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

TRS termination of:
[1] eq(0,0) -> true
[2] eq(0,s(x)) -> false
[3] eq(s(x),0) -> false
[4] eq(s(x),s(y)) -> eq(x,y)
[5] le(0,y) -> true
[6] le(s(x),0) -> false
[7] le(s(x),s(y)) -> le(x,y)
[8] app(nil,y) -> y
[9] app(add(n,x),y) -> add(n,app(x,y))
[10] min(nil) -> 0
[11] min(add(n,x)) -> minIter(add(n,x),add(n,x),0)
[12] minIter(nil,add(n,y),m) -> minIter(add(n,y),add(n,y),s(m))
[13] minIter(add(n,x),y,m) -> if_min(le(n,m),x,y,m)
[14] if_min(true,x,y,m) -> m
[15] if_min(false,x,y,m) -> minIter(x,y,m)
[16] head(add(n,x)) -> n
[17] tail(add(n,x)) -> x
[18] tail(nil) -> nil
[19] null(nil) -> true
[20] null(add(n,x)) -> false
[21] rm(n,nil) -> nil
[22] rm(n,add(m,x)) -> if_rm(eq(n,m),n,add(m,x))
[23] if_rm(true,n,add(m,x)) -> rm(n,x)
[24] if_rm(false,n,add(m,x)) -> add(m,rm(n,x))
[25] minsort(nil,nil) -> nil
[26] minsort(add(n,x),y) -> if_minsort(eq(n,min(add(n,x))),add(n,x),y)
[27] if_minsort(true,add(n,x),y) -> add(n,minsort(app(rm(n,x),y),nil))
[28] if_minsort(false,add(n,x),y) -> minsort(x,add(n,y))


Sub problem: 
guided: 

DP termination of:
<Marked_eq(s(x),s(y)) , Marked_eq(x,y)>
<Marked_le(s(x),s(y)) , Marked_le(x,y)>
<Marked_app(add(n,x),y) , Marked_app(x,y)>
<Marked_min(add(n,x)) , Marked_minIter(add(n,x),add(n,x),0)>
<Marked_minIter(nil,add(n,y),m) , Marked_minIter(add(n,y),add(n,y),s(m))>
<Marked_minIter(add(n,x),y,m) , Marked_if_min(le(n,m),x,y,m)>
<Marked_minIter(add(n,x),y,m) , Marked_le(n,m)>
<Marked_if_min(false,x,y,m) , Marked_minIter(x,y,m)>
<Marked_rm(n,add(m,x)) , Marked_if_rm(eq(n,m),n,add(m,x))>
<Marked_rm(n,add(m,x)) , Marked_eq(n,m)>
<Marked_if_rm(true,n,add(m,x)) , Marked_rm(n,x)>
<Marked_if_rm(false,n,add(m,x)) , Marked_rm(n,x)>
<Marked_minsort(add(n,x),y) , Marked_if_minsort(eq(n,min(add(n,x))),add(n,x),y)>
<Marked_minsort(add(n,x),y) , Marked_eq(n,min(add(n,x)))>
<Marked_minsort(add(n,x),y) , Marked_min(add(n,x))>
<Marked_if_minsort(true,add(n,x),y) , Marked_minsort(app(rm(n,x),y),nil)>
<Marked_if_minsort(true,add(n,x),y) , Marked_app(rm(n,x),y)>
<Marked_if_minsort(true,add(n,x),y) , Marked_rm(n,x)>
<Marked_if_minsort(false,add(n,x),y) , Marked_minsort(x,add(n,y))>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 6 components:

{  <Marked_if_minsort(false,add(n,x),y) , Marked_minsort(x,add(n,y))> 
      --> <Marked_minsort(add(n,x),y) , Marked_if_minsort(eq(n,min(add(n,x))),
                                         add(n,x),y)>
  <Marked_if_minsort(true,add(n,x),y) , Marked_minsort(app(rm(n,x),y),nil)> 
     --> <Marked_minsort(add(n,x),y) , Marked_if_minsort(eq(n,min(add(n,x))),
                                        add(n,x),y)>
  <Marked_minsort(add(n,x),y) , Marked_if_minsort(eq(n,min(add(n,x))),
                                 add(n,x),y)> 
     --> <Marked_if_minsort(false,add(n,x),y) , Marked_minsort(x,add(n,y))>
  <Marked_minsort(add(n,x),y) , Marked_if_minsort(eq(n,min(add(n,x))),
                                 add(n,x),y)> 
     --> <Marked_if_minsort(true,add(n,x),y) , Marked_minsort(app(rm(n,x),y),
                                                nil)>
}

