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