MAYBE

Problem:
 active(f(X)) -> mark(if(X,c(),f(true())))
 active(if(true(),X,Y)) -> mark(X)
 active(if(false(),X,Y)) -> mark(Y)
 active(f(X)) -> f(active(X))
 active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
 active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
 f(mark(X)) -> mark(f(X))
 if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
 if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
 proper(f(X)) -> f(proper(X))
 proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
 proper(c()) -> ok(c())
 proper(true()) -> ok(true())
 proper(false()) -> ok(false())
 f(ok(X)) -> ok(f(X))
 if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 DP Processor:
  DPs:
   active#(f(X)) -> f#(true())
   active#(f(X)) -> if#(X,c(),f(true()))
   active#(f(X)) -> active#(X)
   active#(f(X)) -> f#(active(X))
   active#(if(X1,X2,X3)) -> active#(X1)
   active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3)
   active#(if(X1,X2,X3)) -> active#(X2)
   active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3)
   f#(mark(X)) -> f#(X)
   if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
   if#(X1,mark(X2),X3) -> if#(X1,X2,X3)
   proper#(f(X)) -> proper#(X)
   proper#(f(X)) -> f#(proper(X))
   proper#(if(X1,X2,X3)) -> proper#(X3)
   proper#(if(X1,X2,X3)) -> proper#(X2)
   proper#(if(X1,X2,X3)) -> proper#(X1)
   proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3))
   f#(ok(X)) -> f#(X)
   if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3)
   top#(mark(X)) -> proper#(X)
   top#(mark(X)) -> top#(proper(X))
   top#(ok(X)) -> active#(X)
   top#(ok(X)) -> top#(active(X))
  TRS:
   active(f(X)) -> mark(if(X,c(),f(true())))
   active(if(true(),X,Y)) -> mark(X)
   active(if(false(),X,Y)) -> mark(Y)
   active(f(X)) -> f(active(X))
   active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
   active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
   f(mark(X)) -> mark(f(X))
   if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
   if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
   proper(f(X)) -> f(proper(X))
   proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
   proper(c()) -> ok(c())
   proper(true()) -> ok(true())
   proper(false()) -> ok(false())
   f(ok(X)) -> ok(f(X))
   if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  TDG Processor:
   DPs:
    active#(f(X)) -> f#(true())
    active#(f(X)) -> if#(X,c(),f(true()))
    active#(f(X)) -> active#(X)
    active#(f(X)) -> f#(active(X))
    active#(if(X1,X2,X3)) -> active#(X1)
    active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3)
    active#(if(X1,X2,X3)) -> active#(X2)
    active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3)
    f#(mark(X)) -> f#(X)
    if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
    if#(X1,mark(X2),X3) -> if#(X1,X2,X3)
    proper#(f(X)) -> proper#(X)
    proper#(f(X)) -> f#(proper(X))
    proper#(if(X1,X2,X3)) -> proper#(X3)
    proper#(if(X1,X2,X3)) -> proper#(X2)
    proper#(if(X1,X2,X3)) -> proper#(X1)
    proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3))
    f#(ok(X)) -> f#(X)
    if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3)
    top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X))
   TRS:
    active(f(X)) -> mark(if(X,c(),f(true())))
    active(if(true(),X,Y)) -> mark(X)
    active(if(false(),X,Y)) -> mark(Y)
    active(f(X)) -> f(active(X))
    active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
    active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
    f(mark(X)) -> mark(f(X))
    if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
    if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
    proper(f(X)) -> f(proper(X))
    proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
    proper(c()) -> ok(c())
    proper(true()) -> ok(true())
    proper(false()) -> ok(false())
    f(ok(X)) -> ok(f(X))
    if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
    top(mark(X)) -> top(proper(X))
    top(ok(X)) -> top(active(X))
   graph:
    top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X))
    top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X)) ->
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X)
    top#(ok(X)) -> active#(X) ->
    active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3)
