YES

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

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [if](x0, x1, x2) = x0 + 4x1 + x2,
   
   [active](x0) = x0,
   
   [true] = 0,
   
   [mark](x0) = x0,
   
   [f](x0) = 2x0,
   
   [false] = 1,
   
   [c] = 0
  orientation:
   active(f(X)) = 2X >= X = mark(if(X,c(),f(true())))
   
   active(if(true(),X,Y)) = 4X + Y >= X = mark(X)
   
   active(if(false(),X,Y)) = 4X + Y + 1 >= Y = mark(Y)
   
   mark(f(X)) = 2X >= 2X = active(f(mark(X)))
   
   mark(if(X1,X2,X3)) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = active(if(mark(X1),mark(X2),X3))
   
   mark(c()) = 0 >= 0 = active(c())
   
   mark(true()) = 0 >= 0 = active(true())
   
   mark(false()) = 1 >= 1 = active(false())
   
   f(mark(X)) = 2X >= 2X = f(X)
   
   f(active(X)) = 2X >= 2X = f(X)
   
   if(mark(X1),X2,X3) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = if(X1,X2,X3)
   
   if(X1,mark(X2),X3) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = if(X1,X2,X3)
   
   if(X1,X2,mark(X3)) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = if(X1,X2,X3)
   
   if(active(X1),X2,X3) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = if(X1,X2,X3)
   
   if(X1,active(X2),X3) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = if(X1,X2,X3)
   
   if(X1,X2,active(X3)) = X1 + 4X2 + X3 >= X1 + 4X2 + X3 = if(X1,X2,X3)
  problem:
   active(f(X)) -> mark(if(X,c(),f(true())))
   active(if(true(),X,Y)) -> mark(X)
   mark(f(X)) -> active(f(mark(X)))
   mark(if(X1,X2,X3)) -> active(if(mark(X1),mark(X2),X3))
   mark(c()) -> active(c())
   mark(true()) -> active(true())
   mark(false()) -> active(false())
   f(mark(X)) -> f(X)
   f(active(X)) -> f(X)
   if(mark(X1),X2,X3) -> if(X1,X2,X3)
   if(X1,mark(X2),X3) -> if(X1,X2,X3)
   if(X1,X2,mark(X3)) -> if(X1,X2,X3)
   if(active(X1),X2,X3) -> if(X1,X2,X3)
   if(X1,active(X2),X3) -> if(X1,X2,X3)
   if(X1,X2,active(X3)) -> if(X1,X2,X3)
  Matrix Interpretation Processor: dim=3
   
   interpretation:
                       [1 0 1]     [1 0 1]     [1 1 0]  
    [if](x0, x1, x2) = [0 0 0]x0 + [0 1 0]x1 + [0 0 0]x2
                       [0 0 0]     [0 0 0]     [0 0 0]  ,
    
                   [1 0 1]  
    [active](x0) = [0 1 0]x0
                   [0 0 0]  ,
    
             [0]
    [true] = [0]
             [0],
    
                 [1 0 1]  
    [mark](x0) = [0 1 0]x0
                 [0 0 0]  ,
    
              [1 0 1]     [0]
    [f](x0) = [0 0 0]x0 + [0]
              [0 0 0]     [1],
    
              [0]
    [false] = [0]
              [0],
    
          [0]
    [c] = [0]
          [0]
   orientation:
                   [1 0 1]    [1]    [1 0 1]                             
    active(f(X)) = [0 0 0]X + [0] >= [0 0 0]X = mark(if(X,c(),f(true())))
                   [0 0 0]    [0]    [0 0 0]                             
    
                             [1 0 1]    [1 1 0]     [1 0 1]           
    active(if(true(),X,Y)) = [0 1 0]X + [0 0 0]Y >= [0 1 0]X = mark(X)
                             [0 0 0]    [0 0 0]     [0 0 0]           
    
                 [1 0 1]    [1]    [1 0 1]    [1]                     
    mark(f(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(f(mark(X)))
                 [0 0 0]    [0]    [0 0 0]    [0]                     
    
                         [1 0 1]     [1 0 1]     [1 1 0]      [1 0 1]     [1 0 1]     [1 1 0]                                     
    mark(if(X1,X2,X3)) = [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 = active(if(mark(X1),mark(X2),X3))
                         [0 0 0]     [0 0 0]     [0 0 0]      [0 0 0]     [0 0 0]     [0 0 0]                                     
    
                [0]    [0]              
    mark(c()) = [0] >= [0] = active(c())
                [0]    [0]              
    
