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

TRS termination of:
[1] g(A) -> A
[2] g(B) -> A
[3] g(B) -> B
[4] g(C) -> A
[5] g(C) -> B
[6] g(C) -> C
[7] foldf(x,nil) -> x
[8] foldf(x,cons(y,z)) -> f(foldf(x,z),y)
[9] f(t,x) -> f'(t,g(x))
[10] f'(triple(a,b,c),C) -> triple(a,b,cons(C,c))
[11] f'(triple(a,b,c),B) -> f(triple(a,b,c),A)
[12] f'(triple(a,b,c),A) -> f''(foldf(triple(cons(A,a),nil,c),b))
[13] f''(triple(a,b,c)) -> foldf(triple(a,b,nil),c)


Sub problem: 
guided: 

DP termination of:
<Marked_foldf(x,cons(y,z)) , Marked_f(foldf(x,z),y)>
<Marked_foldf(x,cons(y,z)) , Marked_foldf(x,z)>
<Marked_f(t,x) , Marked_f'(t,g(x))>
<Marked_f(t,x) , Marked_g(x)>
<Marked_f'(triple(a,b,c),B) , Marked_f(triple(a,b,c),A)>
<Marked_f'(triple(a,b,c),A) , Marked_f''(foldf(triple(cons(A,a),nil,c),b))>
<Marked_f'(triple(a,b,c),A) , Marked_foldf(triple(cons(A,a),nil,c),b)>
<Marked_f''(triple(a,b,c)) , Marked_foldf(triple(a,b,nil),c)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 1 components:

{  <Marked_f''(triple(a,b,c)) , Marked_foldf(triple(a,b,nil),c)> 
      --> <Marked_foldf(x,cons(y,z)) , Marked_foldf(x,z)>
  <Marked_f''(triple(a,b,c)) , Marked_foldf(triple(a,b,nil),c)> 
     --> <Marked_foldf(x,cons(y,z)) , Marked_f(foldf(x,z),y)>
  <Marked_f'(triple(a,b,c),A) , Marked_foldf(triple(cons(A,a),nil,c),b)> 
     --> <Marked_foldf(x,cons(y,z)) , Marked_foldf(x,z)>
  <Marked_f'(triple(a,b,c),A) , Marked_foldf(triple(cons(A,a),nil,c),b)> 
     --> <Marked_foldf(x,cons(y,z)) , Marked_f(foldf(x,z),y)>
  <Marked_f'(triple(a,b,c),A) , Marked_f''(foldf(triple(cons(A,a),nil,c),b))> 
     --> <Marked_f''(triple(a,b,c)) , Marked_foldf(triple(a,b,nil),c)>
  <Marked_f'(triple(a,b,c),B) , Marked_f(triple(a,b,c),A)> 
     --> <Marked_f(t,x) , Marked_f'(t,g(x))>
  <Marked_f(t,x) , Marked_f'(t,g(x))> 
     --> <Marked_f'(triple(a,b,c),A) , Marked_foldf(triple(cons(A,a),nil,c),b)>
  <Marked_f(t,x) , Marked_f'(t,g(x))> 
     --> <Marked_f'(triple(a,b,c),A) , Marked_f''(foldf(triple(cons(A,a),nil,c),
                                                   b))>
  <Marked_f(t,x) , Marked_f'(t,g(x))> 
     --> <Marked_f'(triple(a,b,c),B) , Marked_f(triple(a,b,c),A)>
  <Marked_foldf(x,cons(y,z)) , Marked_foldf(x,z)> 
     --> <Marked_foldf(x,cons(y,z)) , Marked_foldf(x,z)>
  <Marked_foldf(x,cons(y,z)) , Marked_foldf(x,z)> 
     --> <Marked_foldf(x,cons(y,z)) , Marked_f(foldf(x,z),y)>
  <Marked_foldf(x,cons(y,z)) , Marked_f(foldf(x,z),y)> 
     --> <Marked_f(t,x) , Marked_f'(t,g(x))>
}
APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  g(A) >= A ;
  g(B) >= A ;
  g(B) >= B ;
  g(C) >= A ;
  g(C) >= B ;
  g(C) >= C ;
  foldf(x,nil) >= x ;
  foldf(x,cons(y,z)) >= f(foldf(x,z),y) ;
  f(t,x) >= f'(t,g(x)) ;
  f'(triple(a,b,c),A) >= f''(foldf(triple(cons(A,a),nil,c),b)) ;
  f'(triple(a,b,c),B) >= f(triple(a,b,c),A) ;
  f'(triple(a,b,c),C) >= triple(a,b,cons(C,c)) ;
  f''(triple(a,b,c)) >= foldf(triple(a,b,nil),c) ;
  Marked_f''(triple(a,b,c)) >= Marked_foldf(triple(a,b,nil),c) ;
  Marked_foldf(x,cons(y,z)) >= Marked_foldf(x,z) ;
  Marked_foldf(x,cons(y,z)) >= Marked_f(foldf(x,z),y) ;
  Marked_f'(triple(a,b,c),A) >= Marked_f''(foldf(triple(cons(A,a),nil,c),b)) ;
  Marked_f'(triple(a,b,c),A) >= Marked_foldf(triple(cons(A,a),nil,c),b) ;
  Marked_f'(triple(a,b,c),B) >= Marked_f(triple(a,b,c),A) ;
  Marked_f(t,x) >= Marked_f'(t,g(x)) ;
 }
 + Disjunctions:{ 
{
  Marked_f''(triple(a,b,c)) > Marked_foldf(triple(a,b,nil),c) ;
 }
{
  Marked_foldf(x,cons(y,z)) > Marked_foldf(x,z) ;
 }
{
  Marked_foldf(x,cons(y,z)) > Marked_f(foldf(x,z),y) ;
 }
{
  Marked_f'(triple(a,b,c),A) > Marked_f''(foldf(triple(cons(A,a),nil,c),b)) ;
 }
{
  Marked_f'(triple(a,b,c),A) > Marked_foldf(triple(cons(A,a),nil,c),b) ;
 }
{
  Marked_f'(triple(a,b,c),B) > Marked_f(triple(a,b,c),A) ;
 }
{
  Marked_f(t,x) > Marked_f'(t,g(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: g(A) >= A
constraint: g(B) >= A
constraint: g(B) >= B
constraint: g(C) >= A
constraint: g(C) >= B
constraint: g(C) >= C
constraint: foldf(x,nil) >= x
constraint: foldf(x,cons(y,z)) >= f(foldf(x,z),y)
constraint: f(t,x) >= f'(t,g(x))
constraint: f'(triple(a,b,c),A) >= f''(foldf(triple(cons(A,a),nil,c),b))
constraint: f'(triple(a,b,c),B) >= f(triple(a,b,c),A)
constraint: f'(triple(a,b,c),C) >= triple(a,b,cons(C,c))
constraint: f''(triple(a,b,c)) >= foldf(triple(a,b,nil),c)
constraint: Marked_f''(triple(a,b,c)) >= Marked_foldf(triple(a,b,nil),c)
constraint: Marked_foldf(x,cons(y,z)) >= Marked_foldf(x,z)
constraint: Marked_foldf(x,cons(y,z)) >= Marked_f(foldf(x,z),y)
constraint: Marked_f'(triple(a,b,c),A) >= Marked_f''(foldf(triple(cons(A,a),
                                                            nil,c),b))
constraint: Marked_f'(triple(a,b,c),A) >= Marked_foldf(triple(cons(A,a),nil,c),
                                           b)
constraint: Marked_f'(triple(a,b,c),B) >= Marked_f(triple(a,b,c),A)
constraint: Marked_f(t,x) >= Marked_f'(t,g(x))
APPLY CRITERIA (Graph splitting)
Found 1 components:

{  <Marked_f''(triple(a,b,c)) , Marked_foldf(triple(a,b,nil),c)> 
      --> <Marked_foldf(x,cons(y,z)) , Marked_foldf(x,z)>
  <Marked_f''(triple(a,b,c)) , Marked_foldf(triple(a,b,nil),c)> 
     --> <Marked_foldf(x,cons(y,z)) , Marked_f(foldf(x,z),y)>
  <Marked_f'(triple(a,b,c),A) , Marked_foldf(triple(cons(A,a),nil,c),b)> 
     --> <Marked_foldf(x,cons(y,z)) , Marked_foldf(x,z)>
  <Marked_f'(triple(a,b,c),A) , Marked_foldf(triple(cons(A,a),nil,c),b)> 
     --> <Marked_foldf(x,cons(y,z)) , Marked_f(foldf(x,z),y)>
  <Marked_f'(triple(a,b,c),A) , Marked_f''(foldf(triple(cons(A,a),nil,c),b))> 
     --> <Marked_f''(triple(a,b,c)) , Marked_foldf(triple(a,b,nil),c)>
  <Marked_f(t,x) , Marked_f'(t,g(x))> 
     --> <Marked_f'(triple(a,b,c),A) , Marked_foldf(triple(cons(A,a),nil,c),b)>
  <Marked_f(t,x) , Marked_f'(t,g(x))> 
     --> <Marked_f'(triple(a,b,c),A) , Marked_f''(foldf(triple(cons(A,a),nil,c),
                                                   b))>
  <Marked_foldf(x,cons(y,z)) , Marked_foldf(x,z)> 
     --> <Marked_foldf(x,cons(y,z)) , Marked_foldf(x,z)>
  <Marked_foldf(x,cons(y,z)) , Marked_foldf(x,z)> 
     --> <Marked_foldf(x,cons(y,z)) , Marked_f(foldf(x,z),y)>
  <Marked_foldf(x,cons(y,z)) , Marked_f(foldf(x,z),y)> 
     --> <Marked_f(t,x) , Marked_f'(t,g(x))>
}
APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  g(A) >= A ;
  g(B) >= A ;
  g(B) >= B ;
  g(C) >= A ;
  g(C) >= B ;
  g(C) >= C ;
  foldf(x,nil) >= x ;
  foldf(x,cons(y,z)) >= f(foldf(x,z),y) ;
  f(t,x) >= f'(t,g(x)) ;
  f'(triple(a,b,c),A) >= f''(foldf(triple(cons(A,a),nil,c),b)) ;
  f'(triple(a,b,c),B) >= f(triple(a,b,c),A) ;
  f'(triple(a,b,c),C) >= triple(a,b,cons(C,c)) ;
  f''(triple(a,b,c)) >= foldf(triple(a,b,nil),c) ;
  Marked_f''(triple(a,b,c)) >= Marked_foldf(triple(a,b,nil),c) ;
  Marked_foldf(x,cons(y,z)) >= Marked_foldf(x,z) ;
  Marked_foldf(x,cons(y,z)) >= Marked_f(foldf(x,z),y) ;
  Marked_f'(triple(a,b,c),A) >= Marked_f''(foldf(triple(cons(A,a),nil,c),b)) ;
  Marked_f'(triple(a,b,c),A) >= Marked_foldf(triple(cons(A,a),nil,c),b) ;
  Marked_f(t,x) >= Marked_f'(t,g(x)) ;
 }
 + Disjunctions:{ 
{
  Marked_f''(triple(a,b,c)) > Marked_foldf(triple(a,b,nil),c) ;
 }
{
  Marked_foldf(x,cons(y,z)) > Marked_foldf(x,z) ;
 }
{
  Marked_foldf(x,cons(y,z)) > Marked_f(foldf(x,z),y) ;
 }
{
  Marked_f'(triple(a,b,c),A) > Marked_f''(foldf(triple(cons(A,a),nil,c),b)) ;
 }
{
  Marked_f'(triple(a,b,c),A) > Marked_foldf(triple(cons(A,a),nil,c),b) ;
 }
{
  Marked_f(t,x) > Marked_f'(t,g(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: g(A) >= A
constraint: g(B) >= A
constraint: g(B) >= B
constraint: g(C) >= A
constraint: g(C) >= B
constraint: g(C) >= C
constraint: foldf(x,nil) >= x
constraint: foldf(x,cons(y,z)) >= f(foldf(x,z),y)
constraint: f(t,x) >= f'(t,g(x))
constraint: f'(triple(a,b,c),A) >= f''(foldf(triple(cons(A,a),nil,c),b))
constraint: f'(triple(a,b,c),B) >= f(triple(a,b,c),A)
constraint: f'(triple(a,b,c),C) >= triple(a,b,cons(C,c))
constraint: f''(triple(a,b,c)) >= foldf(triple(a,b,nil),c)
constraint: Marked_f''(triple(a,b,c)) >= Marked_foldf(triple(a,b,nil),c)
constraint: Marked_foldf(x,cons(y,z)) >= Marked_foldf(x,z)
constraint: Marked_foldf(x,cons(y,z)) >= Marked_f(foldf(x,z),y)
constraint: Marked_f'(triple(a,b,c),A) >= Marked_f''(foldf(triple(cons(A,a),
                                                            nil,c),b))
constraint: Marked_f'(triple(a,b,c),A) >= Marked_foldf(triple(cons(A,a),nil,c),
                                           b)
constraint: Marked_f(t,x) >= Marked_f'(t,g(x))
APPLY CRITERIA (Graph splitting)
Found 0 components:
SOLVED
{
 
