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

TRS termination of:
[1] eq(0,0) -> true
[2] eq(0,s(m)) -> false
[3] eq(s(n),0) -> false
[4] eq(s(n),s(m)) -> eq(n,m)
[5] le(0,m) -> true
[6] le(s(n),0) -> false
[7] le(s(n),s(m)) -> le(n,m)
[8] min(cons(0,nil)) -> 0
[9] min(cons(s(n),nil)) -> s(n)
[10] min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
[11] if_min(true,cons(n,cons(m,x))) -> min(cons(n,x))
[12] if_min(false,cons(n,cons(m,x))) -> min(cons(m,x))
[13] replace(n,m,nil) -> nil
[14] replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
[15] if_replace(true,n,m,cons(k,x)) -> cons(m,x)
[16] if_replace(false,n,m,cons(k,x)) -> cons(k,replace(n,m,x))
[17] sort(nil) -> nil
[18] sort(cons(n,x)) -> cons(min(cons(n,x)),sort(replace(min(cons(n,x)),n,x)))


Sub problem: 
guided: 

DP termination of:
<Marked_eq(s(n),s(m)) , Marked_eq(n,m)>
<Marked_le(s(n),s(m)) , Marked_le(n,m)>
<Marked_min(cons(n,cons(m,x))) , Marked_if_min(le(n,m),cons(n,cons(m,x)))>
<Marked_min(cons(n,cons(m,x))) , Marked_le(n,m)>
<Marked_if_min(true,cons(n,cons(m,x))) , Marked_min(cons(n,x))>
<Marked_if_min(false,cons(n,cons(m,x))) , Marked_min(cons(m,x))>
<Marked_replace(n,m,cons(k,x)) , Marked_if_replace(eq(n,k),n,m,cons(k,x))>
<Marked_replace(n,m,cons(k,x)) , Marked_eq(n,k)>
<Marked_if_replace(false,n,m,cons(k,x)) , Marked_replace(n,m,x)>
<Marked_sort(cons(n,x)) , Marked_min(cons(n,x))>
<Marked_sort(cons(n,x)) , Marked_sort(replace(min(cons(n,x)),n,x))>
<Marked_sort(cons(n,x)) , Marked_replace(min(cons(n,x)),n,x)>
<Marked_sort(cons(n,x)) , Marked_min(cons(n,x))>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 5 components:

{  <Marked_sort(cons(n,x)) , Marked_sort(replace(min(cons(n,x)),n,x))> 
      --> <Marked_sort(cons(n,x)) , Marked_sort(replace(min(cons(n,x)),n,x))>
}

{  <Marked_if_replace(false,n,m,cons(k,x)) , Marked_replace(n,m,x)> 
      --> <Marked_replace(n,m,cons(k,x)) , Marked_if_replace(eq(n,k),
                                            n,m,cons(k,x))>
  <Marked_replace(n,m,cons(k,x)) , Marked_if_replace(eq(n,k),n,m,cons(k,x))> 
     --> <Marked_if_replace(false,n,m,cons(k,x)) , Marked_replace(n,m,x)>
}

{  <Marked_eq(s(n),s(m)) , Marked_eq(n,m)> 
      --> <Marked_eq(s(n),s(m)) , Marked_eq(n,m)>
}

{  <Marked_if_min(false,cons(n,cons(m,x))) , Marked_min(cons(m,x))> 
      --> <Marked_min(cons(n,cons(m,x))) , Marked_if_min(le(n,m),
                                            cons(n,cons(m,x)))>
  <Marked_if_min(true,cons(n,cons(m,x))) , Marked_min(cons(n,x))> 
     --> <Marked_min(cons(n,cons(m,x))) , Marked_if_min(le(n,m),
                                           cons(n,cons(m,x)))>
  <Marked_min(cons(n,cons(m,x))) , Marked_if_min(le(n,m),cons(n,cons(m,x)))> 
     --> <Marked_if_min(false,cons(n,cons(m,x))) , Marked_min(cons(m,x))>
  <Marked_min(cons(n,cons(m,x))) , Marked_if_min(le(n,m),cons(n,cons(m,x)))> 
     --> <Marked_if_min(true,cons(n,cons(m,x))) , Marked_min(cons(n,x))>
}

