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

TRS termination of:
[1] t(N) -> cs(r(q(N)),nt(ns(N)))
[2] q(0) -> 0
[3] q(s(X)) -> s(p(q(X),d(X)))
[4] d(0) -> 0
[5] d(s(X)) -> s(s(d(X)))
[6] p(0,X) -> X
[7] p(X,0) -> X
[8] p(s(X),s(Y)) -> s(s(p(X,Y)))
[9] f(0,X) -> nil
[10] f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
[11] t(X) -> nt(X)
[12] s(X) -> ns(X)
[13] f(X1,X2) -> nf(X1,X2)
[14] a(nt(X)) -> t(a(X))
[15] a(ns(X)) -> s(a(X))
[16] a(nf(X1,X2)) -> f(a(X1),a(X2))
[17] a(X) -> X


Sub problem: 
guided: 

DP termination of:
<Marked_t(N) , Marked_q(N)>
<Marked_q(s(X)) , Marked_s(p(q(X),d(X)))>
<Marked_q(s(X)) , Marked_p(q(X),d(X))>
<Marked_q(s(X)) , Marked_q(X)>
<Marked_q(s(X)) , Marked_d(X)>
<Marked_d(s(X)) , Marked_s(s(d(X)))>
<Marked_d(s(X)) , Marked_s(d(X))>
<Marked_d(s(X)) , Marked_d(X)>
<Marked_p(s(X),s(Y)) , Marked_s(s(p(X,Y)))>
<Marked_p(s(X),s(Y)) , Marked_s(p(X,Y))>
<Marked_p(s(X),s(Y)) , Marked_p(X,Y)>
<Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)>
<Marked_a(nt(X)) , Marked_t(a(X))>
<Marked_a(nt(X)) , Marked_a(X)>
<Marked_a(ns(X)) , Marked_s(a(X))>
<Marked_a(ns(X)) , Marked_a(X)>
<Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))>
<Marked_a(nf(X1,X2)) , Marked_a(X1)>
<Marked_a(nf(X1,X2)) , Marked_a(X2)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 4 components:

{  <Marked_a(nf(X1,X2)) , Marked_a(X2)> 
      --> <Marked_a(nf(X1,X2)) , Marked_a(X2)>
  <Marked_a(nf(X1,X2)) , Marked_a(X2)> --> <Marked_a(nf(X1,X2)) , Marked_a(X1)>
  <Marked_a(nf(X1,X2)) , Marked_a(X2)> 
     --> <Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))>
  <Marked_a(nf(X1,X2)) , Marked_a(X2)> --> <Marked_a(ns(X)) , Marked_a(X)>
  <Marked_a(nf(X1,X2)) , Marked_a(X2)> --> <Marked_a(nt(X)) , Marked_a(X)>
  <Marked_a(nf(X1,X2)) , Marked_a(X1)> --> <Marked_a(nf(X1,X2)) , Marked_a(X2)>
  <Marked_a(nf(X1,X2)) , Marked_a(X1)> --> <Marked_a(nf(X1,X2)) , Marked_a(X1)>
  <Marked_a(nf(X1,X2)) , Marked_a(X1)> 
     --> <Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))>
  <Marked_a(nf(X1,X2)) , Marked_a(X1)> --> <Marked_a(ns(X)) , Marked_a(X)>
  <Marked_a(nf(X1,X2)) , Marked_a(X1)> --> <Marked_a(nt(X)) , Marked_a(X)>
  <Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))> 
     --> <Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)>
  <Marked_a(ns(X)) , Marked_a(X)> --> <Marked_a(nf(X1,X2)) , Marked_a(X2)>
  <Marked_a(ns(X)) , Marked_a(X)> --> <Marked_a(nf(X1,X2)) , Marked_a(X1)>
  <Marked_a(ns(X)) , Marked_a(X)> 
     --> <Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))>
  <Marked_a(ns(X)) , Marked_a(X)> --> <Marked_a(ns(X)) , Marked_a(X)>
  <Marked_a(ns(X)) , Marked_a(X)> --> <Marked_a(nt(X)) , Marked_a(X)>
  <Marked_a(nt(X)) , Marked_a(X)> --> <Marked_a(nf(X1,X2)) , Marked_a(X2)>
  <Marked_a(nt(X)) , Marked_a(X)> --> <Marked_a(nf(X1,X2)) , Marked_a(X1)>
  <Marked_a(nt(X)) , Marked_a(X)> 
     --> <Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))>
  <Marked_a(nt(X)) , Marked_a(X)> --> <Marked_a(ns(X)) , Marked_a(X)>
  <Marked_a(nt(X)) , Marked_a(X)> --> <Marked_a(nt(X)) , Marked_a(X)>
  <Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)> 
     --> <Marked_a(nf(X1,X2)) , Marked_a(X2)>
  <Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)> 
     --> <Marked_a(nf(X1,X2)) , Marked_a(X1)>
  <Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)> 
     --> <Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))>
  <Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)> --> <Marked_a(ns(X)) , Marked_a(X)>
  <Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)> --> <Marked_a(nt(X)) , Marked_a(X)>
}

