YES

Problem:
 b(b(d(d(b(b(x1)))))) -> c(c(d(d(b(b(x1))))))
 b(b(a(a(c(c(x1)))))) -> b(b(c(c(x1))))
 a(a(d(d(x1)))) -> d(d(c(c(x1))))
 b(b(b(b(b(b(x1)))))) -> a(a(b(b(c(c(x1))))))
 d(d(c(c(x1)))) -> b(b(d(d(x1))))
 d(d(c(c(x1)))) -> d(d(b(b(d(d(x1))))))
 d(d(a(a(c(c(x1)))))) -> b(b(b(b(x1))))

Proof:
 Arctic Interpretation Processor:
  dimension: 1
  interpretation:
   [a](x0) = 1x0,
   
   [c](x0) = 1x0,
   
   [d](x0) = x0,
   
   [b](x0) = 1x0
  orientation:
   b(b(d(d(b(b(x1)))))) = 4x1 >= 4x1 = c(c(d(d(b(b(x1))))))
   
   b(b(a(a(c(c(x1)))))) = 6x1 >= 4x1 = b(b(c(c(x1))))
   
   a(a(d(d(x1)))) = 2x1 >= 2x1 = d(d(c(c(x1))))
   
   b(b(b(b(b(b(x1)))))) = 6x1 >= 6x1 = a(a(b(b(c(c(x1))))))
   
   d(d(c(c(x1)))) = 2x1 >= 2x1 = b(b(d(d(x1))))
   
   d(d(c(c(x1)))) = 2x1 >= 2x1 = d(d(b(b(d(d(x1))))))
   
   d(d(a(a(c(c(x1)))))) = 4x1 >= 4x1 = b(b(b(b(x1))))
  problem:
   b(b(d(d(b(b(x1)))))) -> c(c(d(d(b(b(x1))))))
   a(a(d(d(x1)))) -> d(d(c(c(x1))))
   b(b(b(b(b(b(x1)))))) -> a(a(b(b(c(c(x1))))))
   d(d(c(c(x1)))) -> b(b(d(d(x1))))
   d(d(c(c(x1)))) -> d(d(b(b(d(d(x1))))))
   d(d(a(a(c(c(x1)))))) -> b(b(b(b(x1))))
  String Reversal Processor:
   b(b(d(d(b(b(x1)))))) -> b(b(d(d(c(c(x1))))))
   d(d(a(a(x1)))) -> c(c(d(d(x1))))
   b(b(b(b(b(b(x1)))))) -> c(c(b(b(a(a(x1))))))
   c(c(d(d(x1)))) -> d(d(b(b(x1))))
   c(c(d(d(x1)))) -> d(d(b(b(d(d(x1))))))
   c(c(a(a(d(d(x1)))))) -> b(b(b(b(x1))))
   Matrix Interpretation Processor: dim=3
    
    interpretation:
               [1 0 0]     [0]
     [a](x0) = [1 1 1]x0 + [0]
               [0 1 0]     [1],
     
               [1 0 0]  
     [c](x0) = [0 0 0]x0
               [0 0 0]  ,
     
               [1 1 0]  
     [d](x0) = [0 0 0]x0
               [0 0 0]  ,
     
               [1 0 0]  
     [b](x0) = [0 0 0]x0
               [0 0 0]  
    orientation:
                            [1 0 0]      [1 0 0]                         
     b(b(d(d(b(b(x1)))))) = [0 0 0]x1 >= [0 0 0]x1 = b(b(d(d(c(c(x1))))))
                            [0 0 0]      [0 0 0]                         
     
                      [3 2 1]     [1]    [1 1 0]                   
     d(d(a(a(x1)))) = [0 0 0]x1 + [0] >= [0 0 0]x1 = c(c(d(d(x1))))
                      [0 0 0]     [0]    [0 0 0]                   
     
                            [1 0 0]      [1 0 0]                         
     b(b(b(b(b(b(x1)))))) = [0 0 0]x1 >= [0 0 0]x1 = c(c(b(b(a(a(x1))))))
                            [0 0 0]      [0 0 0]                         
     
                      [1 1 0]      [1 0 0]                   
     c(c(d(d(x1)))) = [0 0 0]x1 >= [0 0 0]x1 = d(d(b(b(x1))))
                      [0 0 0]      [0 0 0]                   
     
                      [1 1 0]      [1 1 0]                         
     c(c(d(d(x1)))) = [0 0 0]x1 >= [0 0 0]x1 = d(d(b(b(d(d(x1))))))
                      [0 0 0]      [0 0 0]                         
     
                            [1 1 0]      [1 0 0]                   
     c(c(a(a(d(d(x1)))))) = [0 0 0]x1 >= [0 0 0]x1 = b(b(b(b(x1))))
                            [0 0 0]      [0 0 0]                   
    problem:
     b(b(d(d(b(b(x1)))))) -> b(b(d(d(c(c(x1))))))
     b(b(b(b(b(b(x1)))))) -> c(c(b(b(a(a(x1))))))
     c(c(d(d(x1)))) -> d(d(b(b(x1))))
     c(c(d(d(x1)))) -> d(d(b(b(d(d(x1))))))
     c(c(a(a(d(d(x1)))))) -> b(b(b(b(x1))))
    Bounds Processor:
     bound: 1
     enrichment: match
     automaton:
      final states: {24,18,14,8,1}
      transitions:
       b1(30) -> 31*
       b1(31) -> 32*
       d1(29) -> 30*
       d1(28) -> 29*
       c1(27) -> 28*
       c1(26) -> 27*
       f40() -> 2*
       b0(25) -> 24*
       b0(20) -> 21*
       b0(15) -> 16*
       b0(10) -> 11*
       b0(7) -> 1*
       b0(2) -> 15*
       b0(21) -> 22*
       b0(16) -> 25*
       b0(11) -> 12*
       b0(6) -> 7*
       d0(5) -> 6*
       d0(22) -> 23*
       d0(17) -> 14*
       d0(2) -> 19*
       d0(19) -> 20*
       d0(4) -> 5*
       d0(16) -> 17*
       d0(23) -> 18*
       c0(12) -> 13*
       c0(2) -> 3*
       c0(13) -> 8*
       c0(3) -> 4*
       a0(2) -> 9*
       a0(9) -> 10*
       1 -> 15,16,22
       8 -> 15,16,25,24
       14 -> 3,4
       16 -> 26*
       18 -> 3,4
       24 -> 3,4
       32 -> 1,16,22,15
     problem:
      
     Qed