- : unit = ()
- : unit = ()
h : heuristic = <heuristic>
- : unit = ()
APPLY CRITERIA (Marked dependency pairs)

TRS termination of:
[1] le(0,y) -> true
[2] le(s(x),0) -> false
[3] le(s(x),s(y)) -> le(x,y)
[4] minus(0,y) -> 0
[5] minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
[6] if_minus(true,s(x),y) -> 0
[7] if_minus(false,s(x),y) -> s(minus(x,y))
[8] quot(0,s(y)) -> 0
[9] quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
[10] log(s(0)) -> 0
[11] log(s(s(x))) -> s(log(s(quot(x,s(s(0))))))


Sub problem: 
guided: 

DP termination of:
<Marked_le(s(x),s(y)) , Marked_le(x,y)>
<Marked_minus(s(x),y) , Marked_if_minus(le(s(x),y),s(x),y)>
<Marked_minus(s(x),y) , Marked_le(s(x),y)>
<Marked_if_minus(false,s(x),y) , Marked_minus(x,y)>
<Marked_quot(s(x),s(y)) , Marked_quot(minus(x,y),s(y))>
<Marked_quot(s(x),s(y)) , Marked_minus(x,y)>
<Marked_log(s(s(x))) , Marked_log(s(quot(x,s(s(0)))))>
<Marked_log(s(s(x))) , Marked_quot(x,s(s(0)))>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 4 components:

{  <Marked_log(s(s(x))) , Marked_log(s(quot(x,s(s(0)))))> 
      --> <Marked_log(s(s(x))) , Marked_log(s(quot(x,s(s(0)))))>
}

{  <Marked_quot(s(x),s(y)) , Marked_quot(minus(x,y),s(y))> 
      --> <Marked_quot(s(x),s(y)) , Marked_quot(minus(x,y),s(y))>
}

{  <Marked_if_minus(false,s(x),y) , Marked_minus(x,y)> 
      --> <Marked_minus(s(x),y) , Marked_if_minus(le(s(x),y),s(x),y)>
  <Marked_minus(s(x),y) , Marked_if_minus(le(s(x),y),s(x),y)> 
     --> <Marked_if_minus(false,s(x),y) , Marked_minus(x,y)>
}

{  <Marked_le(s(x),s(y)) , Marked_le(x,y)> 
      --> <Marked_le(s(x),s(y)) , Marked_le(x,y)>
}
APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  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) >= if_minus(le(s(x),y),s(x),y) ;
  if_minus(true,s(x),y) >= 0 ;
  if_minus(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))) ;
  log(s(0)) >= 0 ;
  log(s(s(x))) >= s(log(s(quot(x,s(s(0)))))) ;
  Marked_log(s(s(x))) >= Marked_log(s(quot(x,s(s(0))))) ;
 }
 + Disjunctions:{ 
{
  Marked_log(s(s(x))) > Marked_log(s(quot(x,s(s(0))))) ;
 }
 } 
=== TIMER virtual : 10.000000 ===
Entering poly_solver
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 10.000000 ===
=== STOPING TIMER real ===
Sat solver returned
Sat solver result read
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
constraint: le(0,y) >= true
constraint: le(s(x),0) >= false
constraint: le(s(x),s(y)) >= le(x,y)
constraint: minus(0,y) >= 0
constraint: minus(s(x),y) >= if_minus(le(s(x),y),s(x),y)
constraint: if_minus(true,s(x),y) >= 0
constraint: if_minus(false,s(x),y) >= s(minus(x,y))
constraint: quot(0,s(y)) >= 0
constraint: quot(s(x),s(y)) >= s(quot(minus(x,y),s(y)))
constraint: log(s(0)) >= 0
constraint: log(s(s(x))) >= s(log(s(quot(x,s(s(0))))))
constraint: Marked_log(s(s(x))) >= Marked_log(s(quot(x,s(s(0)))))
APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  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) >= if_minus(le(s(x),y),s(x),y) ;
  if_minus(true,s(x),y) >= 0 ;
  if_minus(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))) ;
  log(s(0)) >= 0 ;
  log(s(s(x))) >= s(log(s(quot(x,s(s(0)))))) ;
  Marked_quot(s(x),s(y)) >= Marked_quot(minus(x,y),s(y)) ;
 }
 + Disjunctions:{ 
{
  Marked_quot(s(x),s(y)) > Marked_quot(minus(x,y),s(y)) ;
 }
 } 
=== TIMER virtual : 10.000000 ===
Entering poly_solver
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 10.000000 ===
=== STOPING TIMER real ===
Sat solver returned
Sat solver result read
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
constraint: le(0,y) >= true
constraint: le(s(x),0) >= false
constraint: le(s(x),s(y)) >= le(x,y)
constraint: minus(0,y) >= 0
constraint: minus(s(x),y) >= if_minus(le(s(x),y),s(x),y)
constraint: if_minus(true,s(x),y) >= 0
constraint: if_minus(false,s(x),y) >= s(minus(x,y))
constraint: quot(0,s(y)) >= 0
constraint: quot(s(x),s(y)) >= s(quot(minus(x,y),s(y)))
constraint: log(s(0)) >= 0
constraint: log(s(s(x))) >= s(log(s(quot(x,s(s(0))))))
constraint: Marked_quot(s(x),s(y)) >= Marked_quot(minus(x,y),s(y))
APPLY CRITERIA (Subterm criterion)

