*** 1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        log(s(0())) -> 0()
        log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
        minus(x,0()) -> x
        minus(x,s(y)) -> pred(minus(x,y))
        pred(s(x)) -> x
        quot(0(),s(y)) -> 0()
        quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {log/1,minus/2,pred/1,quot/2} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {log,minus,pred,quot}/{0,s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        log#(s(0())) -> c_1()
        log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        minus#(x,0()) -> c_3()
        minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
        pred#(s(x)) -> c_5()
        quot#(0(),s(y)) -> c_6()
        quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        log#(s(0())) -> c_1()
        log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        minus#(x,0()) -> c_3()
        minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
        pred#(s(x)) -> c_5()
        quot#(0(),s(y)) -> c_6()
        quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        log(s(0())) -> 0()
        log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
        minus(x,0()) -> x
        minus(x,s(y)) -> pred(minus(x,y))
        pred(s(x)) -> x
        quot(0(),s(y)) -> 0()
        quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
      Signature:
        {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0,c_7/2}
      Obligation:
        Innermost
        basic terms: {log#,minus#,pred#,quot#}/{0,s}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        minus(x,0()) -> x
        minus(x,s(y)) -> pred(minus(x,y))
        pred(s(x)) -> x
        quot(0(),s(y)) -> 0()
        quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        log#(s(0())) -> c_1()
        log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        minus#(x,0()) -> c_3()
        minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
        pred#(s(x)) -> c_5()
        quot#(0(),s(y)) -> c_6()
        quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
*** 1.1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        log#(s(0())) -> c_1()
        log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        minus#(x,0()) -> c_3()
        minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
        pred#(s(x)) -> c_5()
        quot#(0(),s(y)) -> c_6()
        quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        minus(x,0()) -> x
        minus(x,s(y)) -> pred(minus(x,y))
        pred(s(x)) -> x
        quot(0(),s(y)) -> 0()
        quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
      Signature:
        {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0,c_7/2}
      Obligation:
        Innermost
        basic terms: {log#,minus#,pred#,quot#}/{0,s}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {1,3,5,6}
      by application of
        Pre({1,3,5,6}) = {2,4,7}.
      Here rules are labelled as follows:
        1: log#(s(0())) -> c_1()           
        2: log#(s(s(x))) ->                
             c_2(log#(s(quot(x,s(s(0())))))
                ,quot#(x,s(s(0()))))       
        3: minus#(x,0()) -> c_3()          
        4: minus#(x,s(y)) ->               
             c_4(pred#(minus(x,y))         
                ,minus#(x,y))              
        5: pred#(s(x)) -> c_5()            
        6: quot#(0(),s(y)) -> c_6()        
        7: quot#(s(x),s(y)) ->             
             c_7(quot#(minus(x,y),s(y))    
                ,minus#(x,y))              
*** 1.1.1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
        quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
      Strict TRS Rules:
        
      Weak DP Rules:
        log#(s(0())) -> c_1()
        minus#(x,0()) -> c_3()
        pred#(s(x)) -> c_5()
        quot#(0(),s(y)) -> c_6()
      Weak TRS Rules:
        minus(x,0()) -> x
        minus(x,s(y)) -> pred(minus(x,y))
        pred(s(x)) -> x
        quot(0(),s(y)) -> 0()
        quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
      Signature:
        {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0,c_7/2}
      Obligation:
        Innermost
        basic terms: {log#,minus#,pred#,quot#}/{0,s}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
           -->_2 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
           -->_2 quot#(0(),s(y)) -> c_6():7
           -->_1 log#(s(0())) -> c_1():4
           -->_1 log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
        
        2:S:minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
           -->_1 pred#(s(x)) -> c_5():6
           -->_2 minus#(x,0()) -> c_3():5
           -->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2
        
        3:S:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
           -->_1 quot#(0(),s(y)) -> c_6():7
           -->_2 minus#(x,0()) -> c_3():5
           -->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
           -->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2
        
        4:W:log#(s(0())) -> c_1()
           
        
        5:W:minus#(x,0()) -> c_3()
           
        
        6:W:pred#(s(x)) -> c_5()
           
        
        7:W:quot#(0(),s(y)) -> c_6()
           
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        4: log#(s(0())) -> c_1()   
        6: pred#(s(x)) -> c_5()    
        5: minus#(x,0()) -> c_3()  
        7: quot#(0(),s(y)) -> c_6()
*** 1.1.1.1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
        quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        minus(x,0()) -> x
        minus(x,s(y)) -> pred(minus(x,y))
        pred(s(x)) -> x
        quot(0(),s(y)) -> 0()
        quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
      Signature:
        {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0,c_7/2}
      Obligation:
        Innermost
        basic terms: {log#,minus#,pred#,quot#}/{0,s}
    Applied Processor:
      SimplifyRHS
    Proof:
      Consider the dependency graph
        1:S:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
           -->_2 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
           -->_1 log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
        
        2:S:minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
           -->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2
        
        3:S:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
           -->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
           -->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2
        
      Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
        minus#(x,s(y)) -> c_4(minus#(x,y))
*** 1.1.1.1.1.1 Progress [(?,O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        minus#(x,s(y)) -> c_4(minus#(x,y))
        quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        minus(x,0()) -> x
        minus(x,s(y)) -> pred(minus(x,y))
        pred(s(x)) -> x
        quot(0(),s(y)) -> 0()
        quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
      Signature:
        {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
      Obligation:
        Innermost
        basic terms: {log#,minus#,pred#,quot#}/{0,s}
    Applied Processor:
      PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
    Proof:
      We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
        1: log#(s(s(x))) ->                
             c_2(log#(s(quot(x,s(s(0())))))
                ,quot#(x,s(s(0()))))       
        2: minus#(x,s(y)) -> c_4(minus#(x  
                                       ,y))
        3: quot#(s(x),s(y)) ->             
             c_7(quot#(minus(x,y),s(y))    
                ,minus#(x,y))              
        
      The strictly oriented rules are moved into the weak component.
  *** 1.1.1.1.1.1.1 Progress [(?,O(n^2))]  ***
      Considered Problem:
        Strict DP Rules:
          log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
          minus#(x,s(y)) -> c_4(minus#(x,y))
          quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          minus(x,0()) -> x
          minus(x,s(y)) -> pred(minus(x,y))
          pred(s(x)) -> x
          quot(0(),s(y)) -> 0()
          quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        Signature:
          {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        Obligation:
          Innermost
          basic terms: {log#,minus#,pred#,quot#}/{0,s}
      Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
      Proof:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_2) = {1,2},
          uargs(c_4) = {1},
          uargs(c_7) = {1,2}
        
        Following symbols are considered usable:
          {minus,pred,quot,log#,minus#,pred#,quot#}
        TcT has computed the following interpretation:
               p(0) = 0            
             p(log) = x1^2         
           p(minus) = x1           
            p(pred) = x1           
            p(quot) = x1           
               p(s) = 1 + x1       
            p(log#) = 1 + x1 + x1^2
          p(minus#) = x2           
           p(pred#) = 0            
           p(quot#) = 2 + x1*x2    
             p(c_1) = 0            
             p(c_2) = x1 + x2      
             p(c_3) = 1            
             p(c_4) = x1           
             p(c_5) = 0            
             p(c_6) = 1            
             p(c_7) = x1 + x2      
        
        Following rules are strictly oriented:
           log#(s(s(x))) = 7 + 5*x + x^2                 
                         > 5 + 5*x + x^2                 
                         = c_2(log#(s(quot(x,s(s(0())))))
                              ,quot#(x,s(s(0()))))       
        
          minus#(x,s(y)) = 1 + y                         
                         > y                             
                         = c_4(minus#(x,y))              
        
        quot#(s(x),s(y)) = 3 + x + x*y + y               
                         > 2 + x + x*y + y               
                         = c_7(quot#(minus(x,y),s(y))    
                              ,minus#(x,y))              
        
        
        Following rules are (at-least) weakly oriented:
           minus(x,0()) =  x                       
                        >= x                       
                        =  x                       
        
          minus(x,s(y)) =  x                       
                        >= x                       
                        =  pred(minus(x,y))        
        
             pred(s(x)) =  1 + x                   
                        >= x                       
                        =  x                       
        
         quot(0(),s(y)) =  0                       
                        >= 0                       
                        =  0()                     
        
        quot(s(x),s(y)) =  1 + x                   
                        >= 1 + x                   
                        =  s(quot(minus(x,y),s(y)))
        
  *** 1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
          minus#(x,s(y)) -> c_4(minus#(x,y))
          quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
        Weak TRS Rules:
          minus(x,0()) -> x
          minus(x,s(y)) -> pred(minus(x,y))
          pred(s(x)) -> x
          quot(0(),s(y)) -> 0()
          quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        Signature:
          {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        Obligation:
          Innermost
          basic terms: {log#,minus#,pred#,quot#}/{0,s}
      Applied Processor:
        Assumption
      Proof:
        ()
  
  *** 1.1.1.1.1.1.2 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
          minus#(x,s(y)) -> c_4(minus#(x,y))
          quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
        Weak TRS Rules:
          minus(x,0()) -> x
          minus(x,s(y)) -> pred(minus(x,y))
          pred(s(x)) -> x
          quot(0(),s(y)) -> 0()
          quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        Signature:
          {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        Obligation:
          Innermost
          basic terms: {log#,minus#,pred#,quot#}/{0,s}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:W:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
             -->_2 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
             -->_1 log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
          
          2:W:minus#(x,s(y)) -> c_4(minus#(x,y))
             -->_1 minus#(x,s(y)) -> c_4(minus#(x,y)):2
          
          3:W:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
             -->_2 minus#(x,s(y)) -> c_4(minus#(x,y)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: log#(s(s(x))) ->                
               c_2(log#(s(quot(x,s(s(0())))))
                  ,quot#(x,s(s(0()))))       
          3: quot#(s(x),s(y)) ->             
               c_7(quot#(minus(x,y),s(y))    
                  ,minus#(x,y))              
          2: minus#(x,s(y)) -> c_4(minus#(x  
                                         ,y))
  *** 1.1.1.1.1.1.2.1 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          minus(x,0()) -> x
          minus(x,s(y)) -> pred(minus(x,y))
          pred(s(x)) -> x
          quot(0(),s(y)) -> 0()
          quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        Signature:
          {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        Obligation:
          Innermost
          basic terms: {log#,minus#,pred#,quot#}/{0,s}
      Applied Processor:
        EmptyProcessor
      Proof:
        The problem is already closed. The intended complexity is O(1).