* Step 1: MI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            gcd(0(),y) -> y
            gcd(s(x),0()) -> s(x)
            gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
            if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
            if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus} and constructors {0,false,s,true}
    + 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(gcd) = {1},
          uargs(if_gcd) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
               p(0) = [9]                              
           p(false) = [1]                              
             p(gcd) = [1] x_1 + [1] x_2 + [13]         
          p(if_gcd) = [4] x_1 + [1] x_2 + [1] x_3 + [8]
              p(le) = [1]                              
           p(minus) = [1] x_1 + [0]                    
               p(s) = [1] x_1 + [1]                    
            p(true) = [1]                              
        
        Following rules are strictly oriented:
              gcd(0(),y) = [1] y + [22]             
                         > [1] y + [0]              
                         = y                        
        
           gcd(s(x),0()) = [1] x + [23]             
                         > [1] x + [1]              
                         = s(x)                     
        
          gcd(s(x),s(y)) = [1] x + [1] y + [15]     
                         > [1] x + [1] y + [14]     
                         = if_gcd(le(y,x),s(x),s(y))
        
        minus(s(x),s(y)) = [1] x + [1]              
                         > [1] x + [0]              
                         = minus(x,y)               
        
        
        Following rules are (at-least) weakly oriented:
        if_gcd(false(),s(x),s(y)) =  [1] x + [1] y + [14]
                                  >= [1] x + [1] y + [14]
                                  =  gcd(minus(y,x),s(x))
        
         if_gcd(true(),s(x),s(y)) =  [1] x + [1] y + [14]
                                  >= [1] x + [1] y + [14]
                                  =  gcd(minus(x,y),s(y))
        
                        le(0(),y) =  [1]                 
                                  >= [1]                 
                                  =  true()              
        
                     le(s(x),0()) =  [1]                 
                                  >= [1]                 
                                  =  false()             
        
                    le(s(x),s(y)) =  [1]                 
                                  >= [1]                 
                                  =  le(x,y)             
        
                     minus(x,0()) =  [1] x + [0]         
                                  >= [1] x + [0]         
                                  =  x                   
        
* Step 2: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
            if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,0()) -> x
        - Weak TRS:
            gcd(0(),y) -> y
            gcd(s(x),0()) -> s(x)
            gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus} and constructors {0,false,s,true}
    + 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(gcd) = {1},
            uargs(if_gcd) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [6]                           
             p(false) = [1]                           
               p(gcd) = [1] x1 + [1] x2 + [7]         
            p(if_gcd) = [1] x1 + [1] x2 + [1] x3 + [0]
                p(le) = [1]                           
             p(minus) = [1] x1 + [2]                  
                 p(s) = [1] x1 + [1]                  
              p(true) = [4]                           
          
          Following rules are strictly oriented:
          minus(x,0()) = [1] x + [2]
                       > [1] x + [0]
                       = x          
          
          
          Following rules are (at-least) weakly oriented:
                         gcd(0(),y) =  [1] y + [13]             
                                    >= [1] y + [0]              
                                    =  y                        
          
                      gcd(s(x),0()) =  [1] x + [14]             
                                    >= [1] x + [1]              
                                    =  s(x)                     
          
                     gcd(s(x),s(y)) =  [1] x + [1] y + [9]      
                                    >= [1] x + [1] y + [3]      
                                    =  if_gcd(le(y,x),s(x),s(y))
          
          if_gcd(false(),s(x),s(y)) =  [1] x + [1] y + [3]      
                                    >= [1] x + [1] y + [10]     
                                    =  gcd(minus(y,x),s(x))     
          
           if_gcd(true(),s(x),s(y)) =  [1] x + [1] y + [6]      
                                    >= [1] x + [1] y + [10]     
                                    =  gcd(minus(x,y),s(y))     
          
                          le(0(),y) =  [1]                      
                                    >= [4]                      
                                    =  true()                   
          
                       le(s(x),0()) =  [1]                      
                                    >= [1]                      
                                    =  false()                  
          
                      le(s(x),s(y)) =  [1]                      
                                    >= [1]                      
                                    =  le(x,y)                  
          
                   minus(s(x),s(y)) =  [1] x + [3]              
                                    >= [1] x + [2]              
                                    =  minus(x,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:
            if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
            if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
        - Weak TRS:
            gcd(0(),y) -> y
            gcd(s(x),0()) -> s(x)
            gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus} and constructors {0,false,s,true}
    + 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(gcd) = {1},
            uargs(if_gcd) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [0]                           
             p(false) = [4]                           
               p(gcd) = [1] x1 + [1] x2 + [2]         
            p(if_gcd) = [1] x1 + [1] x2 + [1] x3 + [0]
                p(le) = [1]                           
             p(minus) = [1] x1 + [4]                  
                 p(s) = [1] x1 + [4]                  
              p(true) = [4]                           
          
