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