*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        half(0()) -> 0()
        half(s(s(x))) -> s(half(x))
        log(s(0())) -> 0()
        log(s(s(x))) -> s(log(s(half(x))))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {half/1,log/1} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {half,log}/{0,s}
    Applied Processor:
      ToInnermost
    Proof:
      switch to innermost, as the system is overlay and right linear and does not contain weak rules
*** 1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        half(0()) -> 0()
        half(s(s(x))) -> s(half(x))
        log(s(0())) -> 0()
        log(s(s(x))) -> s(log(s(half(x))))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {half/1,log/1} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {half,log}/{0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following nonconstant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(log) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
             p(0) = [1]         
          p(half) = [8]         
           p(log) = [1] x1 + [3]
             p(s) = [1] x1 + [0]
        
        Following rules are strictly oriented:
          half(0()) = [8]
                    > [1]
                    = 0()
        
        log(s(0())) = [4]
                    > [1]
                    = 0()
        
        
        Following rules are (at-least) weakly oriented:
        half(s(s(x))) =  [8]               
                      >= [8]               
                      =  s(half(x))        
        
         log(s(s(x))) =  [1] x + [3]       
                      >= [11]              
                      =  s(log(s(half(x))))
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        half(s(s(x))) -> s(half(x))
        log(s(s(x))) -> s(log(s(half(x))))
      Weak DP Rules:
        
      Weak TRS Rules:
        half(0()) -> 0()
        log(s(0())) -> 0()
      Signature:
        {half/1,log/1} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {half,log}/{0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following nonconstant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(log) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
             p(0) = [10]        
          p(half) = [1] x1 + [4]
           p(log) = [1] x1 + [0]
             p(s) = [1] x1 + [9]
        
        Following rules are strictly oriented:
        half(s(s(x))) = [1] x + [22]
                      > [1] x + [13]
                      = s(half(x))  
        
        
        Following rules are (at-least) weakly oriented:
           half(0()) =  [14]              
                     >= [10]              
                     =  0()               
        
         log(s(0())) =  [19]              
                     >= [10]              
                     =  0()               
        
        log(s(s(x))) =  [1] x + [18]      
                     >= [1] x + [22]      
                     =  s(log(s(half(x))))
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        log(s(s(x))) -> s(log(s(half(x))))
      Weak DP Rules:
        
      Weak TRS Rules:
        half(0()) -> 0()
        half(s(s(x))) -> s(half(x))
        log(s(0())) -> 0()
      Signature:
        {half/1,log/1} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {half,log}/{0,s}
    Applied Processor:
      NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
    Proof:
      We apply a matrix interpretation of kind constructor based matrix interpretation:
      The following argument positions are considered usable:
        uargs(log) = {1},
        uargs(s) = {1}
      
      Following symbols are considered usable:
        {half,log}
      TcT has computed the following interpretation:
           p(0) = [1]         
        p(half) = [1] x1 + [1]
         p(log) = [3] x1 + [0]
           p(s) = [1] x1 + [2]
      
      Following rules are strictly oriented:
      log(s(s(x))) = [3] x + [12]      
                   > [3] x + [11]      
                   = s(log(s(half(x))))
      
      
      Following rules are (at-least) weakly oriented:
          half(0()) =  [2]        
                    >= [1]        
                    =  0()        
      
      half(s(s(x))) =  [1] x + [5]
                    >= [1] x + [3]
                    =  s(half(x)) 
      
        log(s(0())) =  [9]        
                    >= [1]        
                    =  0()        
      
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        half(0()) -> 0()
        half(s(s(x))) -> s(half(x))
        log(s(0())) -> 0()
        log(s(s(x))) -> s(log(s(half(x))))
      Signature:
        {half/1,log/1} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {half,log}/{0,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).