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)
+(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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
+' :: :::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
computeLine :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
computeLine#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 → :::nil:#0:#s:#neg:#pos
computeLine#2 :: :::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 :: :::nil:#0:#s:#neg:#pos
lineMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#2 :: :::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
matrixMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
matrixMult#1 :: :::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:
#add, computeLine, computeLine#1, lineMult, lineMult#1, matrixMult, matrixMult#1, #natmult

They will be analysed ascendingly in the following order:
#add < #natmult
computeLine = computeLine#1
computeLine < matrixMult#1
lineMult < computeLine#1
lineMult = lineMult#1
matrixMult = matrixMult#1

(6) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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
+' :: :::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
computeLine :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
computeLine#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 → :::nil:#0:#s:#neg:#pos
computeLine#2 :: :::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 :: :::nil:#0:#s:#neg:#pos
lineMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#2 :: :::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
matrixMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
matrixMult#1 :: :::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, computeLine, computeLine#1, lineMult, lineMult#1, matrixMult, matrixMult#1, #natmult

They will be analysed ascendingly in the following order:
#add < #natmult
computeLine = computeLine#1
computeLine < matrixMult#1
lineMult < computeLine#1
lineMult = lineMult#1
matrixMult = matrixMult#1

(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)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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
+' :: :::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
computeLine :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
computeLine#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 → :::nil:#0:#s:#neg:#pos
computeLine#2 :: :::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 :: :::nil:#0:#s:#neg:#pos
lineMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#2 :: :::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
matrixMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
matrixMult#1 :: :::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, computeLine, computeLine#1, lineMult, lineMult#1, matrixMult, matrixMult#1

They will be analysed ascendingly in the following order:
computeLine = computeLine#1
computeLine < matrixMult#1
lineMult < computeLine#1
lineMult = lineMult#1
matrixMult = matrixMult#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)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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
+' :: :::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
computeLine :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
computeLine#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 → :::nil:#0:#s:#neg:#pos
computeLine#2 :: :::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 :: :::nil:#0:#s:#neg:#pos
lineMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#2 :: :::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
matrixMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
matrixMult#1 :: :::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:
lineMult#1, computeLine, computeLine#1, lineMult, matrixMult, matrixMult#1

They will be analysed ascendingly in the following order:
computeLine = computeLine#1
computeLine < matrixMult#1
lineMult < computeLine#1
lineMult = lineMult#1
matrixMult = matrixMult#1

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

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

Induction Base:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, 0)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c))

Induction Step:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, +(n112_3, 1))), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) →_R^Ω(1)
lineMult#2(gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c), nil, gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3))) →_R^Ω(1)
::(*'(nil, gen_:::nil:#0:#s:#neg:#pos2_3(c)), lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(c), gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)), nil)) →_R^Ω(1)
::(#mult(nil, gen_:::nil:#0:#s:#neg:#pos2_3(c)), lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(c), gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)), nil)) →_R^Ω(1)
::(#mult(nil, gen_:::nil:#0:#s:#neg:#pos2_3(c)), lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)), nil, gen_:::nil:#0:#s:#neg:#pos2_3(c))) →_IH
::(#mult(nil, gen_:::nil:#0:#s:#neg:#pos2_3(c)), *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)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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
+' :: :::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
computeLine :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
computeLine#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 → :::nil:#0:#s:#neg:#pos
computeLine#2 :: :::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 :: :::nil:#0:#s:#neg:#pos
lineMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#2 :: :::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
matrixMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
matrixMult#1 :: :::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:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) → *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:
lineMult, computeLine, computeLine#1, matrixMult, matrixMult#1

They will be analysed ascendingly in the following order:
computeLine = computeLine#1
computeLine < matrixMult#1
lineMult < computeLine#1
lineMult = lineMult#1
matrixMult = matrixMult#1

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

Proved the following rewrite lemma:
lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n4515426_3), gen_:::nil:#0:#s:#neg:#pos2_3(0)) → *3_3, rt ∈ Ω(n4515426₃)

