YES

Problem:
 f(0()) -> true()
 f(1()) -> false()
 f(s(x)) -> f(x)
 if(true(),x,y) -> x
 if(false(),x,y) -> y
 g(s(x),s(y)) -> if(f(x),s(x),s(y))
 g(x,c(y)) -> g(x,g(s(c(y)),y))

Proof:
 Matrix Interpretation Processor: dim=2
  
  interpretation:
             [2]
   [false] = [0],
   
             [2 0]     [1]
   [c](x0) = [3 1]x0 + [3],
   
             [1 0]     [0]
   [f](x0) = [0 0]x0 + [1],
   
         [2]
   [1] = [0],
   
             [1 0]     [1]
   [s](x0) = [0 0]x0 + [0],
   
         [1]
   [0] = [0],
   
                 [2 0]     [1 1]  
   [g](x0, x1) = [0 0]x0 + [0 0]x1,
   
                      [1 0]     [1 2]     [1 2]     [1]
   [if](x0, x1, x2) = [0 0]x0 + [0 1]x1 + [0 1]x2 + [0],
   
            [0]
   [true] = [1]
  orientation:
            [1]    [0]         
   f(0()) = [1] >= [1] = true()
   
            [2]    [2]          
   f(1()) = [1] >= [0] = false()
   
             [1 0]    [1]    [1 0]    [0]       
   f(s(x)) = [0 0]x + [1] >= [0 0]x + [1] = f(x)
   
                    [1 2]    [1 2]    [1]         
   if(true(),x,y) = [0 1]x + [0 1]y + [0] >= x = x
   
                     [1 2]    [1 2]    [3]         
   if(false(),x,y) = [0 1]x + [0 1]y + [0] >= y = y
   
                  [2 0]    [1 0]    [3]    [2 0]    [1 0]    [3]                     
   g(s(x),s(y)) = [0 0]x + [0 0]y + [0] >= [0 0]x + [0 0]y + [0] = if(f(x),s(x),s(y))
   
               [2 0]    [5 1]    [4]    [2 0]    [5 1]    [4]                    
   g(x,c(y)) = [0 0]x + [0 0]y + [0] >= [0 0]x + [0 0]y + [0] = g(x,g(s(c(y)),y))
  problem:
   f(1()) -> false()
   g(s(x),s(y)) -> if(f(x),s(x),s(y))
   g(x,c(y)) -> g(x,g(s(c(y)),y))
  Matrix Interpretation Processor: dim=3
   
   interpretation:
              [0]
    [false] = [0]
              [0],
    
              [1 1 0]  
    [c](x0) = [1 1 1]x0
              [1 1 1]  ,
    
              [1 0 0]  
    [f](x0) = [1 0 1]x0
              [0 0 0]  ,
    
          [1]
    [1] = [0]
          [1],
    
              [1 0 0]  
    [s](x0) = [1 0 0]x0
              [0 0 1]  ,
    
                  [1 1 0]     [1 1 1]  
    [g](x0, x1) = [0 0 0]x0 + [0 0 0]x1
                  [0 0 0]     [0 0 0]  ,
    
                       [1 0 0]     [1 0 0]     [1 0 1]  
    [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2
                       [0 0 0]     [0 0 0]     [0 0 0]  
   orientation:
             [1]    [0]          
    f(1()) = [2] >= [0] = false()
             [0]    [0]          
    
                   [2 0 0]    [2 0 1]     [2 0 0]    [1 0 1]                      
    g(s(x),s(y)) = [0 0 0]x + [0 0 0]y >= [0 0 0]x + [0 0 0]y = if(f(x),s(x),s(y))
                   [0 0 0]    [0 0 0]     [0 0 0]    [0 0 0]                      
    
                [1 1 0]    [3 3 2]     [1 1 0]    [3 3 1]                     
    g(x,c(y)) = [0 0 0]x + [0 0 0]y >= [0 0 0]x + [0 0 0]y = g(x,g(s(c(y)),y))
                [0 0 0]    [0 0 0]     [0 0 0]    [0 0 0]                     
   problem:
    g(s(x),s(y)) -> if(f(x),s(x),s(y))
    g(x,c(y)) -> g(x,g(s(c(y)),y))
   Matrix Interpretation Processor: dim=3
    
    interpretation:
               [1 0 0]     [0]
     [c](x0) = [1 1 1]x0 + [1]
               [1 0 0]     [0],
     
               [1 0 0]  
     [f](x0) = [0 0 0]x0
               [0 0 0]  ,
     
               [1 0 0]     [0]
     [s](x0) = [1 0 0]x0 + [1]
               [0 1 0]     [0],
     
                   [1 1 0]     [1 1 1]  
     [g](x0, x1) = [0 0 0]x0 + [0 0 0]x1
                   [0 0 0]     [0 0 0]  ,
     
                        [1 0 0]     [1 0 0]     [1 0 0]  
     [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2
                        [0 0 0]     [0 0 0]     [0 0 0]  
    orientation:
                    [2 0 0]    [2 1 0]    [2]    [2 0 0]    [1 0 0]                      
     g(s(x),s(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y = if(f(x),s(x),s(y))
                    [0 0 0]    [0 0 0]    [0]    [0 0 0]    [0 0 0]                      
     
                 [1 1 0]    [3 1 1]    [1]    [1 1 0]    [3 1 1]    [1]                    
     g(x,c(y)) = [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = g(x,g(s(c(y)),y))
                 [0 0 0]    [0 0 0]    [0]    [0 0 0]    [0 0 0]    [0]                    
    problem:
     g(x,c(y)) -> g(x,g(s(c(y)),y))
    Matrix Interpretation Processor: dim=2
     
     interpretation:
                [1 0]     [0]
      [c](x0) = [2 1]x0 + [3],
      
                [2 0]     [0]
      [s](x0) = [0 0]x0 + [1],
      
                         [1 1]     [0]
      [g](x0, x1) = x0 + [0 0]x1 + [1]
     orientation:
                      [3 1]    [3]        [3 1]    [2]                    
      g(x,c(y)) = x + [0 0]y + [1] >= x + [0 0]y + [1] = g(x,g(s(c(y)),y))
     problem:
      
     Qed