{  <Marked_app(add(n,x),y) , Marked_app(x,y)> 
      --> <Marked_app(add(n,x),y) , Marked_app(x,y)>
}

{  <Marked_if_rm(false,n,add(m,x)) , Marked_rm(n,x)> 
      --> <Marked_rm(n,add(m,x)) , Marked_if_rm(eq(n,m),n,add(m,x))>
  <Marked_if_rm(true,n,add(m,x)) , Marked_rm(n,x)> 
     --> <Marked_rm(n,add(m,x)) , Marked_if_rm(eq(n,m),n,add(m,x))>
  <Marked_rm(n,add(m,x)) , Marked_if_rm(eq(n,m),n,add(m,x))> 
     --> <Marked_if_rm(false,n,add(m,x)) , Marked_rm(n,x)>
  <Marked_rm(n,add(m,x)) , Marked_if_rm(eq(n,m),n,add(m,x))> 
     --> <Marked_if_rm(true,n,add(m,x)) , Marked_rm(n,x)>
}

{  <Marked_eq(s(x),s(y)) , Marked_eq(x,y)> 
      --> <Marked_eq(s(x),s(y)) , Marked_eq(x,y)>
}

{  <Marked_if_min(false,x,y,m) , Marked_minIter(x,y,m)> 
      --> <Marked_minIter(add(n,x),y,m) , Marked_if_min(le(n,m),x,y,m)>
  <Marked_if_min(false,x,y,m) , Marked_minIter(x,y,m)> 
     --> <Marked_minIter(nil,add(n,y),m) , Marked_minIter(add(n,y),add(n,y),
                                            s(m))>
  <Marked_minIter(add(n,x),y,m) , Marked_if_min(le(n,m),x,y,m)> 
     --> <Marked_if_min(false,x,y,m) , Marked_minIter(x,y,m)>
  <Marked_minIter(nil,add(n,y),m) , Marked_minIter(add(n,y),add(n,y),s(m))> 
     --> <Marked_minIter(add(n,x),y,m) , Marked_if_min(le(n,m),x,y,m)>
}