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

Induction Step:
lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(+(n4515426_3, 1)), gen_:::nil:#0:#s:#neg:#pos2_3(0)) →_R^Ω(1)
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(n4515426_3, 1)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(a)) →_R^Ω(1)
lineMult#2(gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(a), nil, gen_:::nil:#0:#s:#neg:#pos2_3(n4515426_3)) →_R^Ω(1)
::(*'(nil, gen_:::nil:#0:#s:#neg:#pos2_3(a)), lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n4515426_3), nil)) →_R^Ω(1)
::(#mult(nil, gen_:::nil:#0:#s:#neg:#pos2_3(a)), lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n4515426_3), nil)) →_IH
::(#mult(nil, gen_:::nil:#0:#s:#neg:#pos2_3(a)), *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)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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
+' :: :::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
computeLine :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
computeLine#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 → :::nil:#0:#s:#neg:#pos
computeLine#2 :: :::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 :: :::nil:#0:#s:#neg:#pos
lineMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#2 :: :::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
matrixMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
matrixMult#1 :: :::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:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) → *3_3, rt ∈ Ω(n112₃)
lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n4515426_3), gen_:::nil:#0:#s:#neg:#pos2_3(0)) → *3_3, rt ∈ Ω(n4515426₃)

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:
lineMult#1, computeLine, computeLine#1, matrixMult, matrixMult#1

They will be analysed ascendingly in the following order:
computeLine = computeLine#1
computeLine < matrixMult#1
lineMult < computeLine#1
lineMult = lineMult#1
matrixMult = matrixMult#1

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

Proved the following rewrite lemma:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n9035676_3)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) → *3_3, rt ∈ Ω(n9035676₃)

Induction Base:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, 0)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c))

Induction Step:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, +(n9035676_3, 1))), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) →_R^Ω(1)
lineMult#2(gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c), nil, gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n9035676_3))) →_R^Ω(1)
::(*'(nil, gen_:::nil:#0:#s:#neg:#pos2_3(c)), lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(c), gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n9035676_3)), nil)) →_R^Ω(1)
::(#mult(nil, gen_:::nil:#0:#s:#neg:#pos2_3(c)), lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(c), gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n9035676_3)), nil)) →_R^Ω(1)
::(#mult(nil, gen_:::nil:#0:#s:#neg:#pos2_3(c)), lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n9035676_3)), nil, gen_:::nil:#0:#s:#neg:#pos2_3(c))) →_IH
::(#mult(nil, gen_:::nil:#0:#s:#neg:#pos2_3(c)), *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)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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
+' :: :::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
computeLine :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
computeLine#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 → :::nil:#0:#s:#neg:#pos
computeLine#2 :: :::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 :: :::nil:#0:#s:#neg:#pos
lineMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#2 :: :::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
matrixMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
matrixMult#1 :: :::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:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n9035676_3)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) → *3_3, rt ∈ Ω(n9035676₃)
lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n4515426_3), gen_:::nil:#0:#s:#neg:#pos2_3(0)) → *3_3, rt ∈ Ω(n4515426₃)

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:
computeLine#1, computeLine, matrixMult, matrixMult#1

They will be analysed ascendingly in the following order:
computeLine = computeLine#1
computeLine < matrixMult#1
matrixMult = matrixMult#1

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol computeLine#1.

(21) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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
+' :: :::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
computeLine :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
computeLine#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 → :::nil:#0:#s:#neg:#pos
computeLine#2 :: :::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 :: :::nil:#0:#s:#neg:#pos
lineMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#2 :: :::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
matrixMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
matrixMult#1 :: :::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:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n9035676_3)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) → *3_3, rt ∈ Ω(n9035676₃)
lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n4515426_3), gen_:::nil:#0:#s:#neg:#pos2_3(0)) → *3_3, rt ∈ Ω(n4515426₃)

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:
computeLine, matrixMult, matrixMult#1

They will be analysed ascendingly in the following order:
computeLine = computeLine#1
computeLine < matrixMult#1
matrixMult = matrixMult#1