{  <Marked_q(s(X)) , Marked_q(X)> --> <Marked_q(s(X)) , Marked_q(X)>
}

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

{  <Marked_d(s(X)) , Marked_d(X)> --> <Marked_d(s(X)) , Marked_d(X)>
}
APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  q(0) >= 0 ;
  q(s(X)) >= s(p(q(X),d(X))) ;
  t(N) >= cs(r(q(N)),nt(ns(N))) ;
  t(X) >= nt(X) ;
  s(X) >= ns(X) ;
  p(0,X) >= X ;
  p(s(X),s(Y)) >= s(s(p(X,Y))) ;
  p(X,0) >= X ;
  d(0) >= 0 ;
  d(s(X)) >= s(s(d(X))) ;
  f(0,X) >= nil ;
  f(s(X),cs(Y,Z)) >= cs(Y,nf(X,a(Z))) ;
  f(X1,X2) >= nf(X1,X2) ;
  a(nt(X)) >= t(a(X)) ;
  a(ns(X)) >= s(a(X)) ;
  a(nf(X1,X2)) >= f(a(X1),a(X2)) ;
  a(X) >= X ;
  Marked_a(nt(X)) >= Marked_a(X) ;
  Marked_a(ns(X)) >= Marked_a(X) ;
  Marked_a(nf(X1,X2)) >= Marked_a(X1) ;
  Marked_a(nf(X1,X2)) >= Marked_a(X2) ;
  Marked_a(nf(X1,X2)) >= Marked_f(a(X1),a(X2)) ;
  Marked_f(s(X),cs(Y,Z)) >= Marked_a(Z) ;
 }
 + Disjunctions:{ 
{
  Marked_a(nt(X)) > Marked_a(X) ;
 }
{
  Marked_a(ns(X)) > Marked_a(X) ;
 }
{
  Marked_a(nf(X1,X2)) > Marked_a(X1) ;
 }
{
  Marked_a(nf(X1,X2)) > Marked_a(X2) ;
 }
{
  Marked_a(nf(X1,X2)) > Marked_f(a(X1),a(X2)) ;
 }
{
  Marked_f(s(X),cs(Y,Z)) > Marked_a(Z) ;
 }
 } 
=== 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: q(0) >= 0
constraint: q(s(X)) >= s(p(q(X),d(X)))
constraint: t(N) >= cs(r(q(N)),nt(ns(N)))
constraint: t(X) >= nt(X)
constraint: s(X) >= ns(X)
constraint: p(0,X) >= X
constraint: p(s(X),s(Y)) >= s(s(p(X,Y)))
constraint: p(X,0) >= X
constraint: d(0) >= 0
constraint: d(s(X)) >= s(s(d(X)))
constraint: f(0,X) >= nil
constraint: f(s(X),cs(Y,Z)) >= cs(Y,nf(X,a(Z)))
constraint: f(X1,X2) >= nf(X1,X2)
constraint: a(nt(X)) >= t(a(X))
constraint: a(ns(X)) >= s(a(X))
constraint: a(nf(X1,X2)) >= f(a(X1),a(X2))
constraint: a(X) >= X
constraint: Marked_a(nt(X)) >= Marked_a(X)
constraint: Marked_a(ns(X)) >= Marked_a(X)
constraint: Marked_a(nf(X1,X2)) >= Marked_a(X1)
constraint: Marked_a(nf(X1,X2)) >= Marked_a(X2)
constraint: Marked_a(nf(X1,X2)) >= Marked_f(a(X1),a(X2))
constraint: Marked_f(s(X),cs(Y,Z)) >= Marked_a(Z)
APPLY CRITERIA (Subterm criterion)

