We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict Trs:
  { mult(@n, @l) -> mult#1(@l, @n)
  , mult#1(::(@x, @xs), @n) -> ::(*(@n, @x), mult(@n, @xs))
  , mult#1(nil(), @n) -> nil()
  , *(@x, @y) -> #mult(@x, @y)
  , dyade#1(::(@x, @xs), @l2) -> ::(mult(@x, @l2), dyade(@xs, @l2))
  , dyade#1(nil(), @l2) -> nil()
  , dyade(@l1, @l2) -> dyade#1(@l1, @l2) }
Weak Trs:
  { #natmult(#0(), @y) -> #0()
  , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y))
  , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y))
  , #mult(#pos(@x), #0()) -> #0()
  , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y))
  , #mult(#0(), #pos(@y)) -> #0()
  , #mult(#0(), #0()) -> #0()
  , #mult(#0(), #neg(@y)) -> #0()
  , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y))
  , #mult(#neg(@x), #0()) -> #0()
  , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y))
  , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
  , #succ(#0()) -> #pos(#s(#0()))
  , #succ(#neg(#s(#0()))) -> #0()
  , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
  , #add(#pos(#s(#0())), @y) -> #succ(@y)
  , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
  , #add(#0(), @y) -> @y
  , #add(#neg(#s(#0())), @y) -> #pred(@y)
  , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
  , #pred(#pos(#s(#0()))) -> #0()
  , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
  , #pred(#0()) -> #neg(#s(#0()))
  , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

We add the following dependency tuples:

