* Step 1: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
            =(.(x,y),nil()) -> false()
            =(nil(),.(y,z)) -> false()
            =(nil(),nil()) -> true()
            del(.(x,.(y,z))) -> f(=(x,y),x,y,z)
            f(false(),x,y,z) -> .(x,del(.(y,z)))
            f(true(),x,y,z) -> del(.(y,z))
        - Signature:
            {=/2,del/1,f/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {=,del,f} and constructors {.,and,false,nil,true,u,v}
    + 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(.) = {2},
            uargs(f) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(.) = [1] x2 + [0]         
                p(=) = [0]                  
              p(and) = [1] x1 + [1] x2 + [0]
              p(del) = [0]                  
                p(f) = [1] x1 + [5]         
            p(false) = [0]                  
              p(nil) = [0]                  
             p(true) = [0]                  
                p(u) = [0]                  
                p(v) = [0]                  
          
          Following rules are strictly oriented:
          f(false(),x,y,z) = [5]             
                           > [0]             
                           = .(x,del(.(y,z)))
          
           f(true(),x,y,z) = [5]             
                           > [0]             
                           = del(.(y,z))     
          
          
          Following rules are (at-least) weakly oriented:
          =(.(x,y),.(u(),v())) =  [0]                   
                               >= [0]                   
                               =  and(=(x,u()),=(y,v()))
          
               =(.(x,y),nil()) =  [0]                   
                               >= [0]                   
                               =  false()               
          
               =(nil(),.(y,z)) =  [0]                   
                               >= [0]                   
                               =  false()               
          
                =(nil(),nil()) =  [0]                   
                               >= [0]                   
                               =  true()                
          
              del(.(x,.(y,z))) =  [0]                   
                               >= [5]                   
                               =  f(=(x,y),x,y,z)       
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
            =(.(x,y),nil()) -> false()
            =(nil(),.(y,z)) -> false()
            =(nil(),nil()) -> true()
            del(.(x,.(y,z))) -> f(=(x,y),x,y,z)
        - Weak TRS:
            f(false(),x,y,z) -> .(x,del(.(y,z)))
            f(true(),x,y,z) -> del(.(y,z))
        - Signature:
            {=/2,del/1,f/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {=,del,f} and constructors {.,and,false,nil,true,u,v}
    + 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(.) = {2},
            uargs(f) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(.) = [1] x1 + [1] x2 + [4]                  
                p(=) = [1] x1 + [0]                           
              p(and) = [0]                                    
              p(del) = [2] x1 + [1]                           
                p(f) = [1] x1 + [1] x2 + [2] x3 + [2] x4 + [9]
            p(false) = [4]                                    
              p(nil) = [8]                                    
             p(true) = [0]                                    
                p(u) = [8]                                    
                p(v) = [0]                                    
          
          Following rules are strictly oriented:
          =(.(x,y),.(u(),v())) = [1] x + [1] y + [4]         
                               > [0]                         
                               = and(=(x,u()),=(y,v()))      
          
               =(nil(),.(y,z)) = [8]                         
                               > [4]                         
                               = false()                     
          
                =(nil(),nil()) = [8]                         
                               > [0]                         
                               = true()                      
          
              del(.(x,.(y,z))) = [2] x + [2] y + [2] z + [17]
                               > [2] x + [2] y + [2] z + [9] 
                               = f(=(x,y),x,y,z)             
          
          
          Following rules are (at-least) weakly oriented:
           =(.(x,y),nil()) =  [1] x + [1] y + [4]         
                           >= [4]                         
                           =  false()                     
          
          f(false(),x,y,z) =  [1] x + [2] y + [2] z + [13]
                           >= [1] x + [2] y + [2] z + [13]
                           =  .(x,del(.(y,z)))            
          
           f(true(),x,y,z) =  [1] x + [2] y + [2] z + [9] 
                           >= [2] y + [2] z + [9]         
                           =  del(.(y,z))                 
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            =(.(x,y),nil()) -> false()
        - Weak TRS:
            =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
            =(nil(),.(y,z)) -> false()
            =(nil(),nil()) -> true()
            del(.(x,.(y,z))) -> f(=(x,y),x,y,z)
            f(false(),x,y,z) -> .(x,del(.(y,z)))
            f(true(),x,y,z) -> del(.(y,z))
        - Signature:
            {=/2,del/1,f/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {=,del,f} and constructors {.,and,false,nil,true,u,v}
    + 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(.) = {2},
            uargs(f) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(.) = [1] x1 + [1] x2 + [1]                  
                p(=) = [1] x1 + [0]                           
              p(and) = [1] x1 + [1] x2 + [1]                  
              p(del) = [2] x1 + [2]                           
                p(f) = [1] x1 + [1] x2 + [2] x3 + [2] x4 + [6]
            p(false) = [0]                                    
              p(nil) = [0]                                    
             p(true) = [0]                                    
                p(u) = [1]                                    
                p(v) = [0]                                    
          
          Following rules are strictly oriented:
          =(.(x,y),nil()) = [1] x + [1] y + [1]
                          > [0]                
                          = false()            
          
          
          Following rules are (at-least) weakly oriented:
          =(.(x,y),.(u(),v())) =  [1] x + [1] y + [1]        
                               >= [1] x + [1] y + [1]        
                               =  and(=(x,u()),=(y,v()))     
          
               =(nil(),.(y,z)) =  [0]                        
                               >= [0]                        
                               =  false()                    
          
                =(nil(),nil()) =  [0]                        
                               >= [0]                        
                               =  true()                     
          
              del(.(x,.(y,z))) =  [2] x + [2] y + [2] z + [6]
                               >= [2] x + [2] y + [2] z + [6]
                               =  f(=(x,y),x,y,z)            
          
              f(false(),x,y,z) =  [1] x + [2] y + [2] z + [6]
                               >= [1] x + [2] y + [2] z + [5]
                               =  .(x,del(.(y,z)))           
          
               f(true(),x,y,z) =  [1] x + [2] y + [2] z + [6]
                               >= [2] y + [2] z + [4]        
                               =  del(.(y,z))                
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
            =(.(x,y),nil()) -> false()
            =(nil(),.(y,z)) -> false()
            =(nil(),nil()) -> true()
            del(.(x,.(y,z))) -> f(=(x,y),x,y,z)
            f(false(),x,y,z) -> .(x,del(.(y,z)))
            f(true(),x,y,z) -> del(.(y,z))
        - Signature:
            {=/2,del/1,f/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {=,del,f} and constructors {.,and,false,nil,true,u,v}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))