ST: Marked_q -> 1

APPLY CRITERIA (Subterm criterion)

ST: Marked_p -> 1

APPLY CRITERIA (Subterm criterion)

ST: Marked_d -> 1

APPLY CRITERIA (Graph splitting)
Found 1 components:

{  <Marked_a(nf(X1,X2)) , Marked_a(X2)> 
      --> <Marked_a(nf(X1,X2)) , Marked_a(X2)>
  <Marked_a(nf(X1,X2)) , Marked_a(X2)> --> <Marked_a(nf(X1,X2)) , Marked_a(X1)>
  <Marked_a(nf(X1,X2)) , Marked_a(X2)> 
     --> <Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))>
  <Marked_a(nf(X1,X2)) , Marked_a(X2)> --> <Marked_a(ns(X)) , Marked_a(X)>
  <Marked_a(nf(X1,X2)) , Marked_a(X1)> --> <Marked_a(nf(X1,X2)) , Marked_a(X2)>
  <Marked_a(nf(X1,X2)) , Marked_a(X1)> --> <Marked_a(nf(X1,X2)) , Marked_a(X1)>
  <Marked_a(nf(X1,X2)) , Marked_a(X1)> 
     --> <Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))>
  <Marked_a(nf(X1,X2)) , Marked_a(X1)> --> <Marked_a(ns(X)) , Marked_a(X)>
  <Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))> 
     --> <Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)>
  <Marked_a(ns(X)) , Marked_a(X)> --> <Marked_a(nf(X1,X2)) , Marked_a(X2)>
  <Marked_a(ns(X)) , Marked_a(X)> --> <Marked_a(nf(X1,X2)) , Marked_a(X1)>
  <Marked_a(ns(X)) , Marked_a(X)> 
     --> <Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))>
  <Marked_a(ns(X)) , Marked_a(X)> --> <Marked_a(ns(X)) , Marked_a(X)>
  <Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)> 
     --> <Marked_a(nf(X1,X2)) , Marked_a(X2)>
  <Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)> 
     --> <Marked_a(nf(X1,X2)) , Marked_a(X1)>
  <Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)> 
     --> <Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))>
  <Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)> --> <Marked_a(ns(X)) , Marked_a(X)>
}
APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  q(0) >= 0 ;
  q(s(X)) >= s(p(q(X),d(X))) ;
  t(N) >= cs(r(q(N)),nt(ns(N))) ;
  t(X) >= nt(X) ;
  s(X) >= ns(X) ;
  p(0,X) >= X ;
  p(s(X),s(Y)) >= s(s(p(X,Y))) ;
  p(X,0) >= X ;
  d(0) >= 0 ;
  d(s(X)) >= s(s(d(X))) ;
  f(0,X) >= nil ;
  f(s(X),cs(Y,Z)) >= cs(Y,nf(X,a(Z))) ;
  f(X1,X2) >= nf(X1,X2) ;
  a(nt(X)) >= t(a(X)) ;
  a(ns(X)) >= s(a(X)) ;
  a(nf(X1,X2)) >= f(a(X1),a(X2)) ;
  a(X) >= X ;
  Marked_a(ns(X)) >= Marked_a(X) ;
  Marked_a(nf(X1,X2)) >= Marked_a(X1) ;
  Marked_a(nf(X1,X2)) >= Marked_a(X2) ;
  Marked_a(nf(X1,X2)) >= Marked_f(a(X1),a(X2)) ;
  Marked_f(s(X),cs(Y,Z)) >= Marked_a(Z) ;
 }
 + Disjunctions:{ 
{
  Marked_a(ns(X)) > Marked_a(X) ;
 }
{
  Marked_a(nf(X1,X2)) > Marked_a(X1) ;
 }
{
  Marked_a(nf(X1,X2)) > Marked_a(X2) ;
 }
{
  Marked_a(nf(X1,X2)) > Marked_f(a(X1),a(X2)) ;
 }
{
  Marked_f(s(X),cs(Y,Z)) > Marked_a(Z) ;
 }
 } 
=== 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: q(0) >= 0
constraint: q(s(X)) >= s(p(q(X),d(X)))
constraint: t(N) >= cs(r(q(N)),nt(ns(N)))
constraint: t(X) >= nt(X)
constraint: s(X) >= ns(X)
constraint: p(0,X) >= X
constraint: p(s(X),s(Y)) >= s(s(p(X,Y)))
constraint: p(X,0) >= X
constraint: d(0) >= 0
constraint: d(s(X)) >= s(s(d(X)))
constraint: f(0,X) >= nil
constraint: f(s(X),cs(Y,Z)) >= cs(Y,nf(X,a(Z)))
constraint: f(X1,X2) >= nf(X1,X2)
constraint: a(nt(X)) >= t(a(X))
constraint: a(ns(X)) >= s(a(X))
constraint: a(nf(X1,X2)) >= f(a(X1),a(X2))
constraint: a(X) >= X
constraint: Marked_a(ns(X)) >= Marked_a(X)
constraint: Marked_a(nf(X1,X2)) >= Marked_a(X1)
constraint: Marked_a(nf(X1,X2)) >= Marked_a(X2)
constraint: Marked_a(nf(X1,X2)) >= Marked_f(a(X1),a(X2))
constraint: Marked_f(s(X),cs(Y,Z)) >= Marked_a(Z)
APPLY CRITERIA (Graph splitting)
Found 1 components:

