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

TRS termination of:
[1] eq(0,0) -> true
[2] eq(0,s(Y)) -> 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] min(cons(0,nil)) -> 0
[9] min(cons(s(N),nil)) -> s(N)
[10] min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L)))
[11] ifmin(true,cons(N,cons(M,L))) -> min(cons(N,L))
[12] ifmin(false,cons(N,cons(M,L))) -> min(cons(M,L))
[13] replace(N,M,nil) -> nil
[14] replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L))
[15] ifrepl(true,N,M,cons(K,L)) -> cons(M,L)
[16] ifrepl(false,N,M,cons(K,L)) -> cons(K,replace(N,M,L))
[17] selsort(nil) -> nil
[18] selsort(cons(N,L)) -> ifselsort(eq(N,min(cons(N,L))),cons(N,L))
[19] ifselsort(true,cons(N,L)) -> cons(N,selsort(L))
[20] ifselsort(false,cons(N,L)) ->
 cons(min(cons(N,L)),selsort(replace(min(cons(N,L)),N,L)))


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_min(cons(N,cons(M,L))) , Marked_ifmin(le(N,M),cons(N,cons(M,L)))>
<Marked_min(cons(N,cons(M,L))) , Marked_le(N,M)>
<Marked_ifmin(true,cons(N,cons(M,L))) , Marked_min(cons(N,L))>
<Marked_ifmin(false,cons(N,cons(M,L))) , Marked_min(cons(M,L))>
<Marked_replace(N,M,cons(K,L)) , Marked_ifrepl(eq(N,K),N,M,cons(K,L))>
<Marked_replace(N,M,cons(K,L)) , Marked_eq(N,K)>
<Marked_ifrepl(false,N,M,cons(K,L)) , Marked_replace(N,M,L)>
<Marked_selsort(cons(N,L)) , Marked_ifselsort(eq(N,min(cons(N,L))),cons(N,L))>
<Marked_selsort(cons(N,L)) , Marked_eq(N,min(cons(N,L)))>
<Marked_selsort(cons(N,L)) , Marked_min(cons(N,L))>
<Marked_ifselsort(true,cons(N,L)) , Marked_selsort(L)>
<Marked_ifselsort(false,cons(N,L)) , Marked_min(cons(N,L))>
<Marked_ifselsort(false,cons(N,L)) , Marked_selsort(replace(min(cons(N,L)),N,L))>
<Marked_ifselsort(false,cons(N,L)) , Marked_replace(min(cons(N,L)),N,L)>
<Marked_ifselsort(false,cons(N,L)) , Marked_min(cons(N,L))>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 5 components:

{  <Marked_ifselsort(false,cons(N,L)) , Marked_selsort(replace(min(cons(N,L)),
                                                        N,L))> 
      --> <Marked_selsort(cons(N,L)) , Marked_ifselsort(eq(N,min(cons(N,L))),
                                        cons(N,L))>
  <Marked_ifselsort(true,cons(N,L)) , Marked_selsort(L)> 
     --> <Marked_selsort(cons(N,L)) , Marked_ifselsort(eq(N,min(cons(N,L))),
                                       cons(N,L))>
  <Marked_selsort(cons(N,L)) , Marked_ifselsort(eq(N,min(cons(N,L))),cons(N,L))> 
     --> <Marked_ifselsort(false,cons(N,L)) , Marked_selsort(replace(
                                                              min(cons(N,L)),
                                                              N,L))>
  <Marked_selsort(cons(N,L)) , Marked_ifselsort(eq(N,min(cons(N,L))),cons(N,L))> 
     --> <Marked_ifselsort(true,cons(N,L)) , Marked_selsort(L)>
}