TRS termination of:
[1] g(A) -> A
[2] g(B) -> A
[3] g(B) -> B
[4] g(C) -> A
[5] g(C) -> B
[6] g(C) -> C
[7] foldf(x,nil) -> x
[8] foldf(x,cons(y,z)) -> f(foldf(x,z),y)
[9] f(t,x) -> f'(t,g(x))
[10] f'(triple(a,b,c),C) -> triple(a,b,cons(C,c))
[11] f'(triple(a,b,c),B) -> f(triple(a,b,c),A)
[12] f'(triple(a,b,c),A) -> f''(foldf(triple(cons(A,a),nil,c),b))
[13] f''(triple(a,b,c)) -> foldf(triple(a,b,nil),c)
 ,
 CRITERION: MDP
 [ {
      
     DP termination of:
     <Marked_foldf(x,cons(y,z)) , Marked_f(foldf(x,z),y)>
<Marked_foldf(x,cons(y,z)) , Marked_foldf(x,z)>
<Marked_f(t,x) , Marked_f'(t,g(x))>
<Marked_f(t,x) , Marked_g(x)>
<Marked_f'(triple(a,b,c),B) , Marked_f(triple(a,b,c),A)>
<Marked_f'(triple(a,b,c),A) , Marked_f''(foldf(triple(cons(A,a),nil,c),b))>
<Marked_f'(triple(a,b,c),A) , Marked_foldf(triple(cons(A,a),nil,c),b)>
<Marked_f''(triple(a,b,c)) , Marked_foldf(triple(a,b,nil),c)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_foldf(x,cons(y,z)) , Marked_f(foldf(x,z),y)>
<Marked_foldf(x,cons(y,z)) , Marked_foldf(x,z)>
<Marked_f(t,x) , Marked_f'(t,g(x))>
<Marked_f'(triple(a,b,c),B) , Marked_f(triple(a,b,c),A)>
<Marked_f'(triple(a,b,c),A) , Marked_f''(foldf(triple(cons(A,a),nil,c),b))>
<Marked_f'(triple(a,b,c),A) , Marked_foldf(triple(cons(A,a),nil,c),b)>
<Marked_f''(triple(a,b,c)) , Marked_foldf(triple(a,b,nil),c)>
 ,
 CRITERION: CG using polynomial interpretation = 
 [ A ] () = 0; 
 [ f' ] (X0,X1) = 2*X1 + 1*X0; 
 [ foldf ] (X0,X1) = 1*X1 + 1*X0; 
 [ Marked_foldf ] (X0,X1) = 2*X1 + 2*X0; 
 [ B ] () = 2; 
 [ f'' ] (X0) = 1*X0; 
 [ f ] (X0,X1) = 2*X1 + 1*X0; 
 [ Marked_f ] (X0,X1) = 1*X1 + 2*X0; 
 [ g ] (X0) = 1*X0; 
 [ triple ] (X0,X1,X2) = 1*X2 + 2*X1; 
 [ nil ] () = 0; 
 [ Marked_f' ] (X0,X1) = 1*X1 + 2*X0; 
 [ C ] () = 2; 
 [ Marked_f'' ] (X0) = 2*X0; 
 [ cons ] (X0,X1) = 1*X1 + 2*X0; 
 removing <Marked_f'(triple(a,b,c),B),Marked_f(triple(a,b,c),A)>
 [ {
      
