- : 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] mod(0,y) -> 0
[9] mod(s(x),0) -> 0
[10] mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
[11] if_mod(true,s(x),s(y)) -> mod(minus(x,y),s(y))
[12] if_mod(false,s(x),s(y)) -> s(x)


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_mod(s(x),s(y)) , Marked_if_mod(le(y,x),s(x),s(y))>
<Marked_mod(s(x),s(y)) , Marked_le(y,x)>
<Marked_if_mod(true,s(x),s(y)) , Marked_mod(minus(x,y),s(y))>
<Marked_if_mod(true,s(x),s(y)) , Marked_minus(x,y)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 3 components:

{  <Marked_if_mod(true,s(x),s(y)) , Marked_mod(minus(x,y),s(y))> 
      --> <Marked_mod(s(x),s(y)) , Marked_if_mod(le(y,x),s(x),s(y))>
  <Marked_mod(s(x),s(y)) , Marked_if_mod(le(y,x),s(x),s(y))> 
     --> <Marked_if_mod(true,s(x),s(y)) , Marked_mod(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)) ;
  mod(0,y) >= 0 ;
  mod(s(x),0) >= 0 ;
  mod(s(x),s(y)) >= if_mod(le(y,x),s(x),s(y)) ;
  if_mod(true,s(x),s(y)) >= mod(minus(x,y),s(y)) ;
  if_mod(false,s(x),s(y)) >= s(x) ;
  Marked_if_mod(true,s(x),s(y)) >= Marked_mod(minus(x,y),s(y)) ;
  Marked_mod(s(x),s(y)) >= Marked_if_mod(le(y,x),s(x),s(y)) ;
 }
 + Disjunctions:{ 
{
  Marked_if_mod(true,s(x),s(y)) > Marked_mod(minus(x,y),s(y)) ;
 }
{
  Marked_mod(s(x),s(y)) > Marked_if_mod(le(y,x),s(x),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: mod(0,y) >= 0
constraint: mod(s(x),0) >= 0
constraint: mod(s(x),s(y)) >= if_mod(le(y,x),s(x),s(y))
constraint: if_mod(true,s(x),s(y)) >= mod(minus(x,y),s(y))
constraint: if_mod(false,s(x),s(y)) >= s(x)
constraint: Marked_if_mod(true,s(x),s(y)) >= Marked_mod(minus(x,y),s(y))
constraint: Marked_mod(s(x),s(y)) >= Marked_if_mod(le(y,x),s(x),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:
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] mod(0,y) -> 0
[9] mod(s(x),0) -> 0
[10] mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
[11] if_mod(true,s(x),s(y)) -> mod(minus(x,y),s(y))
[12] if_mod(false,s(x),s(y)) -> s(x)
 ,
 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_mod(s(x),s(y)) , Marked_if_mod(le(y,x),s(x),s(y))>
<Marked_mod(s(x),s(y)) , Marked_le(y,x)>
<Marked_if_mod(true,s(x),s(y)) , Marked_mod(minus(x,y),s(y))>
<Marked_if_mod(true,s(x),s(y)) , Marked_minus(x,y)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_mod(s(x),s(y)) , Marked_if_mod(le(y,x),s(x),s(y))>
<Marked_if_mod(true,s(x),s(y)) , Marked_mod(minus(x,y),s(y))>
 ,
 CRITERION: CG using polynomial interpretation = 
 [ true ] () = 0; 
 [ if_mod ] (X0,X1,X2) = 1*X1; 
 [ s ] (X0) = 2*X0 + 2; 
 [ 0 ] () = 0; 
 [ Marked_mod ] (X0,X1) = 2*X0; 
 [ if_minus ] (X0,X1,X2) = 1*X1; 
 [ le ] (X0,X1) = 0; 
 [ Marked_if_mod ] (X0,X1,X2) = 2*X1; 
 [ minus ] (X0,X1) = 1*X0; 
 [ false ] () = 0; 
 [ mod ] (X0,X1) = 1*X0 + 1; 
 removing <Marked_if_mod(true,s(x),s(y)),Marked_mod(minus(x,y),s(y))>
 [ {
      
     DP termination of:
     <Marked_mod(s(x),s(y)) , Marked_if_mod(le(y,x),s(x),s(y))>
 ,
 CRITERION: SG
 [  ]} 
 ]} 
{
 
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 0.091122 seconds (real time)
Cime Exit Status: 0