YES

Problem:
 implies(not(x),y) -> or(x,y)
 implies(not(x),or(y,z)) -> implies(y,or(x,z))
 implies(x,or(y,z)) -> or(y,implies(x,z))

Proof:
 DP Processor:
  DPs:
   implies#(not(x),or(y,z)) -> implies#(y,or(x,z))
   implies#(x,or(y,z)) -> implies#(x,z)
  TRS:
   implies(not(x),y) -> or(x,y)
   implies(not(x),or(y,z)) -> implies(y,or(x,z))
   implies(x,or(y,z)) -> or(y,implies(x,z))
  Usable Rule Processor:
   DPs:
    implies#(not(x),or(y,z)) -> implies#(y,or(x,z))
    implies#(x,or(y,z)) -> implies#(x,z)
   TRS:
    
   CDG Processor:
    DPs:
     implies#(not(x),or(y,z)) -> implies#(y,or(x,z))
     implies#(x,or(y,z)) -> implies#(x,z)
    TRS:
     
    graph:
     implies#(not(x),or(y,z)) -> implies#(y,or(x,z)) ->
     implies#(x,or(y,z)) -> implies#(x,z)
     implies#(x,or(y,z)) -> implies#(x,z) -> implies#(x,or(y,z)) -> implies#(x,z)
    Restore Modifier:
     DPs:
      implies#(not(x),or(y,z)) -> implies#(y,or(x,z))
      implies#(x,or(y,z)) -> implies#(x,z)
     TRS:
      implies(not(x),y) -> or(x,y)
      implies(not(x),or(y,z)) -> implies(y,or(x,z))
      implies(x,or(y,z)) -> or(y,implies(x,z))
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 2/4
      DPs:
       implies#(x,or(y,z)) -> implies#(x,z)
      TRS:
       implies(not(x),y) -> or(x,y)
       implies(not(x),or(y,z)) -> implies(y,or(x,z))
       implies(x,or(y,z)) -> or(y,implies(x,z))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [implies#](x0, x1) = x1 + 1,
        
        [or](x0, x1) = x0 + x1 + 1,
        
        [implies](x0, x1) = x0 + x1 + 1,
        
        [not](x0) = x0 + 1
       orientation:
        implies#(x,or(y,z)) = y + z + 2 >= z + 1 = implies#(x,z)
        
        implies(not(x),y) = x + y + 2 >= x + y + 1 = or(x,y)
        
        implies(not(x),or(y,z)) = x + y + z + 3 >= x + y + z + 2 = implies(y,or(x,z))
        
        implies(x,or(y,z)) = x + y + z + 2 >= x + y + z + 2 = or(y,implies(x,z))
       problem:
        DPs:
         
        TRS:
         implies(not(x),y) -> or(x,y)
         implies(not(x),or(y,z)) -> implies(y,or(x,z))
         implies(x,or(y,z)) -> or(y,implies(x,z))
       Qed