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