{  <Marked_le(s(x),s(y)) , Marked_le(x,y)> 
      --> <Marked_le(s(x),s(y)) , Marked_le(x,y)>
}
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  eq(0,0) >= true ;
  eq(0,s(x)) >= false ;
  eq(s(x),0) >= false ;
  eq(s(x),s(y)) >= eq(x,y) ;
  le(0,y) >= true ;
  le(s(x),0) >= false ;
  le(s(x),s(y)) >= le(x,y) ;
  app(nil,y) >= y ;
  app(add(n,x),y) >= add(n,app(x,y)) ;
  min(nil) >= 0 ;
  min(add(n,x)) >= minIter(add(n,x),add(n,x),0) ;
  minIter(nil,add(n,y),m) >= minIter(add(n,y),add(n,y),s(m)) ;
  minIter(add(n,x),y,m) >= if_min(le(n,m),x,y,m) ;
  if_min(true,x,y,m) >= m ;
  if_min(false,x,y,m) >= minIter(x,y,m) ;
  head(add(n,x)) >= n ;
  tail(nil) >= nil ;
  tail(add(n,x)) >= x ;
  null(nil) >= true ;
  null(add(n,x)) >= false ;
  rm(n,nil) >= nil ;
  rm(n,add(m,x)) >= if_rm(eq(n,m),n,add(m,x)) ;
  if_rm(true,n,add(m,x)) >= rm(n,x) ;
  if_rm(false,n,add(m,x)) >= add(m,rm(n,x)) ;
  minsort(nil,nil) >= nil ;
  minsort(add(n,x),y) >= if_minsort(eq(n,min(add(n,x))),add(n,x),y) ;
  if_minsort(true,add(n,x),y) >= add(n,minsort(app(rm(n,x),y),nil)) ;
  if_minsort(false,add(n,x),y) >= minsort(x,add(n,y)) ;
  Marked_if_minsort(true,add(n,x),y) >= Marked_minsort(app(rm(n,x),y),nil) ;
  Marked_if_minsort(false,add(n,x),y) >= Marked_minsort(x,add(n,y)) ;
  Marked_minsort(add(n,x),y) >= Marked_if_minsort(eq(n,min(add(n,x))),
                                 add(n,x),y) ;
 }
 + Disjunctions:{ 
{
  Marked_if_minsort(true,add(n,x),y) > Marked_minsort(app(rm(n,x),y),nil) ;
 }
{
  Marked_if_minsort(false,add(n,x),y) > Marked_minsort(x,add(n,y)) ;
 }
{
  Marked_minsort(add(n,x),y) > Marked_if_minsort(eq(n,min(add(n,x))),
                                add(n,x),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: eq(0,0) >= true
constraint: eq(0,s(x)) >= false
constraint: eq(s(x),0) >= false
constraint: eq(s(x),s(y)) >= eq(x,y)
constraint: le(0,y) >= true
constraint: le(s(x),0) >= false
constraint: le(s(x),s(y)) >= le(x,y)
constraint: app(nil,y) >= y
constraint: app(add(n,x),y) >= add(n,app(x,y))
constraint: min(nil) >= 0
constraint: min(add(n,x)) >= minIter(add(n,x),add(n,x),0)
constraint: minIter(nil,add(n,y),m) >= minIter(add(n,y),add(n,y),s(m))
constraint: minIter(add(n,x),y,m) >= if_min(le(n,m),x,y,m)
constraint: if_min(true,x,y,m) >= m
constraint: if_min(false,x,y,m) >= minIter(x,y,m)
constraint: head(add(n,x)) >= n
constraint: tail(nil) >= nil
constraint: tail(add(n,x)) >= x
constraint: null(nil) >= true
constraint: null(add(n,x)) >= false
constraint: rm(n,nil) >= nil
constraint: rm(n,add(m,x)) >= if_rm(eq(n,m),n,add(m,x))
constraint: if_rm(true,n,add(m,x)) >= rm(n,x)
constraint: if_rm(false,n,add(m,x)) >= add(m,rm(n,x))
constraint: minsort(nil,nil) >= nil
constraint: minsort(add(n,x),y) >= if_minsort(eq(n,min(add(n,x))),add(n,x),y)
constraint: if_minsort(true,add(n,x),y) >= add(n,minsort(app(rm(n,x),y),nil))
constraint: if_minsort(false,add(n,x),y) >= minsort(x,add(n,y))
constraint: Marked_if_minsort(true,add(n,x),y) >= Marked_minsort(app(rm(n,x),y),
                                                   nil)
constraint: Marked_if_minsort(false,add(n,x),y) >= Marked_minsort(x,add(n,y))
constraint: Marked_minsort(add(n,x),y) >= Marked_if_minsort(eq(n,min(add(n,x))),
                                           add(n,x),y)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  eq(0,0) >= true ;
  eq(0,s(x)) >= false ;
  eq(s(x),0) >= false ;
  eq(s(x),s(y)) >= eq(x,y) ;
  le(0,y) >= true ;
  le(s(x),0) >= false ;
  le(s(x),s(y)) >= le(x,y) ;
  app(nil,y) >= y ;
  app(add(n,x),y) >= add(n,app(x,y)) ;
  min(nil) >= 0 ;