Strict DPs:
  { mult^#(@n, @l) -> c_1(mult#1^#(@l, @n))
  , mult#1^#(::(@x, @xs), @n) -> c_2(*^#(@n, @x), mult^#(@n, @xs))
  , mult#1^#(nil(), @n) -> c_3()
  , *^#(@x, @y) -> c_4(#mult^#(@x, @y))
  , dyade#1^#(::(@x, @xs), @l2) ->
    c_5(mult^#(@x, @l2), dyade^#(@xs, @l2))
  , dyade#1^#(nil(), @l2) -> c_6()
  , dyade^#(@l1, @l2) -> c_7(dyade#1^#(@l1, @l2)) }
Weak DPs:
  { #mult^#(#pos(@x), #pos(@y)) -> c_10(#natmult^#(@x, @y))
  , #mult^#(#pos(@x), #0()) -> c_11()
  , #mult^#(#pos(@x), #neg(@y)) -> c_12(#natmult^#(@x, @y))
  , #mult^#(#0(), #pos(@y)) -> c_13()
  , #mult^#(#0(), #0()) -> c_14()
  , #mult^#(#0(), #neg(@y)) -> c_15()
  , #mult^#(#neg(@x), #pos(@y)) -> c_16(#natmult^#(@x, @y))
  , #mult^#(#neg(@x), #0()) -> c_17()
  , #mult^#(#neg(@x), #neg(@y)) -> c_18(#natmult^#(@x, @y))
  , #natmult^#(#0(), @y) -> c_8()
  , #natmult^#(#s(@x), @y) ->
    c_9(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y))
  , #add^#(#pos(#s(#0())), @y) -> c_23(#succ^#(@y))
  , #add^#(#pos(#s(#s(@x))), @y) ->
    c_24(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
  , #add^#(#0(), @y) -> c_25()
  , #add^#(#neg(#s(#0())), @y) -> c_26(#pred^#(@y))
  , #add^#(#neg(#s(#s(@x))), @y) ->
    c_27(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
  , #succ^#(#pos(#s(@x))) -> c_19()
  , #succ^#(#0()) -> c_20()
  , #succ^#(#neg(#s(#0()))) -> c_21()
  , #succ^#(#neg(#s(#s(@x)))) -> c_22()
  , #pred^#(#pos(#s(#0()))) -> c_28()
  , #pred^#(#pos(#s(#s(@x)))) -> c_29()
  , #pred^#(#0()) -> c_30()
  , #pred^#(#neg(#s(@x))) -> c_31() }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict DPs:
  { mult^#(@n, @l) -> c_1(mult#1^#(@l, @n))
  , mult#1^#(::(@x, @xs), @n) -> c_2(*^#(@n, @x), mult^#(@n, @xs))
  , mult#1^#(nil(), @n) -> c_3()
  , *^#(@x, @y) -> c_4(#mult^#(@x, @y))
  , dyade#1^#(::(@x, @xs), @l2) ->
    c_5(mult^#(@x, @l2), dyade^#(@xs, @l2))
  , dyade#1^#(nil(), @l2) -> c_6()
  , dyade^#(@l1, @l2) -> c_7(dyade#1^#(@l1, @l2)) }
Weak DPs:
  { #mult^#(#pos(@x), #pos(@y)) -> c_10(#natmult^#(@x, @y))
  , #mult^#(#pos(@x), #0()) -> c_11()
  , #mult^#(#pos(@x), #neg(@y)) -> c_12(#natmult^#(@x, @y))
  , #mult^#(#0(), #pos(@y)) -> c_13()
  , #mult^#(#0(), #0()) -> c_14()
  , #mult^#(#0(), #neg(@y)) -> c_15()
  , #mult^#(#neg(@x), #pos(@y)) -> c_16(#natmult^#(@x, @y))
  , #mult^#(#neg(@x), #0()) -> c_17()
  , #mult^#(#neg(@x), #neg(@y)) -> c_18(#natmult^#(@x, @y))
  , #natmult^#(#0(), @y) -> c_8()
  , #natmult^#(#s(@x), @y) ->
    c_9(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y))
  , #add^#(#pos(#s(#0())), @y) -> c_23(#succ^#(@y))
  , #add^#(#pos(#s(#s(@x))), @y) ->
    c_24(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
  , #add^#(#0(), @y) -> c_25()
  , #add^#(#neg(#s(#0())), @y) -> c_26(#pred^#(@y))
  , #add^#(#neg(#s(#s(@x))), @y) ->
    c_27(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
  , #succ^#(#pos(#s(@x))) -> c_19()
  , #succ^#(#0()) -> c_20()
  , #succ^#(#neg(#s(#0()))) -> c_21()
  , #succ^#(#neg(#s(#s(@x)))) -> c_22()
  , #pred^#(#pos(#s(#0()))) -> c_28()
  , #pred^#(#pos(#s(#s(@x)))) -> c_29()
  , #pred^#(#0()) -> c_30()
  , #pred^#(#neg(#s(@x))) -> c_31() }
Weak Trs:
  { #natmult(#0(), @y) -> #0()
  , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y))
  , mult(@n, @l) -> mult#1(@l, @n)
  , mult#1(::(@x, @xs), @n) -> ::(*(@n, @x), mult(@n, @xs))
  , mult#1(nil(), @n) -> nil()
  , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y))
  , #mult(#pos(@x), #0()) -> #0()
  , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y))
  , #mult(#0(), #pos(@y)) -> #0()
  , #mult(#0(), #0()) -> #0()
  , #mult(#0(), #neg(@y)) -> #0()
  , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y))
  , #mult(#neg(@x), #0()) -> #0()
  , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y))
  , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
  , #succ(#0()) -> #pos(#s(#0()))
  , #succ(#neg(#s(#0()))) -> #0()
  , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
  , #add(#pos(#s(#0())), @y) -> #succ(@y)
  , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
  , #add(#0(), @y) -> @y
  , #add(#neg(#s(#0())), @y) -> #pred(@y)
  , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
  , *(@x, @y) -> #mult(@x, @y)
  , dyade#1(::(@x, @xs), @l2) -> ::(mult(@x, @l2), dyade(@xs, @l2))
  , dyade#1(nil(), @l2) -> nil()
  , #pred(#pos(#s(#0()))) -> #0()
  , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
  , #pred(#0()) -> #neg(#s(#0()))
  , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
  , dyade(@l1, @l2) -> dyade#1(@l1, @l2) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