    top#(ok(X)) -> active#(X) -> active#(if(X1,X2,X3)) -> active#(X2)
    top#(ok(X)) -> active#(X) ->
    active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3)
    top#(ok(X)) -> active#(X) -> active#(if(X1,X2,X3)) -> active#(X1)
    top#(ok(X)) -> active#(X) -> active#(f(X)) -> f#(active(X))
    top#(ok(X)) -> active#(X) -> active#(f(X)) -> active#(X)
    top#(ok(X)) -> active#(X) -> active#(f(X)) -> if#(X,c(),f(true()))
    top#(ok(X)) -> active#(X) -> active#(f(X)) -> f#(true())
    top#(mark(X)) -> top#(proper(X)) ->
    top#(ok(X)) -> top#(active(X))
    top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X)
    top#(mark(X)) -> top#(proper(X)) ->
    top#(mark(X)) -> top#(proper(X))
    top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) ->
    proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3))
    top#(mark(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X2)
    top#(mark(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X3)
    top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X)
    proper#(if(X1,X2,X3)) -> proper#(X3) ->
    proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3))
    proper#(if(X1,X2,X3)) -> proper#(X3) ->
    proper#(if(X1,X2,X3)) -> proper#(X1)
    proper#(if(X1,X2,X3)) -> proper#(X3) ->
    proper#(if(X1,X2,X3)) -> proper#(X2)
    proper#(if(X1,X2,X3)) -> proper#(X3) ->
    proper#(if(X1,X2,X3)) -> proper#(X3)
    proper#(if(X1,X2,X3)) -> proper#(X3) ->
    proper#(f(X)) -> f#(proper(X))
    proper#(if(X1,X2,X3)) -> proper#(X3) ->
    proper#(f(X)) -> proper#(X)
    proper#(if(X1,X2,X3)) -> proper#(X2) ->
    proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3))
    proper#(if(X1,X2,X3)) -> proper#(X2) ->
    proper#(if(X1,X2,X3)) -> proper#(X1)
    proper#(if(X1,X2,X3)) -> proper#(X2) ->
    proper#(if(X1,X2,X3)) -> proper#(X2)
    proper#(if(X1,X2,X3)) -> proper#(X2) ->
    proper#(if(X1,X2,X3)) -> proper#(X3)
    proper#(if(X1,X2,X3)) -> proper#(X2) ->
    proper#(f(X)) -> f#(proper(X))
    proper#(if(X1,X2,X3)) -> proper#(X2) ->
    proper#(f(X)) -> proper#(X)
    proper#(if(X1,X2,X3)) -> proper#(X1) ->
    proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3))
    proper#(if(X1,X2,X3)) -> proper#(X1) ->
    proper#(if(X1,X2,X3)) -> proper#(X1)
    proper#(if(X1,X2,X3)) -> proper#(X1) ->
    proper#(if(X1,X2,X3)) -> proper#(X2)
    proper#(if(X1,X2,X3)) -> proper#(X1) ->
    proper#(if(X1,X2,X3)) -> proper#(X3)
    proper#(if(X1,X2,X3)) -> proper#(X1) ->
    proper#(f(X)) -> f#(proper(X))
    proper#(if(X1,X2,X3)) -> proper#(X1) ->
    proper#(f(X)) -> proper#(X)
    proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) ->
    if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3)
    proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) ->
    if#(X1,mark(X2),X3) -> if#(X1,X2,X3)
    proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3)) ->
    if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
    proper#(f(X)) -> proper#(X) ->
    proper#(if(X1,X2,X3)) -> if#(proper(X1),proper(X2),proper(X3))
    proper#(f(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X1)
    proper#(f(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X2)
    proper#(f(X)) -> proper#(X) -> proper#(if(X1,X2,X3)) -> proper#(X3)
    proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X))
    proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X)
    proper#(f(X)) -> f#(proper(X)) -> f#(ok(X)) -> f#(X)
    proper#(f(X)) -> f#(proper(X)) ->
    f#(mark(X)) -> f#(X)
    if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) ->
    if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3)
    if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) ->
    if#(X1,mark(X2),X3) -> if#(X1,X2,X3)
    if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3) ->
    if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
    if#(mark(X1),X2,X3) -> if#(X1,X2,X3) ->
    if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3)
    if#(mark(X1),X2,X3) -> if#(X1,X2,X3) ->
    if#(X1,mark(X2),X3) -> if#(X1,X2,X3)
    if#(mark(X1),X2,X3) -> if#(X1,X2,X3) ->
    if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
    if#(X1,mark(X2),X3) -> if#(X1,X2,X3) ->
    if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3)
    if#(X1,mark(X2),X3) -> if#(X1,X2,X3) ->
    if#(X1,mark(X2),X3) -> if#(X1,X2,X3)
