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))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [active#](x0) = x0 + 1,
      
      [top](x0) = 0,
      
      [ok](x0) = x0,
      
      [proper](x0) = x0,
      
      [false] = 1,
      
      [mark](x0) = x0,
      
      [if](x0, x1, x2) = x0 + x1 + x2 + 1,
      
      [true] = 0,
      
      [c] = 0,
      
      [active](x0) = x0 + 1,
      
      [f](x0) = x0 + 1
     orientation:
      active#(f(X)) = X + 2 >= X + 1 = active#(X)
      
      active#(if(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = active#(X1)
      
      active#(if(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = active#(X2)
      
      active(f(X)) = X + 2 >= X + 2 = mark(if(X,c(),f(true())))
      
      active(if(true(),X,Y)) = X + Y + 2 >= X = mark(X)
      
      active(if(false(),X,Y)) = X + Y + 3 >= Y = mark(Y)
      
      active(f(X)) = X + 2 >= X + 2 = f(active(X))
      
      active(if(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = if(active(X1),X2,X3)
      
      active(if(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = if(X1,active(X2),X3)
      
      f(mark(X)) = X + 1 >= X + 1 = mark(f(X))
      
      if(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(if(X1,X2,X3))
      
      if(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(if(X1,X2,X3))
      
      proper(f(X)) = X + 1 >= X + 1 = f(proper(X))
      
      proper(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(proper(X1),proper(X2),proper(X3))
      
      proper(c()) = 0 >= 0 = ok(c())
      
      proper(true()) = 0 >= 0 = ok(true())
      
      proper(false()) = 1 >= 1 = ok(false())
      
      f(ok(X)) = X + 1 >= X + 1 = ok(f(X))
      
      if(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = ok(if(X1,X2,X3))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     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))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [proper#](x0) = x0 + 1,
      
      [top](x0) = 0,
      
      [ok](x0) = x0,
      
      [proper](x0) = x0,
      
      [false] = 0,
      
      [mark](x0) = 1,
      
      [if](x0, x1, x2) = x0 + x1 + x2,
      
      [true] = 1,
      
      [c] = 0,
      
      [active](x0) = x0 + 1,
      
      [f](x0) = x0 + 1
     orientation:
      proper#(f(X)) = X + 2 >= X + 1 = proper#(X)
      
      proper#(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X3 + 1 = proper#(X3)
      
      proper#(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X2 + 1 = proper#(X2)
      
      proper#(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + 1 = proper#(X1)
      
      active(f(X)) = X + 2 >= 1 = mark(if(X,c(),f(true())))
      
      active(if(true(),X,Y)) = X + Y + 2 >= 1 = mark(X)
      
      active(if(false(),X,Y)) = X + Y + 1 >= 1 = mark(Y)
      
      active(f(X)) = X + 2 >= X + 2 = f(active(X))
      
      active(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(active(X1),X2,X3)
      
      active(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(X1,active(X2),X3)
      
      f(mark(X)) = 2 >= 1 = mark(f(X))
      
      if(mark(X1),X2,X3) = X2 + X3 + 1 >= 1 = mark(if(X1,X2,X3))
      
      if(X1,mark(X2),X3) = X1 + X3 + 1 >= 1 = mark(if(X1,X2,X3))
      
      proper(f(X)) = X + 1 >= X + 1 = f(proper(X))
      
      proper(if(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = if(proper(X1),proper(X2),proper(X3))
      
      proper(c()) = 0 >= 0 = ok(c())
      
      proper(true()) = 1 >= 1 = ok(true())
      
      proper(false()) = 0 >= 0 = ok(false())
      
      f(ok(X)) = X + 1 >= X + 1 = ok(f(X))
      
      if(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(if(X1,X2,X3))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     problem:
      DPs:
       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))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [proper#](x0) = x0 + 1,
       
       [top](x0) = 1,
       
       [ok](x0) = x0,
       
       [proper](x0) = x0,
       
       [false] = 0,
       
       [mark](x0) = 1,
       
       [if](x0, x1, x2) = x0 + x1 + x2 + 1,
       
       [true] = 0,
       
       [c] = 0,
       
       [active](x0) = x0,
       
       [f](x0) = 1
      orientation:
       proper#(if(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X3 + 1 = proper#(X3)
       
       proper#(if(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = proper#(X2)
       
       proper#(if(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = proper#(X1)
       
       active(f(X)) = 1 >= 1 = mark(if(X,c(),f(true())))
       
       active(if(true(),X,Y)) = X + Y + 1 >= 1 = mark(X)
       
       active(if(false(),X,Y)) = X + Y + 1 >= 1 = mark(Y)
       
       active(f(X)) = 1 >= 1 = f(active(X))
       
