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