                   [0]    [0]                 
    mark(true()) = [0] >= [0] = active(true())
                   [0]    [0]                 
    
                    [0]    [0]                  
    mark(false()) = [0] >= [0] = active(false())
                    [0]    [0]                  
    
                 [1 0 1]    [0]    [1 0 1]    [0]       
    f(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = f(X)
                 [0 0 0]    [1]    [0 0 0]    [1]       
    
                   [1 0 1]    [0]    [1 0 1]    [0]       
    f(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = f(X)
                   [0 0 0]    [1]    [0 0 0]    [1]       
    
                         [1 0 1]     [1 0 1]     [1 1 0]      [1 0 1]     [1 0 1]     [1 1 0]                 
    if(mark(X1),X2,X3) = [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 = if(X1,X2,X3)
                         [0 0 0]     [0 0 0]     [0 0 0]      [0 0 0]     [0 0 0]     [0 0 0]                 
    
                         [1 0 1]     [1 0 1]     [1 1 0]      [1 0 1]     [1 0 1]     [1 1 0]                 
    if(X1,mark(X2),X3) = [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 = if(X1,X2,X3)
                         [0 0 0]     [0 0 0]     [0 0 0]      [0 0 0]     [0 0 0]     [0 0 0]                 
    
                         [1 0 1]     [1 0 1]     [1 1 1]      [1 0 1]     [1 0 1]     [1 1 0]                 
    if(X1,X2,mark(X3)) = [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 = if(X1,X2,X3)
                         [0 0 0]     [0 0 0]     [0 0 0]      [0 0 0]     [0 0 0]     [0 0 0]                 
    
                           [1 0 1]     [1 0 1]     [1 1 0]      [1 0 1]     [1 0 1]     [1 1 0]                 
    if(active(X1),X2,X3) = [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 = if(X1,X2,X3)
                           [0 0 0]     [0 0 0]     [0 0 0]      [0 0 0]     [0 0 0]     [0 0 0]                 
    
                           [1 0 1]     [1 0 1]     [1 1 0]      [1 0 1]     [1 0 1]     [1 1 0]                 
    if(X1,active(X2),X3) = [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 = if(X1,X2,X3)
                           [0 0 0]     [0 0 0]     [0 0 0]      [0 0 0]     [0 0 0]     [0 0 0]                 
    
                           [1 0 1]     [1 0 1]     [1 1 1]      [1 0 1]     [1 0 1]     [1 1 0]                 
    if(X1,X2,active(X3)) = [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 = if(X1,X2,X3)
                           [0 0 0]     [0 0 0]     [0 0 0]      [0 0 0]     [0 0 0]     [0 0 0]                 
   problem:
    active(if(true(),X,Y)) -> mark(X)
    mark(f(X)) -> active(f(mark(X)))
    mark(if(X1,X2,X3)) -> active(if(mark(X1),mark(X2),X3))
    mark(c()) -> active(c())
    mark(true()) -> active(true())
    mark(false()) -> active(false())
    f(mark(X)) -> f(X)
    f(active(X)) -> f(X)
    if(mark(X1),X2,X3) -> if(X1,X2,X3)
    if(X1,mark(X2),X3) -> if(X1,X2,X3)
    if(X1,X2,mark(X3)) -> if(X1,X2,X3)
    if(active(X1),X2,X3) -> if(X1,X2,X3)
    if(X1,active(X2),X3) -> if(X1,X2,X3)
    if(X1,X2,active(X3)) -> if(X1,X2,X3)
   Matrix Interpretation Processor: dim=3
    
    interpretation:
                        [1 0 1]     [1 0 1]     [1 0 0]     [1]
     [if](x0, x1, x2) = [0 1 0]x0 + [0 1 0]x1 + [0 0 0]x2 + [0]
                        [0 1 0]     [0 1 0]     [0 0 0]     [0],
     
                      
     [active](x0) = x0
                      ,
     
              [0]
     [true] = [0]
              [0],
     
                  [1 0 1]  
     [mark](x0) = [0 1 0]x0
                  [0 1 0]  ,
     
               [1 0 1]     [0]
     [f](x0) = [0 1 0]x0 + [1]
               [0 1 0]     [1],
     
               [0]
     [false] = [0]
               [0],
     
           [0]
     [c] = [0]
           [0]
    orientation:
                              [1 0 1]    [1 0 0]    [1]    [1 0 1]           
     active(if(true(),X,Y)) = [0 1 0]X + [0 0 0]Y + [0] >= [0 1 0]X = mark(X)
                              [0 1 0]    [0 0 0]    [0]    [0 1 0]           
     