       active(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(active(X1),X2,X3)
       
       active(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(X1,active(X2),X3)
       
       f(mark(X)) = 1 >= 1 = mark(f(X))
       
       if(mark(X1),X2,X3) = X2 + X3 + 2 >= 1 = mark(if(X1,X2,X3))
       
       if(X1,mark(X2),X3) = X1 + X3 + 2 >= 1 = mark(if(X1,X2,X3))
       
       proper(f(X)) = 1 >= 1 = f(proper(X))
       
       proper(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(proper(X1),proper(X2),proper(X3))
       
       proper(c()) = 0 >= 0 = ok(c())
       
       proper(true()) = 0 >= 0 = ok(true())
       
       proper(false()) = 0 >= 0 = ok(false())
       
       f(ok(X)) = 1 >= 1 = ok(f(X))
       
       if(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = ok(if(X1,X2,X3))
       
       top(mark(X)) = 1 >= 1 = top(proper(X))
       
       top(ok(X)) = 1 >= 1 = top(active(X))
      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))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [if#](x0, x1, x2) = x0 + x1 + 1,
      
      [top](x0) = 0,
      
      [ok](x0) = x0,
      
      [proper](x0) = x0,
      
      [false] = 1,
      
      [mark](x0) = x0 + 1,
      
      [if](x0, x1, x2) = x0 + x1 + x2,
      
      [true] = 0,
      
      [c] = 0,
      
      [active](x0) = x0 + 1,
      
      [f](x0) = x0 + 1
     orientation:
      if#(mark(X1),X2,X3) = X1 + X2 + 2 >= X1 + X2 + 1 = if#(X1,X2,X3)
      
      if#(X1,mark(X2),X3) = X1 + X2 + 2 >= X1 + X2 + 1 = if#(X1,X2,X3)
      
      if#(ok(X1),ok(X2),ok(X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = if#(X1,X2,X3)
      
      active(f(X)) = X + 2 >= X + 2 = mark(if(X,c(),f(true())))
      
      active(if(true(),X,Y)) = X + Y + 1 >= X + 1 = mark(X)
      
      active(if(false(),X,Y)) = X + Y + 2 >= Y + 1 = mark(Y)
      
      active(f(X)) = X + 2 >= X + 2 = f(active(X))
      
      active(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(active(X1),X2,X3)
      
      active(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(X1,active(X2),X3)
      
      f(mark(X)) = X + 2 >= X + 2 = mark(f(X))
      
      if(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(if(X1,X2,X3))
      
      if(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(if(X1,X2,X3))
      
      proper(f(X)) = X + 1 >= X + 1 = f(proper(X))
      
      proper(if(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = if(proper(X1),proper(X2),proper(X3))
      
      proper(c()) = 0 >= 0 = ok(c())
      
      proper(true()) = 0 >= 0 = ok(true())
      
      proper(false()) = 1 >= 1 = ok(false())
      
      f(ok(X)) = X + 1 >= X + 1 = ok(f(X))
      
      if(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(if(X1,X2,X3))
      
      top(mark(X)) = 0 >= 0 = top(proper(X))
      
      top(ok(X)) = 0 >= 0 = top(active(X))
     problem:
      DPs:
       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))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [if#](x0, x1, x2) = x2 + 1,
       
       [top](x0) = 0,
       
       [ok](x0) = x0 + 1,
       
       [proper](x0) = x0 + 1,
       
       [false] = 1,
       
       [mark](x0) = 0,
       
       [if](x0, x1, x2) = x0 + 1,
       
       [true] = 1,
       
       [c] = 1,
       
       [active](x0) = x0,
       
       [f](x0) = x0
      orientation:
       if#(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 1 = if#(X1,X2,X3)
       
       active(f(X)) = X >= 0 = mark(if(X,c(),f(true())))
       
       active(if(true(),X,Y)) = 2 >= 0 = mark(X)
       
       active(if(false(),X,Y)) = 2 >= 0 = mark(Y)
       
       active(f(X)) = X >= X = f(active(X))
       
       active(if(X1,X2,X3)) = X1 + 1 >= X1 + 1 = if(active(X1),X2,X3)
       
       active(if(X1,X2,X3)) = X1 + 1 >= X1 + 1 = if(X1,active(X2),X3)
       
       f(mark(X)) = 0 >= 0 = mark(f(X))
       
       if(mark(X1),X2,X3) = 1 >= 0 = mark(if(X1,X2,X3))
       
       if(X1,mark(X2),X3) = X1 + 1 >= 0 = mark(if(X1,X2,X3))
       
       proper(f(X)) = X + 1 >= X + 1 = f(proper(X))
       
