YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> a#(x1)
   a#(b(x1)) -> b#(a(x1))
   b#(a(x1)) -> b#(x1)
   b#(a(x1)) -> a#(c(b(x1)))
   b#(a(x1)) -> a#(a(c(b(x1))))
  TRS:
   a(b(x1)) -> b(a(x1))
   b(a(x1)) -> a(a(c(b(x1))))
  TDG Processor:
   DPs:
    a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> b#(a(x1))
    b#(a(x1)) -> b#(x1)
    b#(a(x1)) -> a#(c(b(x1)))
    b#(a(x1)) -> a#(a(c(b(x1))))
   TRS:
    a(b(x1)) -> b(a(x1))
    b(a(x1)) -> a(a(c(b(x1))))
   graph:
    b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(a(c(b(x1))))
    b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(c(b(x1)))
    b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1)
    b#(a(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> b#(a(x1))
    b#(a(x1)) -> a#(c(b(x1))) -> a#(b(x1)) -> a#(x1)
    b#(a(x1)) -> a#(a(c(b(x1)))) -> a#(b(x1)) -> b#(a(x1))
    b#(a(x1)) -> a#(a(c(b(x1)))) -> a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> b#(a(x1)) -> b#(a(x1)) -> a#(a(c(b(x1))))
    a#(b(x1)) -> b#(a(x1)) -> b#(a(x1)) -> a#(c(b(x1)))
    a#(b(x1)) -> b#(a(x1)) -> b#(a(x1)) -> b#(x1)
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(x1))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
   EDG Processor:
    DPs:
     a#(b(x1)) -> a#(x1)
     a#(b(x1)) -> b#(a(x1))
     b#(a(x1)) -> b#(x1)
     b#(a(x1)) -> a#(c(b(x1)))
     b#(a(x1)) -> a#(a(c(b(x1))))
    TRS:
     a(b(x1)) -> b(a(x1))
     b(a(x1)) -> a(a(c(b(x1))))
    graph:
     b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1)
     b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(c(b(x1)))
     b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(a(c(b(x1))))
     a#(b(x1)) -> b#(a(x1)) -> b#(a(x1)) -> b#(x1)
     a#(b(x1)) -> b#(a(x1)) -> b#(a(x1)) -> a#(c(b(x1)))
     a#(b(x1)) -> b#(a(x1)) -> b#(a(x1)) -> a#(a(c(b(x1))))
     a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
     a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(x1))
    CDG Processor:
     DPs:
      a#(b(x1)) -> a#(x1)
      a#(b(x1)) -> b#(a(x1))
      b#(a(x1)) -> b#(x1)
      b#(a(x1)) -> a#(c(b(x1)))
      b#(a(x1)) -> a#(a(c(b(x1))))
     TRS:
      a(b(x1)) -> b(a(x1))
      b(a(x1)) -> a(a(c(b(x1))))
     graph:
      a#(b(x1)) -> b#(a(x1)) -> b#(a(x1)) -> a#(a(c(b(x1))))
      a#(b(x1)) -> b#(a(x1)) -> b#(a(x1)) -> a#(c(b(x1)))
      a#(b(x1)) -> b#(a(x1)) -> b#(a(x1)) -> b#(x1)
      a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(a(x1))
      a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 5/25
      DPs:
       a#(b(x1)) -> a#(x1)
      TRS:
       a(b(x1)) -> b(a(x1))
       b(a(x1)) -> a(a(c(b(x1))))
      Arctic Interpretation Processor:
       dimension: 1
       interpretation:
        [a#](x0) = 8x0,
        
        [c](x0) = x0,
        
        [a](x0) = x0,
        
        [b](x0) = 2x0
       orientation:
        a#(b(x1)) = 10x1 >= 8x1 = a#(x1)
        
        a(b(x1)) = 2x1 >= 2x1 = b(a(x1))
        
        b(a(x1)) = 2x1 >= 2x1 = a(a(c(b(x1))))
       problem:
        DPs:
         
        TRS:
         a(b(x1)) -> b(a(x1))
         b(a(x1)) -> a(a(c(b(x1))))
       Qed