*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(X,X) -> c(X)
        f(X,c(X)) -> f(s(X),X)
        f(s(X),X) -> f(X,a(X))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2} / {a/1,c/1,s/1}
      Obligation:
        Innermost
        basic terms: {f}/{a,c,s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        f#(X,X) -> c_1()
        f#(X,c(X)) -> c_2(f#(s(X),X))
        f#(s(X),X) -> c_3(f#(X,a(X)))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(X,X) -> c_1()
        f#(X,c(X)) -> c_2(f#(s(X),X))
        f#(s(X),X) -> c_3(f#(X,a(X)))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        f(X,X) -> c(X)
        f(X,c(X)) -> f(s(X),X)
        f(s(X),X) -> f(X,a(X))
      Signature:
        {f/2,f#/2} / {a/1,c/1,s/1,c_1/0,c_2/1,c_3/1}
      Obligation:
        Innermost
        basic terms: {f#}/{a,c,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        f#(X,X) -> c_1()
        f#(X,c(X)) -> c_2(f#(s(X),X))
        f#(s(X),X) -> c_3(f#(X,a(X)))
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(X,X) -> c_1()
        f#(X,c(X)) -> c_2(f#(s(X),X))
        f#(s(X),X) -> c_3(f#(X,a(X)))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2,f#/2} / {a/1,c/1,s/1,c_1/0,c_2/1,c_3/1}
      Obligation:
        Innermost
        basic terms: {f#}/{a,c,s}
    Applied Processor:
      Trivial
    Proof:
      Consider the dependency graph
        1:S:f#(X,X) -> c_1()
           
        
        2:S:f#(X,c(X)) -> c_2(f#(s(X),X))
           -->_1 f#(s(X),X) -> c_3(f#(X,a(X))):3
        
        3:S:f#(s(X),X) -> c_3(f#(X,a(X)))
           
        
      The dependency graph contains no loops, we remove all dependency pairs.
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2,f#/2} / {a/1,c/1,s/1,c_1/0,c_2/1,c_3/1}
      Obligation:
        Innermost
        basic terms: {f#}/{a,c,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).