{  <Marked_le(s(n),s(m)) , Marked_le(n,m)> 
      --> <Marked_le(s(n),s(m)) , Marked_le(n,m)>
}
APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  eq(0,0) >= true ;
  eq(0,s(m)) >= false ;
  eq(s(n),0) >= false ;
  eq(s(n),s(m)) >= eq(n,m) ;
  le(0,m) >= true ;
  le(s(n),0) >= false ;
  le(s(n),s(m)) >= le(n,m) ;
  min(cons(0,nil)) >= 0 ;
  min(cons(s(n),nil)) >= s(n) ;
  min(cons(n,cons(m,x))) >= if_min(le(n,m),cons(n,cons(m,x))) ;
  if_min(true,cons(n,cons(m,x))) >= min(cons(n,x)) ;
  if_min(false,cons(n,cons(m,x))) >= min(cons(m,x)) ;
  replace(n,m,cons(k,x)) >= if_replace(eq(n,k),n,m,cons(k,x)) ;
  replace(n,m,nil) >= nil ;
  if_replace(true,n,m,cons(k,x)) >= cons(m,x) ;
  if_replace(false,n,m,cons(k,x)) >= cons(k,replace(n,m,x)) ;
  sort(cons(n,x)) >= cons(min(cons(n,x)),sort(replace(min(cons(n,x)),n,x))) ;
  sort(nil) >= nil ;
  Marked_sort(cons(n,x)) >= Marked_sort(replace(min(cons(n,x)),n,x)) ;
 }
 + Disjunctions:{ 
{
  Marked_sort(cons(n,x)) > Marked_sort(replace(min(cons(n,x)),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(m)) >= false
constraint: eq(s(n),0) >= false
constraint: eq(s(n),s(m)) >= eq(n,m)
constraint: le(0,m) >= true
constraint: le(s(n),0) >= false
constraint: le(s(n),s(m)) >= le(n,m)
constraint: min(cons(0,nil)) >= 0
constraint: min(cons(s(n),nil)) >= s(n)
constraint: min(cons(n,cons(m,x))) >= if_min(le(n,m),cons(n,cons(m,x)))
constraint: if_min(true,cons(n,cons(m,x))) >= min(cons(n,x))
constraint: if_min(false,cons(n,cons(m,x))) >= min(cons(m,x))
constraint: replace(n,m,cons(k,x)) >= if_replace(eq(n,k),n,m,cons(k,x))
constraint: replace(n,m,nil) >= nil
constraint: if_replace(true,n,m,cons(k,x)) >= cons(m,x)
constraint: if_replace(false,n,m,cons(k,x)) >= cons(k,replace(n,m,x))
constraint: sort(cons(n,x)) >= cons(min(cons(n,x)),
                                sort(replace(min(cons(n,x)),n,x)))
constraint: sort(nil) >= nil
constraint: Marked_sort(cons(n,x)) >= Marked_sort(replace(min(cons(n,x)),n,x))
APPLY CRITERIA (Subterm criterion)

ST: Marked_if_replace -> 4
Marked_replace -> 3

APPLY CRITERIA (Subterm criterion)