{  <Marked_a(ns(X)) , Marked_a(X)> --> <Marked_a(ns(X)) , Marked_a(X)>
}
APPLY CRITERIA (Subterm criterion)

ST: Marked_a -> 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:
SOLVED
{
 
TRS termination of:
[1] t(N) -> cs(r(q(N)),nt(ns(N)))
[2] q(0) -> 0
[3] q(s(X)) -> s(p(q(X),d(X)))
[4] d(0) -> 0
[5] d(s(X)) -> s(s(d(X)))
[6] p(0,X) -> X
[7] p(X,0) -> X
[8] p(s(X),s(Y)) -> s(s(p(X,Y)))
[9] f(0,X) -> nil
[10] f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
[11] t(X) -> nt(X)
[12] s(X) -> ns(X)
[13] f(X1,X2) -> nf(X1,X2)
[14] a(nt(X)) -> t(a(X))
[15] a(ns(X)) -> s(a(X))
[16] a(nf(X1,X2)) -> f(a(X1),a(X2))
[17] a(X) -> X
 ,
 CRITERION: MDP
 [ {
      
     DP termination of:
     <Marked_t(N) , Marked_q(N)>
<Marked_q(s(X)) , Marked_s(p(q(X),d(X)))>
<Marked_q(s(X)) , Marked_p(q(X),d(X))>
<Marked_q(s(X)) , Marked_q(X)>
<Marked_q(s(X)) , Marked_d(X)>
<Marked_d(s(X)) , Marked_s(s(d(X)))>
<Marked_d(s(X)) , Marked_s(d(X))>
<Marked_d(s(X)) , Marked_d(X)>
<Marked_p(s(X),s(Y)) , Marked_s(s(p(X,Y)))>
<Marked_p(s(X),s(Y)) , Marked_s(p(X,Y))>
<Marked_p(s(X),s(Y)) , Marked_p(X,Y)>
<Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)>
<Marked_a(nt(X)) , Marked_t(a(X))>
<Marked_a(nt(X)) , Marked_a(X)>
<Marked_a(ns(X)) , Marked_s(a(X))>
<Marked_a(ns(X)) , Marked_a(X)>
<Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))>
<Marked_a(nf(X1,X2)) , Marked_a(X1)>
<Marked_a(nf(X1,X2)) , Marked_a(X2)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)>
<Marked_a(nt(X)) , Marked_a(X)>
<Marked_a(ns(X)) , Marked_a(X)>
<Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))>
<Marked_a(nf(X1,X2)) , Marked_a(X1)>
<Marked_a(nf(X1,X2)) , Marked_a(X2)>
 ,
 CRITERION: CG using polynomial interpretation = 
 [ cs ] (X0,X1) = 1*X1; 
 [ p ] (X0,X1) = 2*X1 + 2*X0; 
 [ ns ] (X0) = 1*X0; 
 [ nf ] (X0,X1) = 1*X1 + 1*X0; 
 [ q ] (X0) = 0; 
 [ nil ] () = 0; 
 [ 0 ] () = 0; 
 [ Marked_a ] (X0) = 2*X0; 
 [ r ] (X0) = 0; 
 [ d ] (X0) = 0; 
 [ t ] (X0) = 1*X0 + 2; 
 [ a ] (X0) = 1*X0; 
