*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(x,x,y) -> x
        f(x,y,y) -> y
        f(x,y,g(y)) -> x
        f(f(x,y,z),u,f(x,y,v)) -> f(x,y,f(z,u,v))
        f(g(x),x,y) -> y
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3} / {g/1}
      Obligation:
        Innermost
        basic terms: {f}/{g}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        f#(x,x,y) -> c_1()
        f#(x,y,y) -> c_2()
        f#(x,y,g(y)) -> c_3()
        f#(f(x,y,z),u,f(x,y,v)) -> c_4(f#(x,y,f(z,u,v)),f#(z,u,v))
        f#(g(x),x,y) -> c_5()
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(x,x,y) -> c_1()
        f#(x,y,y) -> c_2()
        f#(x,y,g(y)) -> c_3()
        f#(f(x,y,z),u,f(x,y,v)) -> c_4(f#(x,y,f(z,u,v)),f#(z,u,v))
        f#(g(x),x,y) -> c_5()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        f(x,x,y) -> x
        f(x,y,y) -> y
        f(x,y,g(y)) -> x
        f(f(x,y,z),u,f(x,y,v)) -> f(x,y,f(z,u,v))
        f(g(x),x,y) -> y
      Signature:
        {f/3,f#/3} / {g/1,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0}
      Obligation:
        Innermost
        basic terms: {f#}/{g}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        f#(x,x,y) -> c_1()
        f#(x,y,y) -> c_2()
        f#(x,y,g(y)) -> c_3()
        f#(g(x),x,y) -> c_5()
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(x,x,y) -> c_1()
        f#(x,y,y) -> c_2()
        f#(x,y,g(y)) -> c_3()
        f#(g(x),x,y) -> c_5()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/3,f#/3} / {g/1,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0}
      Obligation:
        Innermost
        basic terms: {f#}/{g}
    Applied Processor:
      Trivial
    Proof:
      Consider the dependency graph
        1:S:f#(x,x,y) -> c_1()
           
        
        2:S:f#(x,y,y) -> c_2()
           
        
        3:S:f#(x,y,g(y)) -> c_3()
           
        
        4:S:f#(g(x),x,y) -> c_5()
           
        
      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/3,f#/3} / {g/1,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0}
      Obligation:
        Innermost
        basic terms: {f#}/{g}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).