  min(add(n,x)) >= minIter(add(n,x),add(n,x),0) ;
  minIter(nil,add(n,y),m) >= minIter(add(n,y),add(n,y),s(m)) ;
  minIter(add(n,x),y,m) >= if_min(le(n,m),x,y,m) ;
  if_min(true,x,y,m) >= m ;
  if_min(false,x,y,m) >= minIter(x,y,m) ;
  head(add(n,x)) >= n ;
  tail(nil) >= nil ;
  tail(add(n,x)) >= x ;
  null(nil) >= true ;
  null(add(n,x)) >= false ;
  rm(n,nil) >= nil ;
  rm(n,add(m,x)) >= if_rm(eq(n,m),n,add(m,x)) ;
  if_rm(true,n,add(m,x)) >= rm(n,x) ;
  if_rm(false,n,add(m,x)) >= add(m,rm(n,x)) ;
  minsort(nil,nil) >= nil ;
  minsort(add(n,x),y) >= if_minsort(eq(n,min(add(n,x))),add(n,x),y) ;
  if_minsort(true,add(n,x),y) >= add(n,minsort(app(rm(n,x),y),nil)) ;
  if_minsort(false,add(n,x),y) >= minsort(x,add(n,y)) ;
  Marked_app(add(n,x),y) >= Marked_app(x,y) ;
 }
 + Disjunctions:{ 
{
  Marked_app(add(n,x),y) > Marked_app(x,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: eq(0,0) >= true
constraint: eq(0,s(x)) >= false
constraint: eq(s(x),0) >= false
constraint: eq(s(x),s(y)) >= eq(x,y)
constraint: le(0,y) >= true
constraint: le(s(x),0) >= false
constraint: le(s(x),s(y)) >= le(x,y)
constraint: app(nil,y) >= y
constraint: app(add(n,x),y) >= add(n,app(x,y))
constraint: min(nil) >= 0
constraint: min(add(n,x)) >= minIter(add(n,x),add(n,x),0)
constraint: minIter(nil,add(n,y),m) >= minIter(add(n,y),add(n,y),s(m))
constraint: minIter(add(n,x),y,m) >= if_min(le(n,m),x,y,m)
constraint: if_min(true,x,y,m) >= m
constraint: if_min(false,x,y,m) >= minIter(x,y,m)
constraint: head(add(n,x)) >= n
constraint: tail(nil) >= nil
constraint: tail(add(n,x)) >= x
constraint: null(nil) >= true
constraint: null(add(n,x)) >= false
constraint: rm(n,nil) >= nil
constraint: rm(n,add(m,x)) >= if_rm(eq(n,m),n,add(m,x))
constraint: if_rm(true,n,add(m,x)) >= rm(n,x)
constraint: if_rm(false,n,add(m,x)) >= add(m,rm(n,x))
constraint: minsort(nil,nil) >= nil
constraint: minsort(add(n,x),y) >= if_minsort(eq(n,min(add(n,x))),add(n,x),y)
constraint: if_minsort(true,add(n,x),y) >= add(n,minsort(app(rm(n,x),y),nil))
constraint: if_minsort(false,add(n,x),y) >= minsort(x,add(n,y))
constraint: Marked_app(add(n,x),y) >= Marked_app(x,y)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  eq(0,0) >= true ;
  eq(0,s(x)) >= false ;
  eq(s(x),0) >= false ;
  eq(s(x),s(y)) >= eq(x,y) ;
  le(0,y) >= true ;
  le(s(x),0) >= false ;
  le(s(x),s(y)) >= le(x,y) ;
  app(nil,y) >= y ;
  app(add(n,x),y) >= add(n,app(x,y)) ;
  min(nil) >= 0 ;
  min(add(n,x)) >= minIter(add(n,x),add(n,x),0) ;
  minIter(nil,add(n,y),m) >= minIter(add(n,y),add(n,y),s(m)) ;
  minIter(add(n,x),y,m) >= if_min(le(n,m),x,y,m) ;
  if_min(true,x,y,m) >= m ;
  if_min(false,x,y,m) >= minIter(x,y,m) ;
  head(add(n,x)) >= n ;
  tail(nil) >= nil ;
  tail(add(n,x)) >= x ;
  null(nil) >= true ;
  null(add(n,x)) >= false ;
  rm(n,nil) >= nil ;
  rm(n,add(m,x)) >= if_rm(eq(n,m),n,add(m,x)) ;
  if_rm(true,n,add(m,x)) >= rm(n,x) ;
  if_rm(false,n,add(m,x)) >= add(m,rm(n,x)) ;
  minsort(nil,nil) >= nil ;
  minsort(add(n,x),y) >= if_minsort(eq(n,min(add(n,x))),add(n,x),y) ;
  if_minsort(true,add(n,x),y) >= add(n,minsort(app(rm(n,x),y),nil)) ;
  if_minsort(false,add(n,x),y) >= minsort(x,add(n,y)) ;
  Marked_rm(n,add(m,x)) >= Marked_if_rm(eq(n,m),n,add(m,x)) ;
  Marked_if_rm(true,n,add(m,x)) >= Marked_rm(n,x) ;
  Marked_if_rm(false,n,add(m,x)) >= Marked_rm(n,x) ;
 }
 + Disjunctions:{ 
{
  Marked_rm(n,add(m,x)) > Marked_if_rm(eq(n,m),n,add(m,x)) ;
 }
{
  Marked_if_rm(true,n,add(m,x)) > Marked_rm(n,x) ;
 }
{
  Marked_if_rm(false,n,add(m,x)) > Marked_rm(n,x) ;
 }
 } 
=== 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: eq(0,0) >= true
