*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        c(b(a(),a()),b(y,z),x) -> b(a(),b(z,z))
        f(b(b(x,f(y)),z)) -> c(z,x,f(b(b(f(a()),y),y)))
        f(c(c(a(),y,a()),b(x,z),a())) -> b(y,f(c(f(a()),z,z)))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {c/3,f/1} / {a/0,b/2}
      Obligation:
        Full
        basic terms: {c,f}/{a,b}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following weak dependency pairs:
      
      Strict DPs
        c#(b(a(),a()),b(y,z),x) -> c_1(z,z)
        f#(b(b(x,f(y)),z)) -> c_2(c#(z,x,f(b(b(f(a()),y),y))))
        f#(c(c(a(),y,a()),b(x,z),a())) -> c_3(y,f#(c(f(a()),z,z)))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        c#(b(a(),a()),b(y,z),x) -> c_1(z,z)
        f#(b(b(x,f(y)),z)) -> c_2(c#(z,x,f(b(b(f(a()),y),y))))
        f#(c(c(a(),y,a()),b(x,z),a())) -> c_3(y,f#(c(f(a()),z,z)))
      Strict TRS Rules:
        c(b(a(),a()),b(y,z),x) -> b(a(),b(z,z))
        f(b(b(x,f(y)),z)) -> c(z,x,f(b(b(f(a()),y),y)))
        f(c(c(a(),y,a()),b(x,z),a())) -> b(y,f(c(f(a()),z,z)))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {c/3,f/1,c#/3,f#/1} / {a/0,b/2,c_1/2,c_2/1,c_3/2}
      Obligation:
        Full
        basic terms: {c#,f#}/{a,b}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        c#(b(a(),a()),b(y,z),x) -> c_1(z,z)
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        c#(b(a(),a()),b(y,z),x) -> c_1(z,z)
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {c/3,f/1,c#/3,f#/1} / {a/0,b/2,c_1/2,c_2/1,c_3/2}
      Obligation:
        Full
        basic terms: {c#,f#}/{a,b}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following constant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          none
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
            p(a) = [0]
            p(b) = [0]
            p(c) = [0]
            p(f) = [0]
           p(c#) = [5]
           p(f#) = [0]
          p(c_1) = [0]
          p(c_2) = [0]
          p(c_3) = [0]
        
        Following rules are strictly oriented:
        c#(b(a(),a()),b(y,z),x) = [5]     
                                > [0]     
                                = c_1(z,z)
        
        
        Following rules are (at-least) weakly oriented:
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        c#(b(a(),a()),b(y,z),x) -> c_1(z,z)
      Weak TRS Rules:
        
      Signature:
        {c/3,f/1,c#/3,f#/1} / {a/0,b/2,c_1/2,c_2/1,c_3/2}
      Obligation:
        Full
        basic terms: {c#,f#}/{a,b}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).