          Following rules are strictly oriented:
          if_gcd(false(),s(x),s(y)) = [1] x + [1] y + [12]
                                    > [1] x + [1] y + [10]
                                    = gcd(minus(y,x),s(x))
          
           if_gcd(true(),s(x),s(y)) = [1] x + [1] y + [12]
                                    > [1] x + [1] y + [10]
                                    = gcd(minus(x,y),s(y))
          
          
          Following rules are (at-least) weakly oriented:
                gcd(0(),y) =  [1] y + [2]              
                           >= [1] y + [0]              
                           =  y                        
          
             gcd(s(x),0()) =  [1] x + [6]              
                           >= [1] x + [4]              
                           =  s(x)                     
          
            gcd(s(x),s(y)) =  [1] x + [1] y + [10]     
                           >= [1] x + [1] y + [9]      
                           =  if_gcd(le(y,x),s(x),s(y))
          
                 le(0(),y) =  [1]                      
                           >= [4]                      
                           =  true()                   
          
              le(s(x),0()) =  [1]                      
                           >= [4]                      
                           =  false()                  
          
             le(s(x),s(y)) =  [1]                      
                           >= [1]                      
                           =  le(x,y)                  
          
              minus(x,0()) =  [1] x + [4]              
                           >= [1] x + [0]              
                           =  x                        
          
          minus(s(x),s(y)) =  [1] x + [8]              
                           >= [1] x + [4]              
                           =  minus(x,y)               
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
        - Weak TRS:
            gcd(0(),y) -> y
            gcd(s(x),0()) -> s(x)
            gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
            if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
            if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus} and constructors {0,false,s,true}
    + 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(gcd) = {1},
            uargs(if_gcd) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [4]                           
             p(false) = [7]                           
               p(gcd) = [1] x1 + [1] x2 + [4]         
            p(if_gcd) = [1] x1 + [1] x2 + [1] x3 + [0]
                p(le) = [4]                           
             p(minus) = [1] x1 + [0]                  
                 p(s) = [1] x1 + [1]                  
              p(true) = [3]                           
          
          Following rules are strictly oriented:
          le(0(),y) = [4]   
                    > [3]   
                    = true()
          
          
          Following rules are (at-least) weakly oriented:
                         gcd(0(),y) =  [1] y + [8]              
                                    >= [1] y + [0]              
                                    =  y                        
          
                      gcd(s(x),0()) =  [1] x + [9]              
                                    >= [1] x + [1]              
                                    =  s(x)                     
          
                     gcd(s(x),s(y)) =  [1] x + [1] y + [6]      
                                    >= [1] x + [1] y + [6]      
                                    =  if_gcd(le(y,x),s(x),s(y))
          
          if_gcd(false(),s(x),s(y)) =  [1] x + [1] y + [9]      
                                    >= [1] x + [1] y + [5]      
                                    =  gcd(minus(y,x),s(x))     
          
           if_gcd(true(),s(x),s(y)) =  [1] x + [1] y + [5]      
                                    >= [1] x + [1] y + [5]      
                                    =  gcd(minus(x,y),s(y))     
          
                       le(s(x),0()) =  [4]                      
                                    >= [7]                      
                                    =  false()                  
          
                      le(s(x),s(y)) =  [4]                      
                                    >= [4]                      
                                    =  le(x,y)                  
          
                       minus(x,0()) =  [1] x + [0]              
                                    >= [1] x + [0]              
                                    =  x                        
          
                   minus(s(x),s(y)) =  [1] x + [1]              
                                    >= [1] x + [0]              
                                    =  minus(x,y)               
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
        - Weak TRS:
            gcd(0(),y) -> y
            gcd(s(x),0()) -> s(x)
            gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
            if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
            if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
            le(0(),y) -> true()
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus} and constructors {0,false,s,true}
    + 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(gcd) = {1},
            uargs(if_gcd) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [5]                           
             p(false) = [4]                           
               p(gcd) = [1] x1 + [1] x2 + [6]         
            p(if_gcd) = [1] x1 + [1] x2 + [1] x3 + [1]
                p(le) = [5]                           
             p(minus) = [1] x1 + [3]                  
                 p(s) = [1] x1 + [4]                  
              p(true) = [4]                           
          