constraint: eq(0,s(x)) >= false
constraint: eq(s(x),0) >= false
constraint: eq(s(x),s(y)) >= eq(x,y)
constraint: le(0,y) >= true
constraint: le(s(x),0) >= false
constraint: le(s(x),s(y)) >= le(x,y)
constraint: app(nil,y) >= y
constraint: app(add(n,x),y) >= add(n,app(x,y))
constraint: min(nil) >= 0
constraint: min(add(n,x)) >= minIter(add(n,x),add(n,x),0)
constraint: minIter(nil,add(n,y),m) >= minIter(add(n,y),add(n,y),s(m))
constraint: minIter(add(n,x),y,m) >= if_min(le(n,m),x,y,m)
constraint: if_min(true,x,y,m) >= m
constraint: if_min(false,x,y,m) >= minIter(x,y,m)
constraint: head(add(n,x)) >= n
constraint: tail(nil) >= nil
constraint: tail(add(n,x)) >= x
constraint: null(nil) >= true
constraint: null(add(n,x)) >= false
constraint: rm(n,nil) >= nil
constraint: rm(n,add(m,x)) >= if_rm(eq(n,m),n,add(m,x))
constraint: if_rm(true,n,add(m,x)) >= rm(n,x)
constraint: if_rm(false,n,add(m,x)) >= add(m,rm(n,x))
constraint: minsort(nil,nil) >= nil
constraint: minsort(add(n,x),y) >= if_minsort(eq(n,min(add(n,x))),add(n,x),y)
constraint: if_minsort(true,add(n,x),y) >= add(n,minsort(app(rm(n,x),y),nil))
constraint: if_minsort(false,add(n,x),y) >= minsort(x,add(n,y))
constraint: Marked_rm(n,add(m,x)) >= Marked_if_rm(eq(n,m),n,add(m,x))
constraint: Marked_if_rm(true,n,add(m,x)) >= Marked_rm(n,x)
constraint: Marked_if_rm(false,n,add(m,x)) >= Marked_rm(n,x)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  eq(0,0) >= true ;
  eq(0,s(x)) >= false ;
  eq(s(x),0) >= false ;
  eq(s(x),s(y)) >= eq(x,y) ;
  le(0,y) >= true ;
  le(s(x),0) >= false ;
  le(s(x),s(y)) >= le(x,y) ;
  app(nil,y) >= y ;
  app(add(n,x),y) >= add(n,app(x,y)) ;
  min(nil) >= 0 ;
  min(add(n,x)) >= minIter(add(n,x),add(n,x),0) ;
  minIter(nil,add(n,y),m) >= minIter(add(n,y),add(n,y),s(m)) ;
  minIter(add(n,x),y,m) >= if_min(le(n,m),x,y,m) ;
  if_min(true,x,y,m) >= m ;
  if_min(false,x,y,m) >= minIter(x,y,m) ;
  head(add(n,x)) >= n ;
  tail(nil) >= nil ;
  tail(add(n,x)) >= x ;
  null(nil) >= true ;
  null(add(n,x)) >= false ;
  rm(n,nil) >= nil ;
  rm(n,add(m,x)) >= if_rm(eq(n,m),n,add(m,x)) ;
  if_rm(true,n,add(m,x)) >= rm(n,x) ;
  if_rm(false,n,add(m,x)) >= add(m,rm(n,x)) ;
  minsort(nil,nil) >= nil ;
  minsort(add(n,x),y) >= if_minsort(eq(n,min(add(n,x))),add(n,x),y) ;
  if_minsort(true,add(n,x),y) >= add(n,minsort(app(rm(n,x),y),nil)) ;
  if_minsort(false,add(n,x),y) >= minsort(x,add(n,y)) ;
  Marked_eq(s(x),s(y)) >= Marked_eq(x,y) ;
 }
 + Disjunctions:{ 
{
  Marked_eq(s(x),s(y)) > Marked_eq(x,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
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found for these parameters.
Entering rpo_solver
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
Time out for these parameters.
=== TIMER virtual : 15.000000 ===
Entering poly_solver
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 15.000000 ===
=== STOPING TIMER real ===
Sat solver returned
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found for these parameters.
=== TIMER virtual : 50.000000 ===
trying sub matrices of size: 1
Matrix interpretation constraints generated.
Search parameters: LINEAR MATRIX 3x3 (strict=1x1) ; time limit: 50.. 
Termination constraints generated.
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 50.000000 ===
=== STOPING TIMER real ===
Sat solver returned
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found for these parameters.
No solution found for these constraints.
APPLY CRITERIA (ID_CRIT)
NOT SOLVED
No proof found
Cime worked for 27.748747 seconds (real time)
Cime Exit Status: 0