     DP termination of:
     <Marked_foldf(x,cons(y,z)) , Marked_f(foldf(x,z),y)>
<Marked_foldf(x,cons(y,z)) , Marked_foldf(x,z)>
<Marked_f(t,x) , Marked_f'(t,g(x))>
<Marked_f'(triple(a,b,c),A) , Marked_f''(foldf(triple(cons(A,a),nil,c),b))>
<Marked_f'(triple(a,b,c),A) , Marked_foldf(triple(cons(A,a),nil,c),b)>
<Marked_f''(triple(a,b,c)) , Marked_foldf(triple(a,b,nil),c)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_foldf(x,cons(y,z)) , Marked_f(foldf(x,z),y)>
<Marked_foldf(x,cons(y,z)) , Marked_foldf(x,z)>
<Marked_f(t,x) , Marked_f'(t,g(x))>
<Marked_f'(triple(a,b,c),A) , Marked_f''(foldf(triple(cons(A,a),nil,c),b))>
<Marked_f'(triple(a,b,c),A) , Marked_foldf(triple(cons(A,a),nil,c),b)>
<Marked_f''(triple(a,b,c)) , Marked_foldf(triple(a,b,nil),c)>
 ,
 CRITERION: CG using polynomial interpretation = 
 [ A ] () = 0; 
 [ f' ] (X0,X1) = 1*X0 + 2; 
 [ foldf ] (X0,X1) = 1*X1 + 1*X0; 
 [ Marked_foldf ] (X0,X1) = 2*X1 + 2*X0; 
 [ B ] () = 0; 
 [ f'' ] (X0) = 1*X0; 
 [ f ] (X0,X1) = 1*X0 + 2; 
 [ Marked_f ] (X0,X1) = 2*X0; 
 [ g ] (X0) = 0; 
 [ triple ] (X0,X1,X2) = 1*X2 + 2*X1; 
 [ nil ] () = 0; 
 [ Marked_f' ] (X0,X1) = 2*X0; 
 [ C ] () = 0; 
 [ Marked_f'' ] (X0) = 2*X0; 
 [ cons ] (X0,X1) = 1*X1 + 2; 
 removing <Marked_foldf(x,cons(y,z)),Marked_f(foldf(x,z),y)><Marked_foldf(
                                                              x,cons(y,z)),
Marked_foldf(x,z)>
 [ {
      
     DP termination of:
     <Marked_f(t,x) , Marked_f'(t,g(x))>
<Marked_f'(triple(a,b,c),A) , Marked_f''(foldf(triple(cons(A,a),nil,c),b))>
<Marked_f'(triple(a,b,c),A) , Marked_foldf(triple(cons(A,a),nil,c),b)>
<Marked_f''(triple(a,b,c)) , Marked_foldf(triple(a,b,nil),c)>
 ,
 CRITERION: SG
 [  ]} 
 ]} 
 ]} 
 ]} 
 ]} 
 ]} 

Cime worked for 0.280801 seconds (real time)
Cime Exit Status: 0