We estimate the number of application of {3,4,6} by applications of
Pre({3,4,6}) = {1,2,7}. Here rules are labeled as follows:

  DPs:
    { 1: mult^#(@n, @l) -> c_1(mult#1^#(@l, @n))
    , 2: mult#1^#(::(@x, @xs), @n) -> c_2(*^#(@n, @x), mult^#(@n, @xs))
    , 3: mult#1^#(nil(), @n) -> c_3()
    , 4: *^#(@x, @y) -> c_4(#mult^#(@x, @y))
    , 5: dyade#1^#(::(@x, @xs), @l2) ->
         c_5(mult^#(@x, @l2), dyade^#(@xs, @l2))
    , 6: dyade#1^#(nil(), @l2) -> c_6()
    , 7: dyade^#(@l1, @l2) -> c_7(dyade#1^#(@l1, @l2))
    , 8: #mult^#(#pos(@x), #pos(@y)) -> c_10(#natmult^#(@x, @y))
    , 9: #mult^#(#pos(@x), #0()) -> c_11()
    , 10: #mult^#(#pos(@x), #neg(@y)) -> c_12(#natmult^#(@x, @y))
    , 11: #mult^#(#0(), #pos(@y)) -> c_13()
    , 12: #mult^#(#0(), #0()) -> c_14()
    , 13: #mult^#(#0(), #neg(@y)) -> c_15()
    , 14: #mult^#(#neg(@x), #pos(@y)) -> c_16(#natmult^#(@x, @y))
    , 15: #mult^#(#neg(@x), #0()) -> c_17()
    , 16: #mult^#(#neg(@x), #neg(@y)) -> c_18(#natmult^#(@x, @y))
    , 17: #natmult^#(#0(), @y) -> c_8()
    , 18: #natmult^#(#s(@x), @y) ->
          c_9(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y))
    , 19: #add^#(#pos(#s(#0())), @y) -> c_23(#succ^#(@y))
    , 20: #add^#(#pos(#s(#s(@x))), @y) ->
          c_24(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
    , 21: #add^#(#0(), @y) -> c_25()
    , 22: #add^#(#neg(#s(#0())), @y) -> c_26(#pred^#(@y))
    , 23: #add^#(#neg(#s(#s(@x))), @y) ->
          c_27(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
    , 24: #succ^#(#pos(#s(@x))) -> c_19()
    , 25: #succ^#(#0()) -> c_20()
    , 26: #succ^#(#neg(#s(#0()))) -> c_21()
    , 27: #succ^#(#neg(#s(#s(@x)))) -> c_22()
    , 28: #pred^#(#pos(#s(#0()))) -> c_28()
    , 29: #pred^#(#pos(#s(#s(@x)))) -> c_29()
    , 30: #pred^#(#0()) -> c_30()
    , 31: #pred^#(#neg(#s(@x))) -> c_31() }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict DPs:
  { mult^#(@n, @l) -> c_1(mult#1^#(@l, @n))
  , mult#1^#(::(@x, @xs), @n) -> c_2(*^#(@n, @x), mult^#(@n, @xs))
  , dyade#1^#(::(@x, @xs), @l2) ->
    c_5(mult^#(@x, @l2), dyade^#(@xs, @l2))
  , dyade^#(@l1, @l2) -> c_7(dyade#1^#(@l1, @l2)) }
Weak DPs:
  { mult#1^#(nil(), @n) -> c_3()
  , *^#(@x, @y) -> c_4(#mult^#(@x, @y))
  , #mult^#(#pos(@x), #pos(@y)) -> c_10(#natmult^#(@x, @y))
  , #mult^#(#pos(@x), #0()) -> c_11()
  , #mult^#(#pos(@x), #neg(@y)) -> c_12(#natmult^#(@x, @y))
  , #mult^#(#0(), #pos(@y)) -> c_13()
  , #mult^#(#0(), #0()) -> c_14()
  , #mult^#(#0(), #neg(@y)) -> c_15()
  , #mult^#(#neg(@x), #pos(@y)) -> c_16(#natmult^#(@x, @y))
  , #mult^#(#neg(@x), #0()) -> c_17()
  , #mult^#(#neg(@x), #neg(@y)) -> c_18(#natmult^#(@x, @y))
  , dyade#1^#(nil(), @l2) -> c_6()
  , #natmult^#(#0(), @y) -> c_8()
  , #natmult^#(#s(@x), @y) ->
    c_9(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y))
  , #add^#(#pos(#s(#0())), @y) -> c_23(#succ^#(@y))
  , #add^#(#pos(#s(#s(@x))), @y) ->
    c_24(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
  , #add^#(#0(), @y) -> c_25()
  , #add^#(#neg(#s(#0())), @y) -> c_26(#pred^#(@y))
  , #add^#(#neg(#s(#s(@x))), @y) ->
    c_27(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
  , #succ^#(#pos(#s(@x))) -> c_19()
  , #succ^#(#0()) -> c_20()
