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)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

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

Heuristically decided to analyse the following defined symbols:
#add, appendreverse, appendreverse#1, bftMult', bftMult'#1, matrixMult, computeLine, computeLine#1, lineMult, lineMult#1, matrixMult#1, #natmult

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

(6) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

The following defined symbols remain to be analysed:
#add, appendreverse, appendreverse#1, bftMult', bftMult'#1, matrixMult, computeLine, computeLine#1, lineMult, lineMult#1, matrixMult#1, #natmult

They will be analysed ascendingly in the following order:
#add < #natmult
appendreverse = appendreverse#1
bftMult' = bftMult'#1
matrixMult < bftMult'#1
matrixMult = matrixMult#1
computeLine = computeLine#1
computeLine < matrixMult#1
lineMult < computeLine#1
lineMult = lineMult#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)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

The following defined symbols remain to be analysed:
#natmult, appendreverse, appendreverse#1, bftMult', bftMult'#1, matrixMult, computeLine, computeLine#1, lineMult, lineMult#1, matrixMult#1

They will be analysed ascendingly in the following order:
appendreverse = appendreverse#1
bftMult' = bftMult'#1
matrixMult < bftMult'#1
matrixMult = matrixMult#1
computeLine = computeLine#1
computeLine < matrixMult#1
lineMult < computeLine#1
lineMult = lineMult#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)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

The following defined symbols remain to be analysed:
lineMult#1, appendreverse, appendreverse#1, bftMult', bftMult'#1, matrixMult, computeLine, computeLine#1, lineMult, matrixMult#1

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

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

Proved the following rewrite lemma:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n112_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n112₆)

Induction Base:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, 0)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c))

Induction Step:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, +(n112_6, 1))), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) →_R^Ω(1)
lineMult#2(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c), nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n112_6))) →_R^Ω(1)
::(*'(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)), lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n112_6)), nil)) →_R^Ω(1)
::(#mult(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)), lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n112_6)), nil)) →_R^Ω(1)
::(#mult(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)), lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n112_6)), nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c))) →_IH
::(#mult(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)), *3_6)

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)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n112_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n112₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

The following defined symbols remain to be analysed:
lineMult, appendreverse, appendreverse#1, bftMult', bftMult'#1, matrixMult, computeLine, computeLine#1, matrixMult#1

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

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

Proved the following rewrite lemma:
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) → *3_6, rt ∈ Ω(n4517226₆)

Induction Base:
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0))

Induction Step:
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(n4517226_6, 1)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) →_R^Ω(1)
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(n4517226_6, 1)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a)) →_R^Ω(1)
lineMult#2(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6)) →_R^Ω(1)
::(*'(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a)), lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), nil)) →_R^Ω(1)
::(#mult(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a)), lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), nil)) →_IH
::(#mult(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a)), *3_6)

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)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n112_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n112₆)
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) → *3_6, rt ∈ Ω(n4517226₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

The following defined symbols remain to be analysed:
lineMult#1, appendreverse, appendreverse#1, bftMult', bftMult'#1, matrixMult, computeLine, computeLine#1, matrixMult#1

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

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

Proved the following rewrite lemma:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)

Induction Base:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, 0)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c))

Induction Step:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, +(n9038516_6, 1))), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) →_R^Ω(1)
lineMult#2(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c), nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6))) →_R^Ω(1)
::(*'(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)), lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), nil)) →_R^Ω(1)
::(#mult(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)), lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), nil)) →_R^Ω(1)
::(#mult(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)), lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c))) →_IH
::(#mult(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)), *3_6)

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)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) → *3_6, rt ∈ Ω(n4517226₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

The following defined symbols remain to be analysed:
computeLine#1, appendreverse, appendreverse#1, bftMult', bftMult'#1, matrixMult, computeLine, matrixMult#1

They will be analysed ascendingly in the following order:
appendreverse = appendreverse#1
bftMult' = bftMult'#1
matrixMult < bftMult'#1
matrixMult = matrixMult#1
computeLine = computeLine#1
computeLine < 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)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) → *3_6, rt ∈ Ω(n4517226₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