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(23) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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
+' :: :::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
computeLine :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
computeLine#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 → :::nil:#0:#s:#neg:#pos
computeLine#2 :: :::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 :: :::nil:#0:#s:#neg:#pos
lineMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#2 :: :::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
matrixMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
matrixMult#1 :: :::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:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n9035676_3)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) → *3_3, rt ∈ Ω(n9035676₃)
lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n4515426_3), gen_:::nil:#0:#s:#neg:#pos2_3(0)) → *3_3, rt ∈ Ω(n4515426₃)

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:
matrixMult#1, matrixMult

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

(24) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
matrixMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(n13643793_3), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → gen_:::nil:#0:#s:#neg:#pos2_3(n13643793_3), rt ∈ Ω(1 + n13643793₃)

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

Induction Step:
matrixMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(n13643793_3, 1)), gen_:::nil:#0:#s:#neg:#pos2_3(b)) →_R^Ω(1)
::(computeLine(nil, gen_:::nil:#0:#s:#neg:#pos2_3(b), nil), matrixMult(gen_:::nil:#0:#s:#neg:#pos2_3(n13643793_3), gen_:::nil:#0:#s:#neg:#pos2_3(b))) →_R^Ω(1)
::(computeLine#1(nil, nil, gen_:::nil:#0:#s:#neg:#pos2_3(b)), matrixMult(gen_:::nil:#0:#s:#neg:#pos2_3(n13643793_3), gen_:::nil:#0:#s:#neg:#pos2_3(b))) →_R^Ω(1)
::(nil, matrixMult(gen_:::nil:#0:#s:#neg:#pos2_3(n13643793_3), gen_:::nil:#0:#s:#neg:#pos2_3(b))) →_R^Ω(1)
::(nil, matrixMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(n13643793_3), gen_:::nil:#0:#s:#neg:#pos2_3(b))) →_IH
::(nil, gen_:::nil:#0:#s:#neg:#pos2_3(c13643794_3))

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

(25) Complex Obligation (BEST)

(26) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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
+' :: :::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
computeLine :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
computeLine#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 → :::nil:#0:#s:#neg:#pos
computeLine#2 :: :::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 :: :::nil:#0:#s:#neg:#pos
lineMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#2 :: :::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
matrixMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
matrixMult#1 :: :::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:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n9035676_3)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) → *3_3, rt ∈ Ω(n9035676₃)
lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n4515426_3), gen_:::nil:#0:#s:#neg:#pos2_3(0)) → *3_3, rt ∈ Ω(n4515426₃)
matrixMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(n13643793_3), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → gen_:::nil:#0:#s:#neg:#pos2_3(n13643793_3), rt ∈ Ω(1 + n13643793₃)

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:
matrixMult

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

(27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(28) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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
+' :: :::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
computeLine :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
computeLine#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 → :::nil:#0:#s:#neg:#pos
computeLine#2 :: :::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 :: :::nil:#0:#s:#neg:#pos
lineMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#2 :: :::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
matrixMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
matrixMult#1 :: :::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:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n9035676_3)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) → *3_3, rt ∈ Ω(n9035676₃)
lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n4515426_3), gen_:::nil:#0:#s:#neg:#pos2_3(0)) → *3_3, rt ∈ Ω(n4515426₃)
matrixMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(n13643793_3), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → gen_:::nil:#0:#s:#neg:#pos2_3(n13643793_3), rt ∈ Ω(1 + n13643793₃)

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.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n9035676_3)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) → *3_3, rt ∈ Ω(n9035676₃)

(30) BOUNDS(n^1, INF)

(31) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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
+' :: :::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
computeLine :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
computeLine#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 → :::nil:#0:#s:#neg:#pos
computeLine#2 :: :::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 :: :::nil:#0:#s:#neg:#pos
lineMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#2 :: :::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
matrixMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
matrixMult#1 :: :::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:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n9035676_3)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) → *3_3, rt ∈ Ω(n9035676₃)
lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n4515426_3), gen_:::nil:#0:#s:#neg:#pos2_3(0)) → *3_3, rt ∈ Ω(n4515426₃)
matrixMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(n13643793_3), gen_:::nil:#0:#s:#neg:#pos2_3(b)) → gen_:::nil:#0:#s:#neg:#pos2_3(n13643793_3), rt ∈ Ω(1 + n13643793₃)

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.