ST: Marked_eq -> 1

APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  eq(0,0) >= true ;
  eq(0,s(m)) >= false ;
  eq(s(n),0) >= false ;
  eq(s(n),s(m)) >= eq(n,m) ;
  le(0,m) >= true ;
  le(s(n),0) >= false ;
  le(s(n),s(m)) >= le(n,m) ;
  min(cons(0,nil)) >= 0 ;
  min(cons(s(n),nil)) >= s(n) ;
  min(cons(n,cons(m,x))) >= if_min(le(n,m),cons(n,cons(m,x))) ;
  if_min(true,cons(n,cons(m,x))) >= min(cons(n,x)) ;
  if_min(false,cons(n,cons(m,x))) >= min(cons(m,x)) ;
  replace(n,m,cons(k,x)) >= if_replace(eq(n,k),n,m,cons(k,x)) ;
  replace(n,m,nil) >= nil ;
  if_replace(true,n,m,cons(k,x)) >= cons(m,x) ;
  if_replace(false,n,m,cons(k,x)) >= cons(k,replace(n,m,x)) ;
  sort(cons(n,x)) >= cons(min(cons(n,x)),sort(replace(min(cons(n,x)),n,x))) ;
  sort(nil) >= nil ;
  Marked_min(cons(n,cons(m,x))) >= Marked_if_min(le(n,m),cons(n,cons(m,x))) ;
  Marked_if_min(true,cons(n,cons(m,x))) >= Marked_min(cons(n,x)) ;
  Marked_if_min(false,cons(n,cons(m,x))) >= Marked_min(cons(m,x)) ;
 }
 + Disjunctions:{ 
{
  Marked_min(cons(n,cons(m,x))) > Marked_if_min(le(n,m),cons(n,cons(m,x))) ;
 }
{
  Marked_if_min(true,cons(n,cons(m,x))) > Marked_min(cons(n,x)) ;
 }
{
  Marked_if_min(false,cons(n,cons(m,x))) > Marked_min(cons(m,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(m)) >= false
constraint: eq(s(n),0) >= false
constraint: eq(s(n),s(m)) >= eq(n,m)
constraint: le(0,m) >= true
constraint: le(s(n),0) >= false
constraint: le(s(n),s(m)) >= le(n,m)
constraint: min(cons(0,nil)) >= 0
constraint: min(cons(s(n),nil)) >= s(n)
constraint: min(cons(n,cons(m,x))) >= if_min(le(n,m),cons(n,cons(m,x)))
constraint: if_min(true,cons(n,cons(m,x))) >= min(cons(n,x))
constraint: if_min(false,cons(n,cons(m,x))) >= min(cons(m,x))
constraint: replace(n,m,cons(k,x)) >= if_replace(eq(n,k),n,m,cons(k,x))
constraint: replace(n,m,nil) >= nil
constraint: if_replace(true,n,m,cons(k,x)) >= cons(m,x)
constraint: if_replace(false,n,m,cons(k,x)) >= cons(k,replace(n,m,x))
constraint: sort(cons(n,x)) >= cons(min(cons(n,x)),
                                sort(replace(min(cons(n,x)),n,x)))
constraint: sort(nil) >= nil
constraint: Marked_min(cons(n,cons(m,x))) >= Marked_if_min(le(n,m),
                                              cons(n,cons(m,x)))
constraint: Marked_if_min(true,cons(n,cons(m,x))) >= Marked_min(cons(n,x))
constraint: Marked_if_min(false,cons(n,cons(m,x))) >= Marked_min(cons(m,x))
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:
APPLY CRITERIA (Graph splitting)
Found 0 components:
SOLVED
{
 
TRS termination of:
[1] eq(0,0) -> true
[2] eq(0,s(m)) -> false
[3] eq(s(n),0) -> false
[4] eq(s(n),s(m)) -> eq(n,m)
[5] le(0,m) -> true
[6] le(s(n),0) -> false
[7] le(s(n),s(m)) -> le(n,m)
[8] min(cons(0,nil)) -> 0
[9] min(cons(s(n),nil)) -> s(n)
[10] min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
[11] if_min(true,cons(n,cons(m,x))) -> min(cons(n,x))
[12] if_min(false,cons(n,cons(m,x))) -> min(cons(m,x))
[13] replace(n,m,nil) -> nil
[14] replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
[15] if_replace(true,n,m,cons(k,x)) -> cons(m,x)
[16] if_replace(false,n,m,cons(k,x)) -> cons(k,replace(n,m,x))
[17] sort(nil) -> nil
[18] sort(cons(n,x)) -> cons(min(cons(n,x)),sort(replace(min(cons(n,x)),n,x)))
 ,
 CRITERION: MDP
 [ {
      