The following defined symbols remain to be analysed:
computeLine, appendreverse, appendreverse#1, bftMult', bftMult'#1, matrixMult, matrixMult#1

They will be analysed ascendingly in the following order:
appendreverse = appendreverse#1
bftMult' = bftMult'#1
matrixMult < bftMult'#1
matrixMult = matrixMult#1
computeLine = computeLine#1
computeLine < 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)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) → *3_6, rt ∈ Ω(n4517226₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

The following defined symbols remain to be analysed:
matrixMult#1, appendreverse, appendreverse#1, bftMult', bftMult'#1, matrixMult

They will be analysed ascendingly in the following order:
appendreverse = appendreverse#1
bftMult' = bftMult'#1
matrixMult < bftMult'#1
matrixMult = matrixMult#1

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

Proved the following rewrite lemma:
matrixMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) → gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), rt ∈ Ω(1 + n13679073₆)

Induction Base:
matrixMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) →_R^Ω(1)
nil

Induction Step:
matrixMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(n13679073_6, 1)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) →_R^Ω(1)
::(computeLine(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b), nil), matrixMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b))) →_R^Ω(1)
::(computeLine#1(nil, nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)), matrixMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b))) →_R^Ω(1)
::(nil, matrixMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b))) →_R^Ω(1)
::(nil, matrixMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b))) →_IH
::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c13679074_6))

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)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) → *3_6, rt ∈ Ω(n4517226₆)
matrixMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) → gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), rt ∈ Ω(1 + n13679073₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

The following defined symbols remain to be analysed:
matrixMult, appendreverse, appendreverse#1, bftMult', bftMult'#1

They will be analysed ascendingly in the following order:
appendreverse = appendreverse#1
bftMult' = bftMult'#1
matrixMult < bftMult'#1
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)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) → *3_6, rt ∈ Ω(n4517226₆)
matrixMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) → gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), rt ∈ Ω(1 + n13679073₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

The following defined symbols remain to be analysed:
bftMult'#1, appendreverse, appendreverse#1, bftMult'

They will be analysed ascendingly in the following order:
appendreverse = appendreverse#1
bftMult' = bftMult'#1

(29) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(30) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) → *3_6, rt ∈ Ω(n4517226₆)
matrixMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) → gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), rt ∈ Ω(1 + n13679073₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

The following defined symbols remain to be analysed:
bftMult', appendreverse, appendreverse#1

They will be analysed ascendingly in the following order:
appendreverse = appendreverse#1
bftMult' = bftMult'#1

(31) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol bftMult'.

(32) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) → *3_6, rt ∈ Ω(n4517226₆)
matrixMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) → gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), rt ∈ Ω(1 + n13679073₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

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

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

(33) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
appendreverse#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13681769_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) → gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(n13681769_6, b)), rt ∈ Ω(1 + n13681769₆)

Induction Base:
appendreverse#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) →_R^Ω(1)
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)

Induction Step:
appendreverse#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(n13681769_6, 1)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) →_R^Ω(1)
appendreverse(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13681769_6), ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b))) →_R^Ω(1)
appendreverse#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13681769_6), ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b))) →_IH
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(+(b, 1), c13681770_6))

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

(34) Complex Obligation (BEST)

(35) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) → *3_6, rt ∈ Ω(n4517226₆)
matrixMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) → gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), rt ∈ Ω(1 + n13679073₆)
appendreverse#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13681769_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) → gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(n13681769_6, b)), rt ∈ Ω(1 + n13681769₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

The following defined symbols remain to be analysed:
appendreverse

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

(36) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(37) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) → *3_6, rt ∈ Ω(n4517226₆)
matrixMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) → gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), rt ∈ Ω(1 + n13679073₆)
appendreverse#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13681769_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) → gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(n13681769_6, b)), rt ∈ Ω(1 + n13681769₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(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:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)

(39) BOUNDS(n^1, INF)

(40) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) → *3_6, rt ∈ Ω(n4517226₆)
matrixMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) → gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), rt ∈ Ω(1 + n13679073₆)
appendreverse#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13681769_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) → gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(n13681769_6, b)), rt ∈ Ω(1 + n13681769₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(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:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)

(42) BOUNDS(n^1, INF)

(43) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) → *3_6, rt ∈ Ω(n4517226₆)
matrixMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(b)) → gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n13679073_6), rt ∈ Ω(1 + n13679073₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

No more defined symbols left to analyse.

