* Step 1: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            log(s(0())) -> 0()
            log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
            min(X,0()) -> X
            min(s(X),s(Y)) -> min(X,Y)
            quot(0(),s(Y)) -> 0()
            quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
        - Signature:
            {log/1,min/2,quot/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        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(quot) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(0) = [4]                  
             p(log) = [1] x1 + [0]         
             p(min) = [1] x1 + [4]         
            p(quot) = [1] x1 + [1] x2 + [0]
               p(s) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          min(X,0()) = [1] X + [4]
                     > [1] X + [0]
                     = X          
          
          
          Following rules are (at-least) weakly oriented:
              log(s(0())) =  [4]                         
                          >= [4]                         
                          =  0()                         
          
             log(s(s(X))) =  [1] X + [0]                 
                          >= [1] X + [4]                 
                          =  s(log(s(quot(X,s(s(0()))))))
          
           min(s(X),s(Y)) =  [1] X + [4]                 
                          >= [1] X + [4]                 
                          =  min(X,Y)                    
          
           quot(0(),s(Y)) =  [1] Y + [4]                 
                          >= [4]                         
                          =  0()                         
          
          quot(s(X),s(Y)) =  [1] X + [1] Y + [0]         
                          >= [1] X + [1] Y + [4]         
                          =  s(quot(min(X,Y),s(Y)))      
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            log(s(0())) -> 0()
            log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
            min(s(X),s(Y)) -> min(X,Y)
            quot(0(),s(Y)) -> 0()
            quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
        - Weak TRS:
            min(X,0()) -> X
        - Signature:
            {log/1,min/2,quot/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        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(quot) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(0) = [14]        
             p(log) = [1] x1 + [8]
             p(min) = [1] x1 + [7]
            p(quot) = [1] x1 + [0]
               p(s) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          log(s(0())) = [22]
                      > [14]
                      = 0() 
          
          
          Following rules are (at-least) weakly oriented:
             log(s(s(X))) =  [1] X + [8]                 
                          >= [1] X + [8]                 
                          =  s(log(s(quot(X,s(s(0()))))))
          
               min(X,0()) =  [1] X + [7]                 
                          >= [1] X + [0]                 
                          =  X                           
          
           min(s(X),s(Y)) =  [1] X + [7]                 
                          >= [1] X + [7]                 
                          =  min(X,Y)                    
          
           quot(0(),s(Y)) =  [14]                        
                          >= [14]                        
                          =  0()                         
          
          quot(s(X),s(Y)) =  [1] X + [0]                 
                          >= [1] X + [7]                 
                          =  s(quot(min(X,Y),s(Y)))      
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
            min(s(X),s(Y)) -> min(X,Y)
            quot(0(),s(Y)) -> 0()
            quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
        - Weak TRS:
            log(s(0())) -> 0()
            min(X,0()) -> X
        - Signature:
            {log/1,min/2,quot/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        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(quot) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(0) = [3]                  
             p(log) = [1] x1 + [0]         
             p(min) = [1] x1 + [0]         
            p(quot) = [1] x1 + [1] x2 + [0]
               p(s) = [1] x1 + [1]         
          
          Following rules are strictly oriented:
          min(s(X),s(Y)) = [1] X + [1]
                         > [1] X + [0]
                         = min(X,Y)   
          
          quot(0(),s(Y)) = [1] Y + [4]
                         > [3]        
                         = 0()        
          
          
          Following rules are (at-least) weakly oriented:
              log(s(0())) =  [4]                         
                          >= [3]                         
                          =  0()                         
          
             log(s(s(X))) =  [1] X + [2]                 
                          >= [1] X + [7]                 
                          =  s(log(s(quot(X,s(s(0()))))))
          
               min(X,0()) =  [1] X + [0]                 
                          >= [1] X + [0]                 
                          =  X                           
          
          quot(s(X),s(Y)) =  [1] X + [1] Y + [2]         
                          >= [1] X + [1] Y + [2]         
                          =  s(quot(min(X,Y),s(Y)))      
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: MI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
            quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
        - Weak TRS:
            log(s(0())) -> 0()
            min(X,0()) -> X
            min(s(X),s(Y)) -> min(X,Y)
            quot(0(),s(Y)) -> 0()
        - Signature:
            {log/1,min/2,quot/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
    + Applied Processor:
        MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
        
