*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        h(x,c(y,z),t(w)) -> h(c(s(y),x),z,t(c(t(w),w)))
        h(c(x,y),c(s(z),z),t(w)) -> h(z,c(y,x),t(t(c(x,c(y,t(w))))))
        h(c(s(x),c(s(0()),y)),z,t(x)) -> h(y,c(s(0()),c(x,z)),t(t(c(x,s(x)))))
        t(x) -> x
        t(x) -> c(0(),c(0(),c(0(),c(0(),c(0(),x)))))
        t(t(x)) -> t(c(t(x),x))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {h/3,t/1} / {0/0,c/2,s/1}
      Obligation:
        Full
        basic terms: {h,t}/{0,c,s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following weak dependency pairs:
      
      Strict DPs
        h#(x,c(y,z),t(w)) -> c_1(h#(c(s(y),x),z,t(c(t(w),w))))
        h#(c(x,y),c(s(z),z),t(w)) -> c_2(h#(z,c(y,x),t(t(c(x,c(y,t(w)))))))
        h#(c(s(x),c(s(0()),y)),z,t(x)) -> c_3(h#(y,c(s(0()),c(x,z)),t(t(c(x,s(x))))))
        t#(x) -> c_4(x)
        t#(x) -> c_5(x)
        t#(t(x)) -> c_6(t#(c(t(x),x)))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        h#(x,c(y,z),t(w)) -> c_1(h#(c(s(y),x),z,t(c(t(w),w))))
        h#(c(x,y),c(s(z),z),t(w)) -> c_2(h#(z,c(y,x),t(t(c(x,c(y,t(w)))))))
        h#(c(s(x),c(s(0()),y)),z,t(x)) -> c_3(h#(y,c(s(0()),c(x,z)),t(t(c(x,s(x))))))
        t#(x) -> c_4(x)
        t#(x) -> c_5(x)
        t#(t(x)) -> c_6(t#(c(t(x),x)))
      Strict TRS Rules:
        h(x,c(y,z),t(w)) -> h(c(s(y),x),z,t(c(t(w),w)))
        h(c(x,y),c(s(z),z),t(w)) -> h(z,c(y,x),t(t(c(x,c(y,t(w))))))
        h(c(s(x),c(s(0()),y)),z,t(x)) -> h(y,c(s(0()),c(x,z)),t(t(c(x,s(x)))))
        t(x) -> x
        t(x) -> c(0(),c(0(),c(0(),c(0(),c(0(),x)))))
        t(t(x)) -> t(c(t(x),x))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {h/3,t/1,h#/3,t#/1} / {0/0,c/2,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
      Obligation:
        Full
        basic terms: {h#,t#}/{0,c,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        t#(x) -> c_4(x)
        t#(x) -> c_5(x)
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        t#(x) -> c_4(x)
        t#(x) -> c_5(x)
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {h/3,t/1,h#/3,t#/1} / {0/0,c/2,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
      Obligation:
        Full
        basic terms: {h#,t#}/{0,c,s}
    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(0) = [0]
            p(c) = [0]
            p(h) = [0]
            p(s) = [0]
            p(t) = [0]
           p(h#) = [0]
           p(t#) = [1]
          p(c_1) = [0]
          p(c_2) = [0]
          p(c_3) = [0]
          p(c_4) = [0]
          p(c_5) = [0]
          p(c_6) = [0]
        
        Following rules are strictly oriented:
        t#(x) = [1]   
              > [0]   
              = c_4(x)
        
        t#(x) = [1]   
              > [0]   
              = c_5(x)
        
        
        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:
        t#(x) -> c_4(x)
        t#(x) -> c_5(x)
      Weak TRS Rules:
        
      Signature:
        {h/3,t/1,h#/3,t#/1} / {0/0,c/2,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
      Obligation:
        Full
        basic terms: {h#,t#}/{0,c,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).