ST: Marked_if_minus -> 2
Marked_minus -> 1

APPLY CRITERIA (Subterm criterion)

ST: Marked_le -> 1

APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
SOLVED
{
 
TRS termination of:
[1] le(0,y) -> true
[2] le(s(x),0) -> false
[3] le(s(x),s(y)) -> le(x,y)
[4] minus(0,y) -> 0
[5] minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
[6] if_minus(true,s(x),y) -> 0
[7] if_minus(false,s(x),y) -> s(minus(x,y))
[8] quot(0,s(y)) -> 0
[9] quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
[10] log(s(0)) -> 0
[11] log(s(s(x))) -> s(log(s(quot(x,s(s(0))))))
 ,
 CRITERION: MDP
 [ {
      
     DP termination of:
     <Marked_le(s(x),s(y)) , Marked_le(x,y)>
<Marked_minus(s(x),y) , Marked_if_minus(le(s(x),y),s(x),y)>
<Marked_minus(s(x),y) , Marked_le(s(x),y)>
<Marked_if_minus(false,s(x),y) , Marked_minus(x,y)>
<Marked_quot(s(x),s(y)) , Marked_quot(minus(x,y),s(y))>
<Marked_quot(s(x),s(y)) , Marked_minus(x,y)>
<Marked_log(s(s(x))) , Marked_log(s(quot(x,s(s(0)))))>
<Marked_log(s(s(x))) , Marked_quot(x,s(s(0)))>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_log(s(s(x))) , Marked_log(s(quot(x,s(s(0)))))>
 ,
 CRITERION: ORD
 [ 
     Solution found:
     polynomial interpretation = 
      [ true ] () = 0; 
 [ log ] (X0) = 2*X0 + 0; 
 [ s ] (X0) = 2 + 2*X0 + 0; 
 [ 0 ] () = 0; 
 [ if_minus ] (X0,X1,X2) = 1*X1 + 0; 
 [ le ] (X0,X1) = 0; 
 [ Marked_log ] (X0) = 1*X0 + 0; 
 [ minus ] (X0,X1) = 1*X0 + 0; 
 [ false ] () = 0; 
 [ quot ] (X0,X1) = 1*X0 + 0; 
 ]} 
{
 
DP termination of:
<Marked_quot(s(x),s(y)) , Marked_quot(minus(x,y),s(y))>
 ,
 CRITERION: ORD
 [ 
     Solution found:
     polynomial interpretation = 
      [ true ] () = 0; 
 [ log ] (X0) = 1*X0 + 0; 
 [ s ] (X0) = 2 + 1*X0 + 0; 
 [ 0 ] () = 0; 
 [ Marked_quot ] (X0,X1) = 3*X0 + 0; 
 [ if_minus ] (X0,X1,X2) = 1*X1 + 0; 
 [ le ] (X0,X1) = 0; 
 [ minus ] (X0,X1) = 1*X0 + 0; 
 [ false ] () = 0; 
 [ quot ] (X0,X1) = 1*X0 + 0; 
 ]} 
{
 
DP termination of:
<Marked_minus(s(x),y) , Marked_if_minus(le(s(x),y),s(x),y)>
<Marked_if_minus(false,s(x),y) , Marked_minus(x,y)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
     <Marked_minus(s(x),y) , Marked_if_minus(le(s(x),y),s(x),y)>
 ,
 CRITERION: SG
 [  ]} 
 ]} 
{
 
DP termination of:
<Marked_le(s(x),s(y)) , Marked_le(x,y)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 

Cime worked for 0.239271 seconds (real time)
Cime Exit Status: 0