          Following rules are strictly oriented:
          le(s(x),0()) = [5]    
                       > [4]    
                       = false()
          
          
          Following rules are (at-least) weakly oriented:
                         gcd(0(),y) =  [1] y + [11]             
                                    >= [1] y + [0]              
                                    =  y                        
          
                      gcd(s(x),0()) =  [1] x + [15]             
                                    >= [1] x + [4]              
                                    =  s(x)                     
          
                     gcd(s(x),s(y)) =  [1] x + [1] y + [14]     
                                    >= [1] x + [1] y + [14]     
                                    =  if_gcd(le(y,x),s(x),s(y))
          
          if_gcd(false(),s(x),s(y)) =  [1] x + [1] y + [13]     
                                    >= [1] x + [1] y + [13]     
                                    =  gcd(minus(y,x),s(x))     
          
           if_gcd(true(),s(x),s(y)) =  [1] x + [1] y + [13]     
                                    >= [1] x + [1] y + [13]     
                                    =  gcd(minus(x,y),s(y))     
          
                          le(0(),y) =  [5]                      
                                    >= [4]                      
                                    =  true()                   
          
                      le(s(x),s(y)) =  [5]                      
                                    >= [5]                      
                                    =  le(x,y)                  
          
                       minus(x,0()) =  [1] x + [3]              
                                    >= [1] x + [0]              
                                    =  x                        
          
                   minus(s(x),s(y)) =  [1] x + [7]              
                                    >= [1] x + [3]              
                                    =  minus(x,y)               
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            le(s(x),s(y)) -> le(x,y)
        - Weak TRS:
            gcd(0(),y) -> y
            gcd(s(x),0()) -> s(x)
            gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
            if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
            if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus} and constructors {0,false,s,true}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(gcd) = {1},
          uargs(if_gcd) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
               p(0) = 0                                    
           p(false) = 0                                    
             p(gcd) = x1 + 4*x1*x2 + 3*x1^2 + 4*x2 + 3*x2^2
          p(if_gcd) = 3 + 4*x1 + 4*x2*x3 + 3*x2^2 + 3*x3^2 
              p(le) = x1                                   
           p(minus) = x1                                   
               p(s) = 1 + x1                               
            p(true) = 0                                    
        
        Following rules are strictly oriented:
        le(s(x),s(y)) = 1 + x  
                      > x      
                      = le(x,y)
        
        
        Following rules are (at-least) weakly oriented:
                       gcd(0(),y) =  4*y + 3*y^2                             
                                  >= y                                       
                                  =  y                                       
        
                    gcd(s(x),0()) =  4 + 7*x + 3*x^2                         
                                  >= 1 + x                                   
                                  =  s(x)                                    
        
                   gcd(s(x),s(y)) =  15 + 11*x + 4*x*y + 3*x^2 + 14*y + 3*y^2
                                  >= 13 + 10*x + 4*x*y + 3*x^2 + 14*y + 3*y^2
                                  =  if_gcd(le(y,x),s(x),s(y))               
        
        if_gcd(false(),s(x),s(y)) =  13 + 10*x + 4*x*y + 3*x^2 + 10*y + 3*y^2
                                  >= 7 + 10*x + 4*x*y + 3*x^2 + 5*y + 3*y^2  
                                  =  gcd(minus(y,x),s(x))                    
        
         if_gcd(true(),s(x),s(y)) =  13 + 10*x + 4*x*y + 3*x^2 + 10*y + 3*y^2
                                  >= 7 + 5*x + 4*x*y + 3*x^2 + 10*y + 3*y^2  
                                  =  gcd(minus(x,y),s(y))                    
        
                        le(0(),y) =  0                                       
                                  >= 0                                       
                                  =  true()                                  
        
                     le(s(x),0()) =  1 + x                                   
                                  >= 0                                       
                                  =  false()                                 
        
                     minus(x,0()) =  x                                       
                                  >= x                                       
                                  =  x                                       
        
                 minus(s(x),s(y)) =  1 + x                                   
                                  >= x                                       
                                  =  minus(x,y)                              
        
* Step 7: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            gcd(0(),y) -> y
            gcd(s(x),0()) -> s(x)
            gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
            if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
            if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {gcd/2,if_gcd/3,le/2,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {gcd,if_gcd,le,minus} and constructors {0,false,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))