        The following argument positions are considered usable:
          uargs(log) = {1},
          uargs(quot) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {log,min,quot}
        TcT has computed the following interpretation:
             p(0) = [0]          
           p(log) = [3] x_1 + [2]
           p(min) = [1] x_1 + [0]
          p(quot) = [1] x_1 + [2]
             p(s) = [1] x_1 + [4]
        
        Following rules are strictly oriented:
        log(s(s(X))) = [3] X + [26]                
                     > [3] X + [24]                
                     = s(log(s(quot(X,s(s(0()))))))
        
        
        Following rules are (at-least) weakly oriented:
            log(s(0())) =  [14]                  
                        >= [0]                   
                        =  0()                   
        
             min(X,0()) =  [1] X + [0]           
                        >= [1] X + [0]           
                        =  X                     
        
         min(s(X),s(Y)) =  [1] X + [4]           
                        >= [1] X + [0]           
                        =  min(X,Y)              
        
         quot(0(),s(Y)) =  [2]                   
                        >= [0]                   
                        =  0()                   
        
        quot(s(X),s(Y)) =  [1] X + [6]           
                        >= [1] X + [6]           
                        =  s(quot(min(X,Y),s(Y)))
        
* Step 5: MI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
        - Weak TRS:
            log(s(0())) -> 0()
            log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
            min(X,0()) -> X
            min(s(X),s(Y)) -> min(X,Y)
            quot(0(),s(Y)) -> 0()
        - Signature:
            {log/1,min/2,quot/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
    + Applied Processor:
        MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
        
        The following argument positions are considered usable:
          uargs(log) = {1},
          uargs(quot) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {log,min,quot}
        TcT has computed the following interpretation:
             p(0) = [0]            
                    [0]            
           p(log) = [2 0] x_1 + [2]
                    [0 1]       [0]
           p(min) = [1 0] x_1 + [0]
                    [0 1]       [0]
          p(quot) = [1 1] x_1 + [2]
                    [0 1]       [0]
             p(s) = [1 2] x_1 + [0]
                    [0 1]       [2]
        
        Following rules are strictly oriented:
        quot(s(X),s(Y)) = [1 3] X + [4]         
                          [0 1]     [2]         
                        > [1 3] X + [2]         
                          [0 1]     [2]         
                        = s(quot(min(X,Y),s(Y)))
        
        
        Following rules are (at-least) weakly oriented:
           log(s(0())) =  [2]                         
                          [2]                         
                       >= [0]                         
                          [0]                         
                       =  0()                         
        
          log(s(s(X))) =  [2 8] X + [10]              
                          [0 1]     [4]               
                       >= [2 8] X + [10]              
                          [0 1]     [4]               
                       =  s(log(s(quot(X,s(s(0()))))))
        
            min(X,0()) =  [1 0] X + [0]               
                          [0 1]     [0]               
                       >= [1 0] X + [0]               
                          [0 1]     [0]               
                       =  X                           
        
        min(s(X),s(Y)) =  [1 2] X + [0]               
                          [0 1]     [2]               
                       >= [1 0] X + [0]               
                          [0 1]     [0]               
                       =  min(X,Y)                    
        
        quot(0(),s(Y)) =  [2]                         
                          [0]                         
                       >= [0]                         
                          [0]                         
                       =  0()                         
        
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            log(s(0())) -> 0()
            log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
            min(X,0()) -> X
            min(s(X),s(Y)) -> min(X,Y)
            quot(0(),s(Y)) -> 0()
            quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
        - Signature:
            {log/1,min/2,quot/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))