{  <Marked_ifrepl(false,N,M,cons(K,L)) , Marked_replace(N,M,L)> 
      --> <Marked_replace(N,M,cons(K,L)) , Marked_ifrepl(eq(N,K),N,M,cons(K,L))>
  <Marked_replace(N,M,cons(K,L)) , Marked_ifrepl(eq(N,K),N,M,cons(K,L))> 
     --> <Marked_ifrepl(false,N,M,cons(K,L)) , Marked_replace(N,M,L)>
}

{  <Marked_eq(s(X),s(Y)) , Marked_eq(X,Y)> 
      --> <Marked_eq(s(X),s(Y)) , Marked_eq(X,Y)>
}

{  <Marked_ifmin(false,cons(N,cons(M,L))) , Marked_min(cons(M,L))> 
      --> <Marked_min(cons(N,cons(M,L))) , Marked_ifmin(le(N,M),
                                            cons(N,cons(M,L)))>
  <Marked_ifmin(true,cons(N,cons(M,L))) , Marked_min(cons(N,L))> 
     --> <Marked_min(cons(N,cons(M,L))) , Marked_ifmin(le(N,M),
                                           cons(N,cons(M,L)))>
  <Marked_min(cons(N,cons(M,L))) , Marked_ifmin(le(N,M),cons(N,cons(M,L)))> 
     --> <Marked_ifmin(false,cons(N,cons(M,L))) , Marked_min(cons(M,L))>
  <Marked_min(cons(N,cons(M,L))) , Marked_ifmin(le(N,M),cons(N,cons(M,L)))> 
     --> <Marked_ifmin(true,cons(N,cons(M,L))) , Marked_min(cons(N,L))>
}

{  <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:
{
  eq(0,0) >= true ;
  eq(0,s(Y)) >= 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) ;
  min(cons(0,nil)) >= 0 ;
  min(cons(s(N),nil)) >= s(N) ;
  min(cons(N,cons(M,L))) >= ifmin(le(N,M),cons(N,cons(M,L))) ;
  ifmin(true,cons(N,cons(M,L))) >= min(cons(N,L)) ;
  ifmin(false,cons(N,cons(M,L))) >= min(cons(M,L)) ;
  replace(N,M,cons(K,L)) >= ifrepl(eq(N,K),N,M,cons(K,L)) ;
  replace(N,M,nil) >= nil ;
  ifrepl(true,N,M,cons(K,L)) >= cons(M,L) ;
  ifrepl(false,N,M,cons(K,L)) >= cons(K,replace(N,M,L)) ;
  selsort(cons(N,L)) >= ifselsort(eq(N,min(cons(N,L))),cons(N,L)) ;
  selsort(nil) >= nil ;
  ifselsort(true,cons(N,L)) >= cons(N,selsort(L)) ;
  ifselsort(false,cons(N,L)) >= cons(min(cons(N,L)),
                                 selsort(replace(min(cons(N,L)),N,L))) ;
  Marked_ifselsort(true,cons(N,L)) >= Marked_selsort(L) ;
  Marked_ifselsort(false,cons(N,L)) >= Marked_selsort(replace(min(cons(N,L)),
                                                       N,L)) ;
  Marked_selsort(cons(N,L)) >= Marked_ifselsort(eq(N,min(cons(N,L))),cons(N,L)) ;
 }
 + Disjunctions:{ 
{
  Marked_ifselsort(true,cons(N,L)) > Marked_selsort(L) ;
 }
{
  Marked_ifselsort(false,cons(N,L)) > Marked_selsort(replace(min(cons(N,L)),
                                                      N,L)) ;
 }
{
  Marked_selsort(cons(N,L)) > Marked_ifselsort(eq(N,min(cons(N,L))),cons(N,L)) ;
 }
 } 
=== 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(Y)) >= 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: min(cons(0,nil)) >= 0
constraint: min(cons(s(N),nil)) >= s(N)
constraint: min(cons(N,cons(M,L))) >= ifmin(le(N,M),cons(N,cons(M,L)))
constraint: ifmin(true,cons(N,cons(M,L))) >= min(cons(N,L))
constraint: ifmin(false,cons(N,cons(M,L))) >= min(cons(M,L))
constraint: replace(N,M,cons(K,L)) >= ifrepl(eq(N,K),N,M,cons(K,L))
constraint: replace(N,M,nil) >= nil
constraint: ifrepl(true,N,M,cons(K,L)) >= cons(M,L)
constraint: ifrepl(false,N,M,cons(K,L)) >= cons(K,replace(N,M,L))
constraint: selsort(cons(N,L)) >= ifselsort(eq(N,min(cons(N,L))),cons(N,L))
constraint: selsort(nil) >= nil
constraint: ifselsort(true,cons(N,L)) >= cons(N,selsort(L))
constraint: ifselsort(false,cons(N,L)) >= cons(min(cons(N,L)),
                                           selsort(replace(min(cons(N,L)),N,L)))
constraint: Marked_ifselsort(true,cons(N,L)) >= Marked_selsort(L)
constraint: Marked_ifselsort(false,cons(N,L)) >= Marked_selsort(replace(
                                                                 min(
                                                                  cons(N,L)),
                                                                 N,L))
constraint: Marked_selsort(cons(N,L)) >= Marked_ifselsort(eq(N,min(cons(N,L))),
                                          cons(N,L))
APPLY CRITERIA (Subterm criterion)