(32) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n9035676_3)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) → *3_3, rt ∈ Ω(n9035676₃)

(33) BOUNDS(n^1, INF)

(34) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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
+' :: :::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
computeLine :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
computeLine#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 → :::nil:#0:#s:#neg:#pos
computeLine#2 :: :::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 :: :::nil:#0:#s:#neg:#pos
lineMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#2 :: :::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
matrixMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
matrixMult#1 :: :::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:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n9035676_3)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) → *3_3, rt ∈ Ω(n9035676₃)
lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n4515426_3), gen_:::nil:#0:#s:#neg:#pos2_3(0)) → *3_3, rt ∈ Ω(n4515426₃)

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.

(35) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n9035676_3)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) → *3_3, rt ∈ Ω(n9035676₃)

(36) BOUNDS(n^1, INF)

(37) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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
+' :: :::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
computeLine :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
computeLine#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 → :::nil:#0:#s:#neg:#pos
computeLine#2 :: :::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 :: :::nil:#0:#s:#neg:#pos
lineMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#2 :: :::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
matrixMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
matrixMult#1 :: :::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:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) → *3_3, rt ∈ Ω(n112₃)
lineMult(gen_:::nil:#0:#s:#neg:#pos2_3(a), gen_:::nil:#0:#s:#neg:#pos2_3(n4515426_3), gen_:::nil:#0:#s:#neg:#pos2_3(0)) → *3_3, rt ∈ Ω(n4515426₃)

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.

(38) LowerBoundsProof (EQUIVALENT transformation)

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

(39) BOUNDS(n^1, INF)

(40) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
computeLine(@line, @m, @acc) → computeLine#1(@line, @acc, @m)
computeLine#1(::(@x, @xs), @acc, @m) → computeLine#2(@m, @acc, @x, @xs)
computeLine#1(nil, @acc, @m) → @acc
computeLine#2(::(@l, @ls), @acc, @x, @xs) → computeLine(@xs, @ls, lineMult(@x, @l, @acc))
computeLine#2(nil, @acc, @x, @xs) → nil
lineMult(@n, @l1, @l2) → lineMult#1(@l1, @l2, @n)
lineMult#1(::(@x, @xs), @l2, @n) → lineMult#2(@l2, @n, @x, @xs)
lineMult#1(nil, @l2, @n) → nil
lineMult#2(::(@y, @ys), @n, @x, @xs) → ::(+'(*'(@x, @n), @y), lineMult(@n, @xs, @ys))
lineMult#2(nil, @n, @x, @xs) → ::(*'(@x, @n), lineMult(@n, @xs, nil))
matrixMult(@m1, @m2) → matrixMult#1(@m1, @m2)
matrixMult#1(::(@l, @ls), @m2) → ::(computeLine(@l, @m2, nil), matrixMult(@ls, @m2))
matrixMult#1(nil, @m2) → 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
+' :: :::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
computeLine :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
computeLine#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 → :::nil:#0:#s:#neg:#pos
computeLine#2 :: :::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 :: :::nil:#0:#s:#neg:#pos
lineMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#1 :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
lineMult#2 :: :::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
matrixMult :: :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos → :::nil:#0:#s:#neg:#pos
matrixMult#1 :: :::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:
lineMult#1(gen_:::nil:#0:#s:#neg:#pos2_3(+(1, n112_3)), gen_:::nil:#0:#s:#neg:#pos2_3(0), gen_:::nil:#0:#s:#neg:#pos2_3(c)) → *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.

(41) LowerBoundsProof (EQUIVALENT transformation)

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

(42) BOUNDS(n^1, INF)