  , #succ^#(#neg(#s(#0()))) -> c_21()
  , #succ^#(#neg(#s(#s(@x)))) -> c_22()
  , #pred^#(#pos(#s(#0()))) -> c_28()
  , #pred^#(#pos(#s(#s(@x)))) -> c_29()
  , #pred^#(#0()) -> c_30()
  , #pred^#(#neg(#s(@x))) -> c_31() }
Weak Trs:
  { #natmult(#0(), @y) -> #0()
  , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y))
  , mult(@n, @l) -> mult#1(@l, @n)
  , mult#1(::(@x, @xs), @n) -> ::(*(@n, @x), mult(@n, @xs))
  , mult#1(nil(), @n) -> nil()
  , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y))
  , #mult(#pos(@x), #0()) -> #0()
  , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y))
  , #mult(#0(), #pos(@y)) -> #0()
  , #mult(#0(), #0()) -> #0()
  , #mult(#0(), #neg(@y)) -> #0()
  , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y))
  , #mult(#neg(@x), #0()) -> #0()
  , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y))
  , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
  , #succ(#0()) -> #pos(#s(#0()))
  , #succ(#neg(#s(#0()))) -> #0()
  , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
  , #add(#pos(#s(#0())), @y) -> #succ(@y)
  , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
  , #add(#0(), @y) -> @y
  , #add(#neg(#s(#0())), @y) -> #pred(@y)
  , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
  , *(@x, @y) -> #mult(@x, @y)
  , dyade#1(::(@x, @xs), @l2) -> ::(mult(@x, @l2), dyade(@xs, @l2))
  , dyade#1(nil(), @l2) -> nil()
  , #pred(#pos(#s(#0()))) -> #0()
  , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
  , #pred(#0()) -> #neg(#s(#0()))
  , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
  , dyade(@l1, @l2) -> dyade#1(@l1, @l2) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ mult#1^#(nil(), @n) -> c_3()
, *^#(@x, @y) -> c_4(#mult^#(@x, @y))
, #mult^#(#pos(@x), #pos(@y)) -> c_10(#natmult^#(@x, @y))
, #mult^#(#pos(@x), #0()) -> c_11()
, #mult^#(#pos(@x), #neg(@y)) -> c_12(#natmult^#(@x, @y))
, #mult^#(#0(), #pos(@y)) -> c_13()
, #mult^#(#0(), #0()) -> c_14()
, #mult^#(#0(), #neg(@y)) -> c_15()
, #mult^#(#neg(@x), #pos(@y)) -> c_16(#natmult^#(@x, @y))
, #mult^#(#neg(@x), #0()) -> c_17()
, #mult^#(#neg(@x), #neg(@y)) -> c_18(#natmult^#(@x, @y))
, dyade#1^#(nil(), @l2) -> c_6()
, #natmult^#(#0(), @y) -> c_8()
, #natmult^#(#s(@x), @y) ->
  c_9(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y))
, #add^#(#pos(#s(#0())), @y) -> c_23(#succ^#(@y))
, #add^#(#pos(#s(#s(@x))), @y) ->
  c_24(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, #add^#(#0(), @y) -> c_25()
, #add^#(#neg(#s(#0())), @y) -> c_26(#pred^#(@y))
, #add^#(#neg(#s(#s(@x))), @y) ->
  c_27(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, #succ^#(#pos(#s(@x))) -> c_19()
, #succ^#(#0()) -> c_20()
, #succ^#(#neg(#s(#0()))) -> c_21()
, #succ^#(#neg(#s(#s(@x)))) -> c_22()
, #pred^#(#pos(#s(#0()))) -> c_28()
, #pred^#(#pos(#s(#s(@x)))) -> c_29()
, #pred^#(#0()) -> c_30()
, #pred^#(#neg(#s(@x))) -> c_31() }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict DPs:
  { mult^#(@n, @l) -> c_1(mult#1^#(@l, @n))
  , mult#1^#(::(@x, @xs), @n) -> c_2(*^#(@n, @x), mult^#(@n, @xs))
  , dyade#1^#(::(@x, @xs), @l2) ->
    c_5(mult^#(@x, @l2), dyade^#(@xs, @l2))
  , dyade^#(@l1, @l2) -> c_7(dyade#1^#(@l1, @l2)) }
Weak Trs:
  { #natmult(#0(), @y) -> #0()
  , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y))
  , mult(@n, @l) -> mult#1(@l, @n)
  , mult#1(::(@x, @xs), @n) -> ::(*(@n, @x), mult(@n, @xs))
  , mult#1(nil(), @n) -> nil()
  , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y))
  , #mult(#pos(@x), #0()) -> #0()
  , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y))
  , #mult(#0(), #pos(@y)) -> #0()
  , #mult(#0(), #0()) -> #0()
  , #mult(#0(), #neg(@y)) -> #0()
  , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y))
  , #mult(#neg(@x), #0()) -> #0()
  , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y))
  , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
  , #succ(#0()) -> #pos(#s(#0()))
  , #succ(#neg(#s(#0()))) -> #0()
  , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
  , #add(#pos(#s(#0())), @y) -> #succ(@y)
  , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
  , #add(#0(), @y) -> @y
  , #add(#neg(#s(#0())), @y) -> #pred(@y)
  , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
  , *(@x, @y) -> #mult(@x, @y)
  , dyade#1(::(@x, @xs), @l2) -> ::(mult(@x, @l2), dyade(@xs, @l2))
  , dyade#1(nil(), @l2) -> nil()
  , #pred(#pos(#s(#0()))) -> #0()
  , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
  , #pred(#0()) -> #neg(#s(#0()))
  , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
  , dyade(@l1, @l2) -> dyade#1(@l1, @l2) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { mult#1^#(::(@x, @xs), @n) -> c_2(*^#(@n, @x), mult^#(@n, @xs)) }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict DPs:
  { mult^#(@n, @l) -> c_1(mult#1^#(@l, @n))
  , mult#1^#(::(@x, @xs), @n) -> c_2(mult^#(@n, @xs))
  , dyade#1^#(::(@x, @xs), @l2) ->
    c_3(mult^#(@x, @l2), dyade^#(@xs, @l2))
  , dyade^#(@l1, @l2) -> c_4(dyade#1^#(@l1, @l2)) }
Weak Trs:
  { #natmult(#0(), @y) -> #0()
  , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y))
  , mult(@n, @l) -> mult#1(@l, @n)
  , mult#1(::(@x, @xs), @n) -> ::(*(@n, @x), mult(@n, @xs))
  , mult#1(nil(), @n) -> nil()
  , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y))
  , #mult(#pos(@x), #0()) -> #0()
  , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y))
  , #mult(#0(), #pos(@y)) -> #0()
  , #mult(#0(), #0()) -> #0()
  , #mult(#0(), #neg(@y)) -> #0()
  , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y))
  , #mult(#neg(@x), #0()) -> #0()
  , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y))
  , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
  , #succ(#0()) -> #pos(#s(#0()))
  , #succ(#neg(#s(#0()))) -> #0()
  , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
  , #add(#pos(#s(#0())), @y) -> #succ(@y)
  , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
  , #add(#0(), @y) -> @y
  , #add(#neg(#s(#0())), @y) -> #pred(@y)
  , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
  , *(@x, @y) -> #mult(@x, @y)
  , dyade#1(::(@x, @xs), @l2) -> ::(mult(@x, @l2), dyade(@xs, @l2))
  , dyade#1(nil(), @l2) -> nil()
  , #pred(#pos(#s(#0()))) -> #0()
  , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
  , #pred(#0()) -> #neg(#s(#0()))
  , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
  , dyade(@l1, @l2) -> dyade#1(@l1, @l2) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

No rule is usable, rules are removed from the input problem.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict DPs:
  { mult^#(@n, @l) -> c_1(mult#1^#(@l, @n))
  , mult#1^#(::(@x, @xs), @n) -> c_2(mult^#(@n, @xs))
  , dyade#1^#(::(@x, @xs), @l2) ->
    c_3(mult^#(@x, @l2), dyade^#(@xs, @l2))
  , dyade^#(@l1, @l2) -> c_4(dyade#1^#(@l1, @l2)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

We use the processor 'custom shape polynomial interpretation' to
orient following rules strictly.

DPs:
  { 1: mult^#(@n, @l) -> c_1(mult#1^#(@l, @n)) }

Sub-proof:
----------
  The following argument positions are considered usable:
    Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1, 2},
    Uargs(c_4) = {1}
  TcT has computed the following constructor-restricted polynomial
  interpretation.
         [::](x1, x2) = 1 + x2                               
                                                             
     [mult^#](x1, x2) = 1 + x2                               
                                                             
   [mult#1^#](x1, x2) = x1                                   
                                                             
  [dyade#1^#](x1, x2) = 1 + x1 + 2*x1*x2 + 2*x1^2 + x2 + x2^2
                                                             
    [dyade^#](x1, x2) = 1 + x1 + 2*x1*x2 + 2*x1^2 + x2 + x2^2
                                                             
            [c_1](x1) = x1                                   
                                                             
            [c_2](x1) = x1                                   
                                                             
        [c_3](x1, x2) = 1 + 2*x1 + x2                        
                                                             
            [c_4](x1) = x1                                   
                                                             
  
  This order satisfies the following ordering constraints.
  
                 [mult^#(@n, @l)] =  1 + @l                                         
                                  >  @l                                             
                                  =  [c_1(mult#1^#(@l, @n))]                        
                                                                                    
      [mult#1^#(::(@x, @xs), @n)] =  1 + @xs                                        
                                  >= 1 + @xs                                        
                                  =  [c_2(mult^#(@n, @xs))]                         
                                                                                    
    [dyade#1^#(::(@x, @xs), @l2)] =  4 + 5*@xs + 3*@l2 + 2*@xs*@l2 + 2*@xs^2 + @l2^2
                                  >= 4 + 3*@l2 + @xs + 2*@xs*@l2 + 2*@xs^2 + @l2^2  
                                  =  [c_3(mult^#(@x, @l2), dyade^#(@xs, @l2))]      
                                                                                    
              [dyade^#(@l1, @l2)] =  1 + @l1 + 2*@l1*@l2 + 2*@l1^2 + @l2 + @l2^2    
                                  >= 1 + @l1 + 2*@l1*@l2 + 2*@l1^2 + @l2 + @l2^2    
                                  =  [c_4(dyade#1^#(@l1, @l2))]                     
                                                                                    

