WORST_CASE(?,O(n^2))
* Step 1: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            foldr#3(x8,x12,Nil()) -> x12
            foldr#3(lam1_n(x24),x14,Cons(x32,x6)) -> Cons(Pair(x24,x32),foldr#3(lam1_n(x24),x14,x6))
            foldr#3(lam2_ms(x3),Nil(),Cons(x2,x1)) -> foldr#3(lam1_n(x2),foldr#3(lam2_ms(x3),Nil(),x1),x3)
            main(x2,x1) -> foldr#3(lam2_ms(x2),Nil(),x1)
        - Signature:
            {foldr#3/3,main/2} / {Cons/2,Nil/0,Pair/2,lam1_n/1,lam2_ms/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {foldr#3,main} and constructors {Cons,Nil,Pair,lam1_n
            ,lam2_ms}
    + 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(Cons) = {2},
            uargs(foldr#3) = {2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(Cons) = [1] x1 + [1] x2 + [0]
                p(Nil) = [0]                  
               p(Pair) = [12]                 
            p(foldr#3) = [4] x1 + [1] x2 + [6]
             p(lam1_n) = [0]                  
            p(lam2_ms) = [2]                  
               p(main) = [15]                 
          
          Following rules are strictly oriented:
          foldr#3(x8,x12,Nil()) = [1] x12 + [4] x8 + [6]       
                                > [1] x12 + [0]                
                                = x12                          
          
                    main(x2,x1) = [15]                         
                                > [14]                         
                                = foldr#3(lam2_ms(x2),Nil(),x1)
          
          
          Following rules are (at-least) weakly oriented:
           foldr#3(lam1_n(x24),x14,Cons(x32,x6)) =  [1] x14 + [6]                                       
                                                 >= [1] x14 + [18]                                      
                                                 =  Cons(Pair(x24,x32),foldr#3(lam1_n(x24),x14,x6))     
          
          foldr#3(lam2_ms(x3),Nil(),Cons(x2,x1)) =  [14]                                                
                                                 >= [20]                                                
                                                 =  foldr#3(lam1_n(x2),foldr#3(lam2_ms(x3),Nil(),x1),x3)
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            foldr#3(lam1_n(x24),x14,Cons(x32,x6)) -> Cons(Pair(x24,x32),foldr#3(lam1_n(x24),x14,x6))
            foldr#3(lam2_ms(x3),Nil(),Cons(x2,x1)) -> foldr#3(lam1_n(x2),foldr#3(lam2_ms(x3),Nil(),x1),x3)
        - Weak TRS:
            foldr#3(x8,x12,Nil()) -> x12
            main(x2,x1) -> foldr#3(lam2_ms(x2),Nil(),x1)
        - Signature:
            {foldr#3/3,main/2} / {Cons/2,Nil/0,Pair/2,lam1_n/1,lam2_ms/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {foldr#3,main} and constructors {Cons,Nil,Pair,lam1_n
            ,lam2_ms}
    + 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(Cons) = {2},
          uargs(foldr#3) = {2}
        
        Following symbols are considered usable:
          {foldr#3,main}
        TcT has computed the following interpretation:
             p(Cons) = 2 + x2                   
              p(Nil) = 2                        
             p(Pair) = 0                        
          p(foldr#3) = x1*x3 + x2               
           p(lam1_n) = 1                        
          p(lam2_ms) = 2 + x1                   
             p(main) = 4 + 4*x1*x2 + x1^2 + 3*x2
        
        Following rules are strictly oriented:
        foldr#3(lam2_ms(x3),Nil(),Cons(x2,x1)) = 6 + 2*x1 + x1*x3 + 2*x3                             
                                               > 2 + 2*x1 + x1*x3 + x3                               
                                               = foldr#3(lam1_n(x2),foldr#3(lam2_ms(x3),Nil(),x1),x3)
        
