*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        0() -> 1()
        f(s(x)) -> f(g(x,x))
        g(0(),1()) -> s(0())
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {0/0,f/1,g/2} / {1/0,s/1}
      Obligation:
        Innermost
        basic terms: {0,f,g}/{1,s}
    Applied Processor:
      InnermostRuleRemoval
    Proof:
      Arguments of following rules are not normal-forms.
        g(0(),1()) -> s(0())
      All above mentioned rules can be savely removed.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        0() -> 1()
        f(s(x)) -> f(g(x,x))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {0/0,f/1,g/2} / {1/0,s/1}
      Obligation:
        Innermost
        basic terms: {0,f,g}/{1,s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        0#() -> c_1()
        f#(s(x)) -> c_2(f#(g(x,x)),g#(x,x))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        0#() -> c_1()
        f#(s(x)) -> c_2(f#(g(x,x)),g#(x,x))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        0() -> 1()
        f(s(x)) -> f(g(x,x))
      Signature:
        {0/0,f/1,g/2,0#/0,f#/1,g#/2} / {1/0,s/1,c_1/0,c_2/2}
      Obligation:
        Innermost
        basic terms: {0#,f#,g#}/{1,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        0#() -> c_1()
        f#(s(x)) -> c_2(f#(g(x,x)),g#(x,x))
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        0#() -> c_1()
        f#(s(x)) -> c_2(f#(g(x,x)),g#(x,x))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {0/0,f/1,g/2,0#/0,f#/1,g#/2} / {1/0,s/1,c_1/0,c_2/2}
      Obligation:
        Innermost
        basic terms: {0#,f#,g#}/{1,s}
    Applied Processor:
      Trivial
    Proof:
      Consider the dependency graph
        1:S:0#() -> c_1()
           
        
        2:S:f#(s(x)) -> c_2(f#(g(x,x)),g#(x,x))
           
        
      The dependency graph contains no loops, we remove all dependency pairs.
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {0/0,f/1,g/2,0#/0,f#/1,g#/2} / {1/0,s/1,c_1/0,c_2/2}
      Obligation:
        Innermost
        basic terms: {0#,f#,g#}/{1,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).