       proper(if(X1,X2,X3)) = X1 + 2 >= X1 + 2 = if(proper(X1),proper(X2),proper(X3))
       
       proper(c()) = 2 >= 2 = ok(c())
       
       proper(true()) = 2 >= 2 = ok(true())
       
       proper(false()) = 2 >= 2 = ok(false())
       
       f(ok(X)) = X + 1 >= X + 1 = ok(f(X))
       
       if(ok(X1),ok(X2),ok(X3)) = X1 + 2 >= X1 + 2 = ok(if(X1,X2,X3))
       
       top(mark(X)) = 0 >= 0 = top(proper(X))
       
       top(ok(X)) = 0 >= 0 = top(active(X))
      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))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [f#](x0) = x0 + 1,
      
      [top](x0) = 1,
      
      [ok](x0) = x0,
      
      [proper](x0) = 0,
      
      [false] = 0,
      
      [mark](x0) = x0 + 1,
      
      [if](x0, x1, x2) = x0 + x1 + x2,
      
      [true] = 0,
      
      [c] = 0,
      
      [active](x0) = x0 + 1,
      
      [f](x0) = x0
     orientation:
      f#(mark(X)) = X + 2 >= X + 1 = f#(X)
      
      f#(ok(X)) = X + 1 >= X + 1 = f#(X)
      
      active(f(X)) = X + 1 >= X + 1 = mark(if(X,c(),f(true())))
      
      active(if(true(),X,Y)) = X + Y + 1 >= X + 1 = mark(X)
      
      active(if(false(),X,Y)) = X + Y + 1 >= Y + 1 = mark(Y)
      
      active(f(X)) = X + 1 >= X + 1 = f(active(X))
      
      active(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(active(X1),X2,X3)
      
      active(if(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = if(X1,active(X2),X3)
      
      f(mark(X)) = X + 1 >= X + 1 = mark(f(X))
      
      if(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(if(X1,X2,X3))
      
      if(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(if(X1,X2,X3))
      
      proper(f(X)) = 0 >= 0 = f(proper(X))
      
      proper(if(X1,X2,X3)) = 0 >= 0 = if(proper(X1),proper(X2),proper(X3))
      
      proper(c()) = 0 >= 0 = ok(c())
      
      proper(true()) = 0 >= 0 = ok(true())
      
      proper(false()) = 0 >= 0 = ok(false())
      
      f(ok(X)) = X >= X = ok(f(X))
      
      if(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(if(X1,X2,X3))
      
      top(mark(X)) = 1 >= 1 = top(proper(X))
      
      top(ok(X)) = 1 >= 1 = top(active(X))
     problem:
      DPs:
       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))
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [f#](x0) = x0 + 1,
       
       [top](x0) = 0,
       
       [ok](x0) = x0 + 1,
       
       [proper](x0) = x0 + 1,
       
       [false] = 0,
       
       [mark](x0) = 0,
       
       [if](x0, x1, x2) = x2 + 1,
       
       [true] = 0,
       
       [c] = 1,
       
       [active](x0) = x0,
       
       [f](x0) = x0
      orientation:
       f#(ok(X)) = X + 2 >= X + 1 = f#(X)
       
       active(f(X)) = X >= 0 = mark(if(X,c(),f(true())))
       
       active(if(true(),X,Y)) = Y + 1 >= 0 = mark(X)
       
       active(if(false(),X,Y)) = Y + 1 >= 0 = mark(Y)
       
       active(f(X)) = X >= X = f(active(X))
       
       active(if(X1,X2,X3)) = X3 + 1 >= X3 + 1 = if(active(X1),X2,X3)
       
       active(if(X1,X2,X3)) = X3 + 1 >= X3 + 1 = if(X1,active(X2),X3)
       
       f(mark(X)) = 0 >= 0 = mark(f(X))
       
       if(mark(X1),X2,X3) = X3 + 1 >= 0 = mark(if(X1,X2,X3))
       
       if(X1,mark(X2),X3) = X3 + 1 >= 0 = mark(if(X1,X2,X3))
       
       proper(f(X)) = X + 1 >= X + 1 = f(proper(X))
       
       proper(if(X1,X2,X3)) = X3 + 2 >= X3 + 2 = if(proper(X1),proper(X2),proper(X3))
       
       proper(c()) = 2 >= 2 = ok(c())
       
       proper(true()) = 1 >= 1 = ok(true())
       
       proper(false()) = 1 >= 1 = ok(false())
       
       f(ok(X)) = X + 1 >= X + 1 = ok(f(X))
       
       if(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 2 = ok(if(X1,X2,X3))
       
       top(mark(X)) = 0 >= 0 = top(proper(X))
       
       top(ok(X)) = 0 >= 0 = top(active(X))
      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