ST: Marked_ifrepl -> 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(Y)) >= 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) ;
  min(cons(0,nil)) >= 0 ;
  min(cons(s(N),nil)) >= s(N) ;
  min(cons(N,cons(M,L))) >= ifmin(le(N,M),cons(N,cons(M,L))) ;
  ifmin(true,cons(N,cons(M,L))) >= min(cons(N,L)) ;
  ifmin(false,cons(N,cons(M,L))) >= min(cons(M,L)) ;
  replace(N,M,cons(K,L)) >= ifrepl(eq(N,K),N,M,cons(K,L)) ;
  replace(N,M,nil) >= nil ;
  ifrepl(true,N,M,cons(K,L)) >= cons(M,L) ;
  ifrepl(false,N,M,cons(K,L)) >= cons(K,replace(N,M,L)) ;
  selsort(cons(N,L)) >= ifselsort(eq(N,min(cons(N,L))),cons(N,L)) ;
  selsort(nil) >= nil ;
  ifselsort(true,cons(N,L)) >= cons(N,selsort(L)) ;
  ifselsort(false,cons(N,L)) >= cons(min(cons(N,L)),
                                 selsort(replace(min(cons(N,L)),N,L))) ;
  Marked_min(cons(N,cons(M,L))) >= Marked_ifmin(le(N,M),cons(N,cons(M,L))) ;
  Marked_ifmin(true,cons(N,cons(M,L))) >= Marked_min(cons(N,L)) ;
  Marked_ifmin(false,cons(N,cons(M,L))) >= Marked_min(cons(M,L)) ;
 }
 + Disjunctions:{ 
{
  Marked_min(cons(N,cons(M,L))) > Marked_ifmin(le(N,M),cons(N,cons(M,L))) ;
 }
{
  Marked_ifmin(true,cons(N,cons(M,L))) > Marked_min(cons(N,L)) ;
 }
{
  Marked_ifmin(false,cons(N,cons(M,L))) > Marked_min(cons(M,L)) ;
 }
 } 
=== 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(Y)) >= 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: min(cons(0,nil)) >= 0
constraint: min(cons(s(N),nil)) >= s(N)
constraint: min(cons(N,cons(M,L))) >= ifmin(le(N,M),cons(N,cons(M,L)))
constraint: ifmin(true,cons(N,cons(M,L))) >= min(cons(N,L))
constraint: ifmin(false,cons(N,cons(M,L))) >= min(cons(M,L))
constraint: replace(N,M,cons(K,L)) >= ifrepl(eq(N,K),N,M,cons(K,L))
constraint: replace(N,M,nil) >= nil
constraint: ifrepl(true,N,M,cons(K,L)) >= cons(M,L)
constraint: ifrepl(false,N,M,cons(K,L)) >= cons(K,replace(N,M,L))
constraint: selsort(cons(N,L)) >= ifselsort(eq(N,min(cons(N,L))),cons(N,L))
constraint: selsort(nil) >= nil
constraint: ifselsort(true,cons(N,L)) >= cons(N,selsort(L))
constraint: ifselsort(false,cons(N,L)) >= cons(min(cons(N,L)),
                                           selsort(replace(min(cons(N,L)),N,L)))
constraint: Marked_min(cons(N,cons(M,L))) >= Marked_ifmin(le(N,M),
                                              cons(N,cons(M,L)))
constraint: Marked_ifmin(true,cons(N,cons(M,L))) >= Marked_min(cons(N,L))
constraint: Marked_ifmin(false,cons(N,cons(M,L))) >= Marked_min(cons(M,L))
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(Y)) -> 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] min(cons(0,nil)) -> 0
[9] min(cons(s(N),nil)) -> s(N)
[10] min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L)))
[11] ifmin(true,cons(N,cons(M,L))) -> min(cons(N,L))
[12] ifmin(false,cons(N,cons(M,L))) -> min(cons(M,L))
[13] replace(N,M,nil) -> nil
[14] replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L))
[15] ifrepl(true,N,M,cons(K,L)) -> cons(M,L)
[16] ifrepl(false,N,M,cons(K,L)) -> cons(K,replace(N,M,L))
[17] selsort(nil) -> nil
[18] selsort(cons(N,L)) -> ifselsort(eq(N,min(cons(N,L))),cons(N,L))
[19] ifselsort(true,cons(N,L)) -> cons(N,selsort(L))
[20] ifselsort(false,cons(N,L)) ->
 cons(min(cons(N,L)),selsort(replace(min(cons(N,L)),N,L)))
 ,
 CRITERION: MDP
 [ {
      
