YES

Problem:
 a__g(X) -> a__h(X)
 a__c() -> d()
 a__h(d()) -> a__g(c())
 mark(g(X)) -> a__g(X)
 mark(h(X)) -> a__h(X)
 mark(c()) -> a__c()
 mark(d()) -> d()
 a__g(X) -> g(X)
 a__h(X) -> h(X)
 a__c() -> c()

Proof:
 DP Processor:
  DPs:
   a__g#(X) -> a__h#(X)
   a__h#(d()) -> a__g#(c())
   mark#(g(X)) -> a__g#(X)
   mark#(h(X)) -> a__h#(X)
   mark#(c()) -> a__c#()
  TRS:
   a__g(X) -> a__h(X)
   a__c() -> d()
   a__h(d()) -> a__g(c())
   mark(g(X)) -> a__g(X)
   mark(h(X)) -> a__h(X)
   mark(c()) -> a__c()
   mark(d()) -> d()
   a__g(X) -> g(X)
   a__h(X) -> h(X)
   a__c() -> c()
  CDG Processor:
   DPs:
    a__g#(X) -> a__h#(X)
    a__h#(d()) -> a__g#(c())
    mark#(g(X)) -> a__g#(X)
    mark#(h(X)) -> a__h#(X)
    mark#(c()) -> a__c#()
   TRS:
    a__g(X) -> a__h(X)
    a__c() -> d()
    a__h(d()) -> a__g(c())
    mark(g(X)) -> a__g(X)
    mark(h(X)) -> a__h(X)
    mark(c()) -> a__c()
    mark(d()) -> d()
    a__g(X) -> g(X)
    a__h(X) -> h(X)
    a__c() -> c()
   graph:
    mark#(h(X)) -> a__h#(X) -> a__h#(d()) -> a__g#(c())
    mark#(g(X)) -> a__g#(X) -> a__g#(X) -> a__h#(X)
    a__h#(d()) -> a__g#(c()) -> a__g#(X) -> a__h#(X)
    a__g#(X) -> a__h#(X) -> a__h#(d()) -> a__g#(c())
   SCC Processor:
    #sccs: 1
    #rules: 2
    #arcs: 4/25
    DPs:
     a__h#(d()) -> a__g#(c())
     a__g#(X) -> a__h#(X)
    TRS:
     a__g(X) -> a__h(X)
     a__c() -> d()
     a__h(d()) -> a__g(c())
     mark(g(X)) -> a__g(X)
     mark(h(X)) -> a__h(X)
     mark(c()) -> a__c()
     mark(d()) -> d()
     a__g(X) -> g(X)
     a__h(X) -> h(X)
     a__c() -> c()
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [a__h#](x0) = x0,
      
      [a__g#](x0) = x0 + 1,
      
      [h](x0) = 0,
      
      [mark](x0) = 1,
      
      [g](x0) = 0,
      
      [c] = 0,
      
      [d] = 1,
      
      [a__c] = 1,
      
      [a__h](x0) = 0,
      
      [a__g](x0) = 0
     orientation:
      a__h#(d()) = 1 >= 1 = a__g#(c())
      
      a__g#(X) = X + 1 >= X = a__h#(X)
      
      a__g(X) = 0 >= 0 = a__h(X)
      
      a__c() = 1 >= 1 = d()
      
      a__h(d()) = 0 >= 0 = a__g(c())
      
      mark(g(X)) = 1 >= 0 = a__g(X)
      
      mark(h(X)) = 1 >= 0 = a__h(X)
      
      mark(c()) = 1 >= 1 = a__c()
      
      mark(d()) = 1 >= 1 = d()
      
      a__g(X) = 0 >= 0 = g(X)
      
      a__h(X) = 0 >= 0 = h(X)
      
      a__c() = 1 >= 0 = c()
     problem:
      DPs:
       a__h#(d()) -> a__g#(c())
      TRS:
       a__g(X) -> a__h(X)
       a__c() -> d()
       a__h(d()) -> a__g(c())
       mark(g(X)) -> a__g(X)
       mark(h(X)) -> a__h(X)
       mark(c()) -> a__c()
       mark(d()) -> d()
       a__g(X) -> g(X)
       a__h(X) -> h(X)
       a__c() -> c()
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [a__h#](x0) = 1,
       
       [a__g#](x0) = 0,
       
       [h](x0) = 0,
       
       [mark](x0) = 0,
       
       [g](x0) = 0,
       
       [c] = 0,
       
       [d] = 0,
       
       [a__c] = 0,
       
       [a__h](x0) = 0,
       
       [a__g](x0) = 0
      orientation:
       a__h#(d()) = 1 >= 0 = a__g#(c())
       
       a__g(X) = 0 >= 0 = a__h(X)
       
       a__c() = 0 >= 0 = d()
       
       a__h(d()) = 0 >= 0 = a__g(c())
       
       mark(g(X)) = 0 >= 0 = a__g(X)
       
       mark(h(X)) = 0 >= 0 = a__h(X)
       
       mark(c()) = 0 >= 0 = a__c()
       
       mark(d()) = 0 >= 0 = d()
       
       a__g(X) = 0 >= 0 = g(X)
       
       a__h(X) = 0 >= 0 = h(X)
       
       a__c() = 0 >= 0 = c()
      problem:
       DPs:
        
       TRS:
        a__g(X) -> a__h(X)
        a__c() -> d()
        a__h(d()) -> a__g(c())
        mark(g(X)) -> a__g(X)
        mark(h(X)) -> a__h(X)
        mark(c()) -> a__c()
        mark(d()) -> d()
        a__g(X) -> g(X)
        a__h(X) -> h(X)
        a__c() -> c()
      Qed