    if#(X1,mark(X2),X3) -> if#(X1,X2,X3) -> if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
    f#(ok(X)) -> f#(X) -> f#(ok(X)) -> f#(X)
    f#(ok(X)) -> f#(X) -> f#(mark(X)) -> f#(X)
    f#(mark(X)) -> f#(X) -> f#(ok(X)) -> f#(X)
    f#(mark(X)) -> f#(X) -> f#(mark(X)) -> f#(X)
    active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) ->
    if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3)
    active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) ->
    if#(X1,mark(X2),X3) -> if#(X1,X2,X3)
    active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3) ->
    if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
    active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3) ->
    if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3)
    active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3) ->
    if#(X1,mark(X2),X3) -> if#(X1,X2,X3)
    active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3) ->
    if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
    active#(if(X1,X2,X3)) -> active#(X2) ->
    active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3)
    active#(if(X1,X2,X3)) -> active#(X2) ->
    active#(if(X1,X2,X3)) -> active#(X2)
    active#(if(X1,X2,X3)) -> active#(X2) ->
    active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3)
    active#(if(X1,X2,X3)) -> active#(X2) ->
    active#(if(X1,X2,X3)) -> active#(X1)
    active#(if(X1,X2,X3)) -> active#(X2) ->
    active#(f(X)) -> f#(active(X))
    active#(if(X1,X2,X3)) -> active#(X2) ->
    active#(f(X)) -> active#(X)
    active#(if(X1,X2,X3)) -> active#(X2) ->
    active#(f(X)) -> if#(X,c(),f(true()))
    active#(if(X1,X2,X3)) -> active#(X2) ->
    active#(f(X)) -> f#(true())
    active#(if(X1,X2,X3)) -> active#(X1) ->
    active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3)
    active#(if(X1,X2,X3)) -> active#(X1) ->
    active#(if(X1,X2,X3)) -> active#(X2)
    active#(if(X1,X2,X3)) -> active#(X1) ->
    active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3)
    active#(if(X1,X2,X3)) -> active#(X1) ->
    active#(if(X1,X2,X3)) -> active#(X1)
    active#(if(X1,X2,X3)) -> active#(X1) ->
    active#(f(X)) -> f#(active(X))
    active#(if(X1,X2,X3)) -> active#(X1) ->
    active#(f(X)) -> active#(X)
    active#(if(X1,X2,X3)) -> active#(X1) ->
    active#(f(X)) -> if#(X,c(),f(true()))
    active#(if(X1,X2,X3)) -> active#(X1) ->
    active#(f(X)) -> f#(true())
    active#(f(X)) -> if#(X,c(),f(true())) ->
    if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3)
    active#(f(X)) -> if#(X,c(),f(true())) ->
    if#(X1,mark(X2),X3) -> if#(X1,X2,X3)
    active#(f(X)) -> if#(X,c(),f(true())) ->
    if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
    active#(f(X)) -> f#(true()) -> f#(ok(X)) -> f#(X)
    active#(f(X)) -> f#(true()) -> f#(mark(X)) -> f#(X)
    active#(f(X)) -> f#(active(X)) -> f#(ok(X)) -> f#(X)
    active#(f(X)) -> f#(active(X)) -> f#(mark(X)) -> f#(X)
    active#(f(X)) -> active#(X) ->
    active#(if(X1,X2,X3)) -> if#(X1,active(X2),X3)
    active#(f(X)) -> active#(X) -> active#(if(X1,X2,X3)) -> active#(X2)
    active#(f(X)) -> active#(X) ->
    active#(if(X1,X2,X3)) -> if#(active(X1),X2,X3)
    active#(f(X)) -> active#(X) -> active#(if(X1,X2,X3)) -> active#(X1)
    active#(f(X)) -> active#(X) -> active#(f(X)) -> f#(active(X))
    active#(f(X)) -> active#(X) -> active#(f(X)) -> active#(X)
    active#(f(X)) -> active#(X) -> active#(f(X)) -> if#(X,c(),f(true()))
    active#(f(X)) -> active#(X) -> active#(f(X)) -> f#(true())
   SCC Processor:
    #sccs: 5
    #rules: 14
    #arcs: 101/529
    DPs:
     top#(ok(X)) -> top#(active(X))
     top#(mark(X)) -> top#(proper(X))
    TRS:
     active(f(X)) -> mark(if(X,c(),f(true())))
     active(if(true(),X,Y)) -> mark(X)
     active(if(false(),X,Y)) -> mark(Y)
     active(f(X)) -> f(active(X))
     active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
     active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
     f(mark(X)) -> mark(f(X))
     if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
     if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
     proper(f(X)) -> f(proper(X))
     proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
     proper(c()) -> ok(c())
     proper(true()) -> ok(true())
     proper(false()) -> ok(false())