     DP termination of:
     <Marked_eq(s(X),s(Y)) , Marked_eq(X,Y)>
<Marked_le(s(X),s(Y)) , Marked_le(X,Y)>
<Marked_min(cons(N,cons(M,L))) , Marked_ifmin(le(N,M),cons(N,cons(M,L)))>
<Marked_min(cons(N,cons(M,L))) , Marked_le(N,M)>
<Marked_ifmin(true,cons(N,cons(M,L))) , Marked_min(cons(N,L))>
<Marked_ifmin(false,cons(N,cons(M,L))) , Marked_min(cons(M,L))>
<Marked_replace(N,M,cons(K,L)) , Marked_ifrepl(eq(N,K),N,M,cons(K,L))>
<Marked_replace(N,M,cons(K,L)) , Marked_eq(N,K)>
<Marked_ifrepl(false,N,M,cons(K,L)) , Marked_replace(N,M,L)>
<Marked_selsort(cons(N,L)) , Marked_ifselsort(eq(N,min(cons(N,L))),cons(N,L))>
<Marked_selsort(cons(N,L)) , Marked_eq(N,min(cons(N,L)))>
<Marked_selsort(cons(N,L)) , Marked_min(cons(N,L))>
<Marked_ifselsort(true,cons(N,L)) , Marked_selsort(L)>
<Marked_ifselsort(false,cons(N,L)) , Marked_min(cons(N,L))>
<Marked_ifselsort(false,cons(N,L)) , Marked_selsort(replace(min(cons(N,L)),N,L))>
<Marked_ifselsort(false,cons(N,L)) , Marked_replace(min(cons(N,L)),N,L)>
<Marked_ifselsort(false,cons(N,L)) , Marked_min(cons(N,L))>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_selsort(cons(N,L)) , Marked_ifselsort(eq(N,min(cons(N,L))),
                                   cons(N,L))>
<Marked_ifselsort(true,cons(N,L)) , Marked_selsort(L)>
<Marked_ifselsort(false,cons(N,L)) , Marked_selsort(replace(min(cons(N,L)),N,L))>
 ,
 CRITERION: CG using polynomial interpretation = 
 [ true ] () = 0; 
 [ Marked_selsort ] (X0) = 1*X0 + 1; 
 [ nil ] () = 0; 
 [ s ] (X0) = 0; 
 [ selsort ] (X0) = 3*X0 + 1; 
 [ 0 ] () = 0; 
 [ replace ] (X0,X1,X2) = 1*X2; 
 [ min ] (X0) = 0; 
 [ Marked_ifselsort ] (X0,X1) = 1*X1 + 1; 
 [ eq ] (X0,X1) = 0; 
 [ ifmin ] (X0,X1) = 0; 
 [ le ] (X0,X1) = 0; 
 [ ifselsort ] (X0,X1) = 3*X1 + 1; 
 [ false ] () = 0; 
 [ ifrepl ] (X0,X1,X2,X3) = 1*X3; 
 [ cons ] (X0,X1) = 1*X1 + 1; 
 removing <Marked_ifselsort(true,cons(N,L)),Marked_selsort(L)><Marked_ifselsort(
                                                                false,
                                                                cons(N,L)),
Marked_selsort(replace(min(cons(N,L)),N,L))>
 [ {
      
