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

TRS termination of:
[1] ack_in(0,n) -> ack_out(s(n))
[2] ack_in(s(m),0) -> u11(ack_in(m,s(0)))
[3] u11(ack_out(n)) -> ack_out(n)
[4] ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
[5] u21(ack_out(n),m) -> u22(ack_in(m,n))
[6] u22(ack_out(n)) -> ack_out(n)


Sub problem: 
guided: 

DP termination of:
<Marked_ack_in(s(m),0) , Marked_u11(ack_in(m,s(0)))>
<Marked_ack_in(s(m),0) , Marked_ack_in(m,s(0))>
<Marked_ack_in(s(m),s(n)) , Marked_u21(ack_in(s(m),n),m)>
<Marked_ack_in(s(m),s(n)) , Marked_ack_in(s(m),n)>
<Marked_u21(ack_out(n),m) , Marked_u22(ack_in(m,n))>
<Marked_u21(ack_out(n),m) , Marked_ack_in(m,n)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 1 components:

{  <Marked_u21(ack_out(n),m) , Marked_ack_in(m,n)> 
      --> <Marked_ack_in(s(m),s(n)) , Marked_ack_in(s(m),n)>
  <Marked_u21(ack_out(n),m) , Marked_ack_in(m,n)> 
     --> <Marked_ack_in(s(m),s(n)) , Marked_u21(ack_in(s(m),n),m)>
  <Marked_u21(ack_out(n),m) , Marked_ack_in(m,n)> 
     --> <Marked_ack_in(s(m),0) , Marked_ack_in(m,s(0))>
  <Marked_ack_in(s(m),s(n)) , Marked_ack_in(s(m),n)> 
     --> <Marked_ack_in(s(m),s(n)) , Marked_ack_in(s(m),n)>
  <Marked_ack_in(s(m),s(n)) , Marked_ack_in(s(m),n)> 
     --> <Marked_ack_in(s(m),s(n)) , Marked_u21(ack_in(s(m),n),m)>
  <Marked_ack_in(s(m),s(n)) , Marked_ack_in(s(m),n)> 
     --> <Marked_ack_in(s(m),0) , Marked_ack_in(m,s(0))>
  <Marked_ack_in(s(m),s(n)) , Marked_u21(ack_in(s(m),n),m)> 
     --> <Marked_u21(ack_out(n),m) , Marked_ack_in(m,n)>
  <Marked_ack_in(s(m),0) , Marked_ack_in(m,s(0))> 
     --> <Marked_ack_in(s(m),s(n)) , Marked_ack_in(s(m),n)>
  <Marked_ack_in(s(m),0) , Marked_ack_in(m,s(0))> 
     --> <Marked_ack_in(s(m),s(n)) , Marked_u21(ack_in(s(m),n),m)>
}
APPLY CRITERIA (Subterm criterion)

ST: Marked_u21 -> 2
Marked_ack_in -> 1

APPLY CRITERIA (Graph splitting)
Found 1 components:

{  <Marked_ack_in(s(m),s(n)) , Marked_ack_in(s(m),n)> 
      --> <Marked_ack_in(s(m),s(n)) , Marked_ack_in(s(m),n)>
}
APPLY CRITERIA (Subterm criterion)

ST: Marked_ack_in -> 2

APPLY CRITERIA (Graph splitting)
Found 0 components:
SOLVED
{
 
TRS termination of:
[1] ack_in(0,n) -> ack_out(s(n))
[2] ack_in(s(m),0) -> u11(ack_in(m,s(0)))
[3] u11(ack_out(n)) -> ack_out(n)
[4] ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
[5] u21(ack_out(n),m) -> u22(ack_in(m,n))
[6] u22(ack_out(n)) -> ack_out(n)
 ,
 CRITERION: MDP
 [ {
      
     DP termination of:
     <Marked_ack_in(s(m),0) , Marked_u11(ack_in(m,s(0)))>
<Marked_ack_in(s(m),0) , Marked_ack_in(m,s(0))>
<Marked_ack_in(s(m),s(n)) , Marked_u21(ack_in(s(m),n),m)>
<Marked_ack_in(s(m),s(n)) , Marked_ack_in(s(m),n)>
<Marked_u21(ack_out(n),m) , Marked_u22(ack_in(m,n))>
<Marked_u21(ack_out(n),m) , Marked_ack_in(m,n)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_ack_in(s(m),0) , Marked_ack_in(m,s(0))>
<Marked_ack_in(s(m),s(n)) , Marked_u21(ack_in(s(m),n),m)>
<Marked_ack_in(s(m),s(n)) , Marked_ack_in(s(m),n)>
<Marked_u21(ack_out(n),m) , Marked_ack_in(m,n)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
     <Marked_u21(ack_out(n),m) , Marked_ack_in(m,n)>
<Marked_ack_in(s(m),s(n)) , Marked_ack_in(s(m),n)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_ack_in(s(m),s(n)) , Marked_ack_in(s(m),n)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 
 ]} 
 ]} 

Cime worked for 0.007565 seconds (real time)
Cime Exit Status: 0