(44) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)

(45) BOUNDS(n^1, INF)

(46) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) → *3_6, rt ∈ Ω(n4517226₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

No more defined symbols left to analyse.

(47) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n9038516_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n9038516₆)

(48) BOUNDS(n^1, INF)

(49) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n112_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n112₆)
lineMult(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(a), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(n4517226_6), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0)) → *3_6, rt ∈ Ω(n4517226₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

No more defined symbols left to analyse.

(50) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n112_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n112₆)

(51) BOUNDS(n^1, INF)

(52) Obligation:

Innermost TRS:
Rules:
*'(@x, @y) → #mult(@x, @y)
+'(@x, @y) → #add(@x, @y)
appendreverse(@toreverse, @sofar) → appendreverse#1(@toreverse, @sofar)
appendreverse#1(::(@a, @as), @sofar) → appendreverse(@as, ::(@a, @sofar))
appendreverse#1(nil, @sofar) → @sofar
bftMult(@t, @acc) → bftMult'(tuple#2(::(@t, nil), nil), @acc)
bftMult'(@queue, @acc) → bftMult'#1(bftMult'#2(@queue), @acc)
bftMult'#1(tuple#2(@elem, @queue), @acc) → bftMult'#3(@elem, @acc, @queue)
bftMult'#2(tuple#2(@dequeue@1, @dequeue@2)) → dequeue(@dequeue@1, @dequeue@2)
bftMult'#3(::(@t, @_@3), @acc, @queue) → bftMult'#4(@t, @acc, @queue)
bftMult'#3(nil, @acc, @queue) → @acc
bftMult'#4(leaf, @acc, @queue) → bftMult'(@queue, @acc)
bftMult'#4(node(@y, @t1, @t2), @acc, @queue) → bftMult'#5(enqueue(@t2, enqueue(@t1, @queue)), @acc, @y)
bftMult'#5(@queue', @acc, @y) → bftMult'(@queue', matrixMult(@acc, @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
dequeue(@outq, @inq) → dequeue#1(@outq, @inq)
dequeue#1(::(@t, @ts), @inq) → tuple#2(::(@t, nil), tuple#2(@ts, @inq))
dequeue#1(nil, @inq) → dequeue#2(reverse(@inq))
dequeue#2(::(@t, @ts)) → tuple#2(::(@t, nil), tuple#2(@ts, nil))
dequeue#2(nil) → tuple#2(nil, tuple#2(nil, nil))
enqueue(@t, @queue) → enqueue#1(@queue, @t)
enqueue#1(tuple#2(@outq, @inq), @t) → tuple#2(@outq, ::(@t, @inq))
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
reverse(@xs) → appendreverse(@xs, 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:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#mult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
+' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#add :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
appendreverse#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
:: :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
nil :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult' :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
tuple#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#3 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#4 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
leaf :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
node :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
bftMult'#5 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
computeLine#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
dequeue#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
reverse :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
enqueue#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
lineMult#2 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
matrixMult#1 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#0 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#neg :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#s :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pred :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#pos :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#succ :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
#natmult :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
hole_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos1_6 :: :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6 :: Nat → :::nil:tuple#2:leaf:node:#0:#s:#neg:#pos

Lemmas:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n112_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n112₆)

Generator Equations:
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0) ⇔ nil
gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(x, 1)) ⇔ ::(nil, gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(x))

No more defined symbols left to analyse.

(53) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lineMult#1(gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(+(1, n112_6)), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(0), gen_:::nil:tuple#2:leaf:node:#0:#s:#neg:#pos2_6(c)) → *3_6, rt ∈ Ω(n112₆)

(54) BOUNDS(n^1, INF)