        
        Following rules are (at-least) weakly oriented:
                        foldr#3(x8,x12,Nil()) =  x12 + 2*x8                                     
                                              >= x12                                            
                                              =  x12                                            
        
        foldr#3(lam1_n(x24),x14,Cons(x32,x6)) =  2 + x14 + x6                                   
                                              >= 2 + x14 + x6                                   
                                              =  Cons(Pair(x24,x32),foldr#3(lam1_n(x24),x14,x6))
        
                                  main(x2,x1) =  4 + 3*x1 + 4*x1*x2 + x2^2                      
                                              >= 2 + 2*x1 + x1*x2                               
                                              =  foldr#3(lam2_ms(x2),Nil(),x1)                  
        
* Step 3: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            foldr#3(lam1_n(x24),x14,Cons(x32,x6)) -> Cons(Pair(x24,x32),foldr#3(lam1_n(x24),x14,x6))
        - Weak TRS:
            foldr#3(x8,x12,Nil()) -> x12
            foldr#3(lam2_ms(x3),Nil(),Cons(x2,x1)) -> foldr#3(lam1_n(x2),foldr#3(lam2_ms(x3),Nil(),x1),x3)
            main(x2,x1) -> foldr#3(lam2_ms(x2),Nil(),x1)
        - Signature:
            {foldr#3/3,main/2} / {Cons/2,Nil/0,Pair/2,lam1_n/1,lam2_ms/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {foldr#3,main} and constructors {Cons,Nil,Pair,lam1_n
            ,lam2_ms}
    + 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(Cons) = {2},
          uargs(foldr#3) = {2}
        
        Following symbols are considered usable:
          {foldr#3,main}
        TcT has computed the following interpretation:
             p(Cons) = 2 + x1 + x2                              
              p(Nil) = 0                                        
             p(Pair) = 4                                        
          p(foldr#3) = 2*x1*x3 + x1^2 + x2                      
           p(lam1_n) = 2                                        
          p(lam2_ms) = 1 + x1                                   
             p(main) = 1 + 5*x1 + 4*x1*x2 + 2*x1^2 + 2*x2 + x2^2
        
        Following rules are strictly oriented:
        foldr#3(lam1_n(x24),x14,Cons(x32,x6)) = 12 + x14 + 4*x32 + 4*x6                        
                                              > 10 + x14 + 4*x6                                
                                              = Cons(Pair(x24,x32),foldr#3(lam1_n(x24),x14,x6))
        
        
        Following rules are (at-least) weakly oriented:
                         foldr#3(x8,x12,Nil()) =  x12 + x8^2                                          
                                               >= x12                                                 
                                               =  x12                                                 
        
        foldr#3(lam2_ms(x3),Nil(),Cons(x2,x1)) =  5 + 2*x1 + 2*x1*x3 + 2*x2 + 2*x2*x3 + 6*x3 + x3^2   
                                               >= 5 + 2*x1 + 2*x1*x3 + 6*x3 + x3^2                    
                                               =  foldr#3(lam1_n(x2),foldr#3(lam2_ms(x3),Nil(),x1),x3)
        
                                   main(x2,x1) =  1 + 2*x1 + 4*x1*x2 + x1^2 + 5*x2 + 2*x2^2           
                                               >= 1 + 2*x1 + 2*x1*x2 + 2*x2 + x2^2                    
                                               =  foldr#3(lam2_ms(x2),Nil(),x1)                       
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            foldr#3(x8,x12,Nil()) -> x12
            foldr#3(lam1_n(x24),x14,Cons(x32,x6)) -> Cons(Pair(x24,x32),foldr#3(lam1_n(x24),x14,x6))
            foldr#3(lam2_ms(x3),Nil(),Cons(x2,x1)) -> foldr#3(lam1_n(x2),foldr#3(lam2_ms(x3),Nil(),x1),x3)
            main(x2,x1) -> foldr#3(lam2_ms(x2),Nil(),x1)
        - Signature:
            {foldr#3/3,main/2} / {Cons/2,Nil/0,Pair/2,lam1_n/1,lam2_ms/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {foldr#3,main} and constructors {Cons,Nil,Pair,lam1_n
            ,lam2_ms}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))