     DP termination of:
     <Marked_selsort(cons(N,L)) , Marked_ifselsort(eq(N,min(cons(N,L))),
                                   cons(N,L))>
 ,
 CRITERION: SG
 [  ]} 
 ]} 
{
 
DP termination of:
<Marked_replace(N,M,cons(K,L)) , Marked_ifrepl(eq(N,K),N,M,cons(K,L))>
<Marked_ifrepl(false,N,M,cons(K,L)) , Marked_replace(N,M,L)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
     <Marked_replace(N,M,cons(K,L)) , Marked_ifrepl(eq(N,K),N,M,cons(K,L))>
 ,
 CRITERION: SG
 [  ]} 
 ]} 
{
 
DP termination of:
<Marked_eq(s(X),s(Y)) , Marked_eq(X,Y)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_min(cons(N,cons(M,L))) , Marked_ifmin(le(N,M),cons(N,cons(M,L)))>
<Marked_ifmin(true,cons(N,cons(M,L))) , Marked_min(cons(N,L))>
<Marked_ifmin(false,cons(N,cons(M,L))) , Marked_min(cons(M,L))>
 ,
 CRITERION: CG using polynomial interpretation = 
 [ true ] () = 0; 
 [ nil ] () = 0; 
 [ s ] (X0) = 0; 
 [ Marked_ifmin ] (X0,X1) = 1*X1 + 1; 
 [ selsort ] (X0) = 2*X0; 
 [ 0 ] () = 0; 
 [ replace ] (X0,X1,X2) = 1*X2; 
 [ min ] (X0) = 0; 
 [ eq ] (X0,X1) = 0; 
 [ ifmin ] (X0,X1) = 0; 
 [ le ] (X0,X1) = 0; 
 [ ifselsort ] (X0,X1) = 2*X1; 
 [ false ] () = 0; 
 [ ifrepl ] (X0,X1,X2,X3) = 1*X3; 
 [ cons ] (X0,X1) = 2*X1 + 1; 
 [ Marked_min ] (X0) = 2*X0 + 2; 
 removing <Marked_min(cons(N,cons(M,L))),Marked_ifmin(le(N,M),
                                          cons(N,cons(M,L)))>
 [ {
      
     DP termination of:
     <Marked_ifmin(true,cons(N,cons(M,L))) , Marked_min(cons(N,L))>
<Marked_ifmin(false,cons(N,cons(M,L))) , Marked_min(cons(M,L))>
 ,
 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 0.387439 seconds (real time)
Cime Exit Status: 0