                  [1 1 1]    [1]    [1 1 1]    [0]                     
     mark(f(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = active(f(mark(X)))
                  [0 1 0]    [1]    [0 1 0]    [1]                     
     
                          [1 1 1]     [1 1 1]     [1 0 0]     [1]    [1 1 1]     [1 1 1]     [1 0 0]     [1]                                   
     mark(if(X1,X2,X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] = active(if(mark(X1),mark(X2),X3))
                          [0 1 0]     [0 1 0]     [0 0 0]     [0]    [0 1 0]     [0 1 0]     [0 0 0]     [0]                                   
     
                 [0]    [0]              
     mark(c()) = [0] >= [0] = active(c())
                 [0]    [0]              
     
                    [0]    [0]                 
     mark(true()) = [0] >= [0] = active(true())
                    [0]    [0]                 
     
                     [0]    [0]                  
     mark(false()) = [0] >= [0] = active(false())
                     [0]    [0]                  
     
                  [1 1 1]    [0]    [1 0 1]    [0]       
     f(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = f(X)
                  [0 1 0]    [1]    [0 1 0]    [1]       
     
                    [1 0 1]    [0]    [1 0 1]    [0]       
     f(active(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = f(X)
                    [0 1 0]    [1]    [0 1 0]    [1]       
     
                          [1 1 1]     [1 0 1]     [1 0 0]     [1]    [1 0 1]     [1 0 1]     [1 0 0]     [1]               
     if(mark(X1),X2,X3) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3)
                          [0 1 0]     [0 1 0]     [0 0 0]     [0]    [0 1 0]     [0 1 0]     [0 0 0]     [0]               
     
                          [1 0 1]     [1 1 1]     [1 0 0]     [1]    [1 0 1]     [1 0 1]     [1 0 0]     [1]               
     if(X1,mark(X2),X3) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3)
                          [0 1 0]     [0 1 0]     [0 0 0]     [0]    [0 1 0]     [0 1 0]     [0 0 0]     [0]               
     
                          [1 0 1]     [1 0 1]     [1 0 1]     [1]    [1 0 1]     [1 0 1]     [1 0 0]     [1]               
     if(X1,X2,mark(X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3)
                          [0 1 0]     [0 1 0]     [0 0 0]     [0]    [0 1 0]     [0 1 0]     [0 0 0]     [0]               
     
                            [1 0 1]     [1 0 1]     [1 0 0]     [1]    [1 0 1]     [1 0 1]     [1 0 0]     [1]               
     if(active(X1),X2,X3) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3)
                            [0 1 0]     [0 1 0]     [0 0 0]     [0]    [0 1 0]     [0 1 0]     [0 0 0]     [0]               
     
                            [1 0 1]     [1 0 1]     [1 0 0]     [1]    [1 0 1]     [1 0 1]     [1 0 0]     [1]               
     if(X1,active(X2),X3) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3)
                            [0 1 0]     [0 1 0]     [0 0 0]     [0]    [0 1 0]     [0 1 0]     [0 0 0]     [0]               
     
                            [1 0 1]     [1 0 1]     [1 0 0]     [1]    [1 0 1]     [1 0 1]     [1 0 0]     [1]               
     if(X1,X2,active(X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3)
                            [0 1 0]     [0 1 0]     [0 0 0]     [0]    [0 1 0]     [0 1 0]     [0 0 0]     [0]               
    problem:
     mark(if(X1,X2,X3)) -> active(if(mark(X1),mark(X2),X3))
     mark(c()) -> active(c())
     mark(true()) -> active(true())
     mark(false()) -> active(false())
     f(mark(X)) -> f(X)
     f(active(X)) -> f(X)
     if(mark(X1),X2,X3) -> if(X1,X2,X3)
     if(X1,mark(X2),X3) -> if(X1,X2,X3)
     if(X1,X2,mark(X3)) -> if(X1,X2,X3)
     if(active(X1),X2,X3) -> if(X1,X2,X3)
     if(X1,active(X2),X3) -> if(X1,X2,X3)
     if(X1,X2,active(X3)) -> if(X1,X2,X3)
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [if](x0, x1, x2) = x0 + 4x1 + 4x2,
      
