MAYBE

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict Trs:
  { le(0(), Y) -> true()
  , le(s(X), 0()) -> false()
  , le(s(X), s(Y)) -> le(X, Y)
  , minus(0(), Y) -> 0()
  , minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
  , ifMinus(true(), s(X), Y) -> 0()
  , ifMinus(false(), s(X), Y) -> s(minus(X, Y))
  , quot(0(), s(Y)) -> 0()
  , quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { le^#(0(), Y) -> c_1()
  , le^#(s(X), 0()) -> c_2()
  , le^#(s(X), s(Y)) -> c_3(le^#(X, Y))
  , minus^#(0(), Y) -> c_4()
  , minus^#(s(X), Y) ->
    c_5(ifMinus^#(le(s(X), Y), s(X), Y), le^#(s(X), Y))
  , ifMinus^#(true(), s(X), Y) -> c_6()
  , ifMinus^#(false(), s(X), Y) -> c_7(minus^#(X, Y))
  , quot^#(0(), s(Y)) -> c_8()
  , quot^#(s(X), s(Y)) ->
    c_9(quot^#(minus(X, Y), s(Y)), minus^#(X, Y)) }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { le^#(0(), Y) -> c_1()
  , le^#(s(X), 0()) -> c_2()
  , le^#(s(X), s(Y)) -> c_3(le^#(X, Y))
  , minus^#(0(), Y) -> c_4()
  , minus^#(s(X), Y) ->
    c_5(ifMinus^#(le(s(X), Y), s(X), Y), le^#(s(X), Y))
  , ifMinus^#(true(), s(X), Y) -> c_6()
  , ifMinus^#(false(), s(X), Y) -> c_7(minus^#(X, Y))
  , quot^#(0(), s(Y)) -> c_8()
  , quot^#(s(X), s(Y)) ->
    c_9(quot^#(minus(X, Y), s(Y)), minus^#(X, Y)) }
Weak Trs:
  { le(0(), Y) -> true()
  , le(s(X), 0()) -> false()
  , le(s(X), s(Y)) -> le(X, Y)
  , minus(0(), Y) -> 0()
  , minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
  , ifMinus(true(), s(X), Y) -> 0()
  , ifMinus(false(), s(X), Y) -> s(minus(X, Y))
  , quot(0(), s(Y)) -> 0()
  , quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,4,6,8} by
applications of Pre({1,2,4,6,8}) = {3,5,7,9}. Here rules are
labeled as follows:

  DPs:
    { 1: le^#(0(), Y) -> c_1()
    , 2: le^#(s(X), 0()) -> c_2()
    , 3: le^#(s(X), s(Y)) -> c_3(le^#(X, Y))
    , 4: minus^#(0(), Y) -> c_4()
    , 5: minus^#(s(X), Y) ->
         c_5(ifMinus^#(le(s(X), Y), s(X), Y), le^#(s(X), Y))
    , 6: ifMinus^#(true(), s(X), Y) -> c_6()
    , 7: ifMinus^#(false(), s(X), Y) -> c_7(minus^#(X, Y))
    , 8: quot^#(0(), s(Y)) -> c_8()
    , 9: quot^#(s(X), s(Y)) ->
         c_9(quot^#(minus(X, Y), s(Y)), minus^#(X, Y)) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { le^#(s(X), s(Y)) -> c_3(le^#(X, Y))
  , minus^#(s(X), Y) ->
    c_5(ifMinus^#(le(s(X), Y), s(X), Y), le^#(s(X), Y))
  , ifMinus^#(false(), s(X), Y) -> c_7(minus^#(X, Y))
  , quot^#(s(X), s(Y)) ->
    c_9(quot^#(minus(X, Y), s(Y)), minus^#(X, Y)) }
Weak DPs:
  { le^#(0(), Y) -> c_1()
  , le^#(s(X), 0()) -> c_2()
  , minus^#(0(), Y) -> c_4()
  , ifMinus^#(true(), s(X), Y) -> c_6()
  , quot^#(0(), s(Y)) -> c_8() }
Weak Trs:
  { le(0(), Y) -> true()
  , le(s(X), 0()) -> false()
  , le(s(X), s(Y)) -> le(X, Y)
  , minus(0(), Y) -> 0()
  , minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
  , ifMinus(true(), s(X), Y) -> 0()
  , ifMinus(false(), s(X), Y) -> s(minus(X, Y))
  , quot(0(), s(Y)) -> 0()
  , quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

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

{ le^#(0(), Y) -> c_1()
, le^#(s(X), 0()) -> c_2()
, minus^#(0(), Y) -> c_4()
, ifMinus^#(true(), s(X), Y) -> c_6()
, quot^#(0(), s(Y)) -> c_8() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { le^#(s(X), s(Y)) -> c_3(le^#(X, Y))
  , minus^#(s(X), Y) ->
    c_5(ifMinus^#(le(s(X), Y), s(X), Y), le^#(s(X), Y))
  , ifMinus^#(false(), s(X), Y) -> c_7(minus^#(X, Y))
  , quot^#(s(X), s(Y)) ->
    c_9(quot^#(minus(X, Y), s(Y)), minus^#(X, Y)) }
Weak Trs:
  { le(0(), Y) -> true()
  , le(s(X), 0()) -> false()
  , le(s(X), s(Y)) -> le(X, Y)
  , minus(0(), Y) -> 0()
  , minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
  , ifMinus(true(), s(X), Y) -> 0()
  , ifMinus(false(), s(X), Y) -> s(minus(X, Y))
  , quot(0(), s(Y)) -> 0()
  , quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { le(0(), Y) -> true()
    , le(s(X), 0()) -> false()
    , le(s(X), s(Y)) -> le(X, Y)
    , minus(0(), Y) -> 0()
    , minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
    , ifMinus(true(), s(X), Y) -> 0()
    , ifMinus(false(), s(X), Y) -> s(minus(X, Y)) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { le^#(s(X), s(Y)) -> c_3(le^#(X, Y))
  , minus^#(s(X), Y) ->
    c_5(ifMinus^#(le(s(X), Y), s(X), Y), le^#(s(X), Y))
  , ifMinus^#(false(), s(X), Y) -> c_7(minus^#(X, Y))
  , quot^#(s(X), s(Y)) ->
    c_9(quot^#(minus(X, Y), s(Y)), minus^#(X, Y)) }
Weak Trs:
  { le(0(), Y) -> true()
  , le(s(X), 0()) -> false()
  , le(s(X), s(Y)) -> le(X, Y)
  , minus(0(), Y) -> 0()
  , minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
  , ifMinus(true(), s(X), Y) -> 0()
  , ifMinus(false(), s(X), Y) -> s(minus(X, Y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

None of the processors succeeded.

Details of failed attempt(s):
-----------------------------
1) 'matrices' failed due to the following reason:
   
   None of the processors succeeded.
   
   Details of failed attempt(s):
   -----------------------------
   1) 'matrix interpretation of dimension 4' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   2) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   3) 'matrix interpretation of dimension 3' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   4) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   5) 'matrix interpretation of dimension 2' failed due to the
      following reason:
      
      The input cannot be shown compatible
   
   6) 'matrix interpretation of dimension 1' failed due to the
      following reason:
      
      The input cannot be shown compatible
   

2) 'empty' failed due to the following reason:
   
   Empty strict component of the problem is NOT empty.


Arrrr..