@y)) → #pos(#natmult(@x, @y))
#mult(#neg(@x), #pos(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #0) → #0
#mult(#pos(@x), #neg(@y)) → #neg(#natmult(@x, @y))
#mult(#pos(@x), #pos(@y)) → #pos(#natmult(@x, @y))
#natmult(#0, @y) → #0
#natmult(#s(@x), @y) → #add(#pos(@y), #natmult(@x, @y))
#pred(#0) → #neg(#s(#0))
#pred(#neg(#s(@x))) → #neg(#s(#s(@x)))
#pred(#pos(#s(#0))) → #0
#pred(#pos(#s(#s(@x)))) → #pos(#s(@x))
#succ(#0) → #pos(#s(#0))
#succ(#neg(#s(#0))) → #0
#succ(#neg(#s(#s(@x)))) → #neg(#s(@x))
#succ(#pos(#s(@x))) → #pos(#s(#s(@x)))

Types:
*' :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
dyade#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
:: :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
mult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
nil :: :::nil:#0:#s:#neg:#pos
mult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#add :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#0 :: :::nil:#0:#s:#neg:#pos
#neg :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#s :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pred :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#pos :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#succ :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
#natmult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
hole_:::nil:#0:#s:#neg:#pos1_3 :: :::nil:#0:#s:#neg:#pos
gen_:::nil:#0:#s:#neg:#pos2_3 :: Nat → :::nil:#0:#s:#neg:#pos

Lemmas:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n112₃)

Generator Equations:
gen_:::nil:#0:#s:#neg:#pos2_3(0) ⇔ nil
gen_:::nil:#0:#s:#neg:#pos2_3(+(x, 1)) ⇔ ::(nil, gen_:::nil:#0:#s:#neg:#pos2_3(x))

No more defined symbols left to analyse.

(37) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → *3_3, rt ∈ Ω(n112₃)

(38) BOUNDS(n^1, INF)