      [active](x0) = x0,
      
      [true] = 4,
      
      [mark](x0) = 4x0,
      
      [f](x0) = 2x0,
      
      [false] = 6,
      
      [c] = 1
     orientation:
      mark(if(X1,X2,X3)) = 4X1 + 16X2 + 16X3 >= 4X1 + 16X2 + 4X3 = active(if(mark(X1),mark(X2),X3))
      
      mark(c()) = 4 >= 1 = active(c())
      
      mark(true()) = 16 >= 4 = active(true())
      
      mark(false()) = 24 >= 6 = active(false())
      
      f(mark(X)) = 8X >= 2X = f(X)
      
      f(active(X)) = 2X >= 2X = f(X)
      
      if(mark(X1),X2,X3) = 4X1 + 4X2 + 4X3 >= X1 + 4X2 + 4X3 = if(X1,X2,X3)
      
      if(X1,mark(X2),X3) = X1 + 16X2 + 4X3 >= X1 + 4X2 + 4X3 = if(X1,X2,X3)
      
      if(X1,X2,mark(X3)) = X1 + 4X2 + 16X3 >= X1 + 4X2 + 4X3 = if(X1,X2,X3)
      
      if(active(X1),X2,X3) = X1 + 4X2 + 4X3 >= X1 + 4X2 + 4X3 = if(X1,X2,X3)
      
      if(X1,active(X2),X3) = X1 + 4X2 + 4X3 >= X1 + 4X2 + 4X3 = if(X1,X2,X3)
      
      if(X1,X2,active(X3)) = X1 + 4X2 + 4X3 >= X1 + 4X2 + 4X3 = if(X1,X2,X3)
     problem:
      mark(if(X1,X2,X3)) -> active(if(mark(X1),mark(X2),X3))
      f(mark(X)) -> f(X)
      f(active(X)) -> f(X)
      if(mark(X1),X2,X3) -> if(X1,X2,X3)
      if(X1,mark(X2),X3) -> if(X1,X2,X3)
      if(X1,X2,mark(X3)) -> if(X1,X2,X3)
      if(active(X1),X2,X3) -> if(X1,X2,X3)
      if(X1,active(X2),X3) -> if(X1,X2,X3)
      if(X1,X2,active(X3)) -> if(X1,X2,X3)
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [if](x0, x1, x2) = x0 + x1 + x2 + 4,
       
       [active](x0) = x0,
       
       [mark](x0) = 2x0 + 2,
       
       [f](x0) = 4x0
      orientation:
       mark(if(X1,X2,X3)) = 2X1 + 2X2 + 2X3 + 10 >= 2X1 + 2X2 + X3 + 8 = active(if(mark(X1),mark(X2),X3))
       
       f(mark(X)) = 8X + 8 >= 4X = f(X)
       
       f(active(X)) = 4X >= 4X = f(X)
       
       if(mark(X1),X2,X3) = 2X1 + X2 + X3 + 6 >= X1 + X2 + X3 + 4 = if(X1,X2,X3)
       
       if(X1,mark(X2),X3) = X1 + 2X2 + X3 + 6 >= X1 + X2 + X3 + 4 = if(X1,X2,X3)
       
       if(X1,X2,mark(X3)) = X1 + X2 + 2X3 + 6 >= X1 + X2 + X3 + 4 = if(X1,X2,X3)
       
       if(active(X1),X2,X3) = X1 + X2 + X3 + 4 >= X1 + X2 + X3 + 4 = if(X1,X2,X3)
       
       if(X1,active(X2),X3) = X1 + X2 + X3 + 4 >= X1 + X2 + X3 + 4 = if(X1,X2,X3)
       
       if(X1,X2,active(X3)) = X1 + X2 + X3 + 4 >= X1 + X2 + X3 + 4 = if(X1,X2,X3)
      problem:
       f(active(X)) -> f(X)
       if(active(X1),X2,X3) -> if(X1,X2,X3)
       if(X1,active(X2),X3) -> if(X1,X2,X3)
       if(X1,X2,active(X3)) -> if(X1,X2,X3)
      Matrix Interpretation Processor: dim=1
       
       interpretation:
        [if](x0, x1, x2) = 2x0 + 5x1 + x2 + 4,
        
        [active](x0) = x0 + 4,
        
        [f](x0) = x0 + 5
       orientation:
        f(active(X)) = X + 9 >= X + 5 = f(X)
        
        if(active(X1),X2,X3) = 2X1 + 5X2 + X3 + 12 >= 2X1 + 5X2 + X3 + 4 = if(X1,X2,X3)
        
        if(X1,active(X2),X3) = 2X1 + 5X2 + X3 + 24 >= 2X1 + 5X2 + X3 + 4 = if(X1,X2,X3)
        
        if(X1,X2,active(X3)) = 2X1 + 5X2 + X3 + 8 >= 2X1 + 5X2 + X3 + 4 = if(X1,X2,X3)
       problem:
        
       Qed