     DP termination of:
     <Marked_eq(s(n),s(m)) , Marked_eq(n,m)>
<Marked_le(s(n),s(m)) , Marked_le(n,m)>
<Marked_min(cons(n,cons(m,x))) , Marked_if_min(le(n,m),cons(n,cons(m,x)))>
<Marked_min(cons(n,cons(m,x))) , Marked_le(n,m)>
<Marked_if_min(true,cons(n,cons(m,x))) , Marked_min(cons(n,x))>
<Marked_if_min(false,cons(n,cons(m,x))) , Marked_min(cons(m,x))>
<Marked_replace(n,m,cons(k,x)) , Marked_if_replace(eq(n,k),n,m,cons(k,x))>
<Marked_replace(n,m,cons(k,x)) , Marked_eq(n,k)>
<Marked_if_replace(false,n,m,cons(k,x)) , Marked_replace(n,m,x)>
<Marked_sort(cons(n,x)) , Marked_min(cons(n,x))>
<Marked_sort(cons(n,x)) , Marked_sort(replace(min(cons(n,x)),n,x))>
<Marked_sort(cons(n,x)) , Marked_replace(min(cons(n,x)),n,x)>
<Marked_sort(cons(n,x)) , Marked_min(cons(n,x))>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_sort(cons(n,x)) , Marked_sort(replace(min(cons(n,x)),n,x))>
 ,
 CRITERION: ORD
 [ 
     Solution found:
     polynomial interpretation = 
      [ true ] () = 0; 
 [ nil ] () = 0; 
 [ s ] (X0) = 0; 
 [ sort ] (X0) = 2*X0 + 0; 
 [ 0 ] () = 0; 
 [ replace ] (X0,X1,X2) = 1 + 1*X2 + 0; 
 [ min ] (X0) = 0; 
 [ eq ] (X0,X1) = 0; 
 [ if_min ] (X0,X1) = 0; 
 [ le ] (X0,X1) = 0; 
 [ Marked_sort ] (X0) = 1*X0 + 0; 
 [ false ] () = 0; 
 [ if_replace ] (X0,X1,X2,X3) = 1 + 1*X3 + 0; 
 [ cons ] (X0,X1) = 2 + 1*X1 + 0; 
 ]} 
{
 
DP termination of:
<Marked_replace(n,m,cons(k,x)) , Marked_if_replace(eq(n,k),n,m,cons(k,x))>
<Marked_if_replace(false,n,m,cons(k,x)) , Marked_replace(n,m,x)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
     <Marked_replace(n,m,cons(k,x)) , Marked_if_replace(eq(n,k),n,m,cons(k,x))>
 ,
 CRITERION: SG
 [  ]} 
 ]} 
{
 
DP termination of:
<Marked_eq(s(n),s(m)) , Marked_eq(n,m)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_min(cons(n,cons(m,x))) , Marked_if_min(le(n,m),cons(n,cons(m,x)))>
<Marked_if_min(true,cons(n,cons(m,x))) , Marked_min(cons(n,x))>
<Marked_if_min(false,cons(n,cons(m,x))) , Marked_min(cons(m,x))>
 ,
 CRITERION: ORD
 [ 
     Solution found:
     polynomial interpretation = 
      [ true ] () = 0; 
 [ nil ] () = 0; 
 [ s ] (X0) = 0; 
 [ sort ] (X0) = 2*X0 + 0; 
 [ 0 ] () = 0; 
 [ Marked_if_min ] (X0,X1) = 1*X1 + 0; 
 [ replace ] (X0,X1,X2) = 1*X2 + 0; 
 [ min ] (X0) = 3*X0 + 0; 
 [ Marked_min ] (X0) = 2*X0 + 0; 
 [ eq ] (X0,X1) = 0; 
 [ if_min ] (X0,X1) = 2 + 2*X1 + 0; 
 [ le ] (X0,X1) = 0; 
 [ false ] () = 0; 
 [ if_replace ] (X0,X1,X2,X3) = 1*X3 + 0; 
 [ cons ] (X0,X1) = 2 + 3*X1 + 0; 
 ]} 
{
 
DP termination of:
<Marked_le(s(n),s(m)) , Marked_le(n,m)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 

Cime worked for 0.347306 seconds (real time)
Cime Exit Status: 0