 [ nt ] (X0) = 1*X0 + 2; 
 [ f ] (X0,X1) = 1*X1 + 1*X0; 
 [ s ] (X0) = 1*X0; 
 [ Marked_f ] (X0,X1) = 2*X1; 
 removing <Marked_a(nt(X)),Marked_a(X)>
 [ {
      
     DP termination of:
     <Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)>
<Marked_a(ns(X)) , Marked_a(X)>
<Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))>
<Marked_a(nf(X1,X2)) , Marked_a(X1)>
<Marked_a(nf(X1,X2)) , Marked_a(X2)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_f(s(X),cs(Y,Z)) , Marked_a(Z)>
<Marked_a(ns(X)) , Marked_a(X)>
<Marked_a(nf(X1,X2)) , Marked_f(a(X1),a(X2))>
<Marked_a(nf(X1,X2)) , Marked_a(X1)>
<Marked_a(nf(X1,X2)) , Marked_a(X2)>
 ,
 CRITERION: CG using polynomial interpretation = 
 [ cs ] (X0,X1) = 1*X1; 
 [ p ] (X0,X1) = 1*X1 + 1*X0; 
 [ ns ] (X0) = 1*X0; 
 [ nf ] (X0,X1) = 1*X1 + 2*X0 + 2; 
 [ q ] (X0) = 0; 
 [ nil ] () = 2; 
 [ 0 ] () = 0; 
 [ Marked_a ] (X0) = 2*X0; 
 [ r ] (X0) = 0; 
 [ d ] (X0) = 0; 
 [ t ] (X0) = 0; 
 [ a ] (X0) = 1*X0; 
 [ nt ] (X0) = 0; 
 [ f ] (X0,X1) = 1*X1 + 2*X0 + 2; 
 [ s ] (X0) = 1*X0; 
 [ Marked_f ] (X0,X1) = 2*X1 + 2*X0 + 2; 
 removing <Marked_f(s(X),cs(Y,Z)),Marked_a(Z)><Marked_a(nf(X1,X2)),Marked_f(
                                                                    a(
                                                                    X1),
                                                                    a(X2))><
Marked_a(nf(X1,X2)),Marked_a(X1)><Marked_a(nf(X1,X2)),Marked_a(X2)>
 [ {
      
     DP termination of:
     <Marked_a(ns(X)) , Marked_a(X)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_a(ns(X)) , Marked_a(X)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 
 ]} 
 ]} 
{
 
DP termination of:
<Marked_q(s(X)) , Marked_q(X)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_p(s(X),s(Y)) , Marked_p(X,Y)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_d(s(X)) , Marked_d(X)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 

Cime worked for 0.192558 seconds (real time)
Cime Exit Status: 0