We return to the main proof. Consider the set of all dependency
pairs

:
  { 1: mult^#(@n, @l) -> c_1(mult#1^#(@l, @n))
  , 2: mult#1^#(::(@x, @xs), @n) -> c_2(mult^#(@n, @xs))
  , 3: dyade#1^#(::(@x, @xs), @l2) ->
       c_3(mult^#(@x, @l2), dyade^#(@xs, @l2))
  , 4: dyade^#(@l1, @l2) -> c_4(dyade#1^#(@l1, @l2)) }

Processor 'custom shape polynomial interpretation' induces the
complexity certificate YES(?,O(n^2)) on application of dependency
pairs {1}. These cover all (indirect) predecessors of dependency
pairs {1,2}, their number of application is equally bounded. The
dependency pairs are shifted into the weak component.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs:
  { dyade#1^#(::(@x, @xs), @l2) ->
    c_3(mult^#(@x, @l2), dyade^#(@xs, @l2))
  , dyade^#(@l1, @l2) -> c_4(dyade#1^#(@l1, @l2)) }
Weak DPs:
  { mult^#(@n, @l) -> c_1(mult#1^#(@l, @n))
  , mult#1^#(::(@x, @xs), @n) -> c_2(mult^#(@n, @xs)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ mult^#(@n, @l) -> c_1(mult#1^#(@l, @n))
, mult#1^#(::(@x, @xs), @n) -> c_2(mult^#(@n, @xs)) }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs:
  { dyade#1^#(::(@x, @xs), @l2) ->
    c_3(mult^#(@x, @l2), dyade^#(@xs, @l2))
  , dyade^#(@l1, @l2) -> c_4(dyade#1^#(@l1, @l2)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { dyade#1^#(::(@x, @xs), @l2) ->
    c_3(mult^#(@x, @l2), dyade^#(@xs, @l2)) }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs:
  { dyade#1^#(::(@x, @xs), @l2) -> c_1(dyade^#(@xs, @l2))
  , dyade^#(@l1, @l2) -> c_2(dyade#1^#(@l1, @l2)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.

DPs:
  { 1: dyade#1^#(::(@x, @xs), @l2) -> c_1(dyade^#(@xs, @l2))
  , 2: dyade^#(@l1, @l2) -> c_2(dyade#1^#(@l1, @l2)) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_1) = {1}, Uargs(c_2) = {1}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
           [::](x1, x2) = [1] x2 + [2]
                                      
       [mult^#](x1, x2) = [0]         
                                      
     [mult#1^#](x1, x2) = [0]         
                                      
    [dyade#1^#](x1, x2) = [4] x1 + [0]
                                      
      [dyade^#](x1, x2) = [4] x1 + [4]
                                      
              [c_1](x1) = [0]         
                                      
              [c_2](x1) = [0]         
                                      
          [c_3](x1, x2) = [0]         
                                      
              [c_4](x1) = [0]         
                                      
                    [c] = [0]         
                                      
              [c_1](x1) = [1] x1 + [3]
                                      
              [c_2](x1) = [1] x1 + [0]
  
  The order satisfies the following ordering constraints:
  
    [dyade#1^#(::(@x, @xs), @l2)] = [4] @xs + [8]             
                                  > [4] @xs + [7]             
                                  = [c_1(dyade^#(@xs, @l2))]  
                                                              
              [dyade^#(@l1, @l2)] = [4] @l1 + [4]             
                                  > [4] @l1 + [0]             
                                  = [c_2(dyade#1^#(@l1, @l2))]
                                                              

The strictly oriented rules are moved into the weak component.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Weak DPs:
  { dyade#1^#(::(@x, @xs), @l2) -> c_1(dyade^#(@xs, @l2))
  , dyade^#(@l1, @l2) -> c_2(dyade#1^#(@l1, @l2)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ dyade#1^#(::(@x, @xs), @l2) -> c_1(dyade^#(@xs, @l2))
, dyade^#(@l1, @l2) -> c_2(dyade#1^#(@l1, @l2)) }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Rules: Empty
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

Hurray, we answered YES(O(1),O(n^2))