     f(ok(X)) -> ok(f(X))
     if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Open
    
    DPs:
     active#(f(X)) -> active#(X)
     active#(if(X1,X2,X3)) -> active#(X1)
     active#(if(X1,X2,X3)) -> active#(X2)
    TRS:
     active(f(X)) -> mark(if(X,c(),f(true())))
     active(if(true(),X,Y)) -> mark(X)
     active(if(false(),X,Y)) -> mark(Y)
     active(f(X)) -> f(active(X))
     active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
     active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
     f(mark(X)) -> mark(f(X))
     if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
     if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
     proper(f(X)) -> f(proper(X))
     proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
     proper(c()) -> ok(c())
     proper(true()) -> ok(true())
     proper(false()) -> ok(false())
     f(ok(X)) -> ok(f(X))
     if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(active#) = 0
     problem:
      DPs:
       
      TRS:
       active(f(X)) -> mark(if(X,c(),f(true())))
       active(if(true(),X,Y)) -> mark(X)
       active(if(false(),X,Y)) -> mark(Y)
       active(f(X)) -> f(active(X))
       active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
       active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
       f(mark(X)) -> mark(f(X))
       if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
       if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
       proper(f(X)) -> f(proper(X))
       proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
       proper(c()) -> ok(c())
       proper(true()) -> ok(true())
       proper(false()) -> ok(false())
       f(ok(X)) -> ok(f(X))
       if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     proper#(f(X)) -> proper#(X)
     proper#(if(X1,X2,X3)) -> proper#(X3)
     proper#(if(X1,X2,X3)) -> proper#(X2)
     proper#(if(X1,X2,X3)) -> proper#(X1)
    TRS:
     active(f(X)) -> mark(if(X,c(),f(true())))
     active(if(true(),X,Y)) -> mark(X)
     active(if(false(),X,Y)) -> mark(Y)
     active(f(X)) -> f(active(X))
     active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
     active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
     f(mark(X)) -> mark(f(X))
     if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
     if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
     proper(f(X)) -> f(proper(X))
     proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
     proper(c()) -> ok(c())
     proper(true()) -> ok(true())
     proper(false()) -> ok(false())
     f(ok(X)) -> ok(f(X))
     if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(proper#) = 0
     problem:
      DPs:
       
      TRS:
       active(f(X)) -> mark(if(X,c(),f(true())))
       active(if(true(),X,Y)) -> mark(X)
       active(if(false(),X,Y)) -> mark(Y)
       active(f(X)) -> f(active(X))
       active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
       active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
       f(mark(X)) -> mark(f(X))
       if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
       if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
       proper(f(X)) -> f(proper(X))
       proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
       proper(c()) -> ok(c())
       proper(true()) -> ok(true())
       proper(false()) -> ok(false())
       f(ok(X)) -> ok(f(X))
       if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
     if#(X1,mark(X2),X3) -> if#(X1,X2,X3)
     if#(ok(X1),ok(X2),ok(X3)) -> if#(X1,X2,X3)
    TRS:
     active(f(X)) -> mark(if(X,c(),f(true())))
     active(if(true(),X,Y)) -> mark(X)
     active(if(false(),X,Y)) -> mark(Y)
     active(f(X)) -> f(active(X))
     active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
     active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
     f(mark(X)) -> mark(f(X))
     if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
     if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
     proper(f(X)) -> f(proper(X))
     proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
     proper(c()) -> ok(c())
     proper(true()) -> ok(true())
     proper(false()) -> ok(false())
     f(ok(X)) -> ok(f(X))
     if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(if#) = 2
     problem:
      DPs:
       if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
       if#(X1,mark(X2),X3) -> if#(X1,X2,X3)
      TRS:
       active(f(X)) -> mark(if(X,c(),f(true())))
       active(if(true(),X,Y)) -> mark(X)
       active(if(false(),X,Y)) -> mark(Y)
       active(f(X)) -> f(active(X))
       active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
       active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
       f(mark(X)) -> mark(f(X))
       if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
       if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
       proper(f(X)) -> f(proper(X))
       proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
       proper(c()) -> ok(c())
       proper(true()) -> ok(true())
       proper(false()) -> ok(false())
       f(ok(X)) -> ok(f(X))
       if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(if#) = 1
      problem:
       DPs:
        if#(mark(X1),X2,X3) -> if#(X1,X2,X3)
       TRS:
        active(f(X)) -> mark(if(X,c(),f(true())))
        active(if(true(),X,Y)) -> mark(X)
        active(if(false(),X,Y)) -> mark(Y)
        active(f(X)) -> f(active(X))
        active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
        active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
        f(mark(X)) -> mark(f(X))
        if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
        if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
        proper(f(X)) -> f(proper(X))
        proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
        proper(c()) -> ok(c())
        proper(true()) -> ok(true())
        proper(false()) -> ok(false())
        f(ok(X)) -> ok(f(X))
        if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Subterm Criterion Processor:
       simple projection:
        pi(if#) = 0
       problem:
        DPs:
         
        TRS:
         active(f(X)) -> mark(if(X,c(),f(true())))
         active(if(true(),X,Y)) -> mark(X)
         active(if(false(),X,Y)) -> mark(Y)
         active(f(X)) -> f(active(X))
         active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
         active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
         f(mark(X)) -> mark(f(X))
         if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
         if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
         proper(f(X)) -> f(proper(X))
         proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
         proper(c()) -> ok(c())
         proper(true()) -> ok(true())
         proper(false()) -> ok(false())
         f(ok(X)) -> ok(f(X))
         if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
         top(mark(X)) -> top(proper(X))
         top(ok(X)) -> top(active(X))
       Qed
    
    DPs:
     f#(mark(X)) -> f#(X)
     f#(ok(X)) -> f#(X)
    TRS:
     active(f(X)) -> mark(if(X,c(),f(true())))
     active(if(true(),X,Y)) -> mark(X)
     active(if(false(),X,Y)) -> mark(Y)
     active(f(X)) -> f(active(X))
     active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
     active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
     f(mark(X)) -> mark(f(X))
     if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
     if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
     proper(f(X)) -> f(proper(X))
     proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
     proper(c()) -> ok(c())
     proper(true()) -> ok(true())
     proper(false()) -> ok(false())
     f(ok(X)) -> ok(f(X))
     if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(f#) = 0
     problem:
      DPs:
       
      TRS:
       active(f(X)) -> mark(if(X,c(),f(true())))
       active(if(true(),X,Y)) -> mark(X)
       active(if(false(),X,Y)) -> mark(Y)
       active(f(X)) -> f(active(X))
       active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
       active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
       f(mark(X)) -> mark(f(X))
       if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
       if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
       proper(f(X)) -> f(proper(X))
       proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
       proper(c()) -> ok(c())
       proper(true()) -> ok(true())
       proper(false()) -> ok(false())
       f(ok(X)) -> ok(f(X))
       if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed