WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs 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:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          =#(.(x,y),.(u(),v())) -> c_1(=#(x,u()),=#(y,v()))
          =#(.(x,y),nil()) -> c_2()
          =#(nil(),.(y,z)) -> c_3()
          =#(nil(),nil()) -> c_4()
          del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z),=#(x,y))
          f#(false(),x,y,z) -> c_6(del#(.(y,z)))
          f#(true(),x,y,z) -> c_7(del#(.(y,z)))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            =#(.(x,y),.(u(),v())) -> c_1(=#(x,u()),=#(y,v()))
            =#(.(x,y),nil()) -> c_2()
            =#(nil(),.(y,z)) -> c_3()
            =#(nil(),nil()) -> c_4()
            del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z),=#(x,y))
            f#(false(),x,y,z) -> c_6(del#(.(y,z)))
            f#(true(),x,y,z) -> c_7(del#(.(y,z)))
        - 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,del#/1,f#/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/2
            ,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {=#,del#,f#} and constructors {.,and,false,nil,true,u,v}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,3,4}
        by application of
          Pre({1,2,3,4}) = {5}.
        Here rules are labelled as follows:
          1: =#(.(x,y),.(u(),v())) -> c_1(=#(x,u()),=#(y,v()))
          2: =#(.(x,y),nil()) -> c_2()
          3: =#(nil(),.(y,z)) -> c_3()
          4: =#(nil(),nil()) -> c_4()
          5: del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z),=#(x,y))
          6: f#(false(),x,y,z) -> c_6(del#(.(y,z)))
          7: f#(true(),x,y,z) -> c_7(del#(.(y,z)))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z),=#(x,y))
            f#(false(),x,y,z) -> c_6(del#(.(y,z)))
            f#(true(),x,y,z) -> c_7(del#(.(y,z)))
        - Weak DPs:
            =#(.(x,y),.(u(),v())) -> c_1(=#(x,u()),=#(y,v()))
            =#(.(x,y),nil()) -> c_2()
            =#(nil(),.(y,z)) -> c_3()
            =#(nil(),nil()) -> c_4()
        - 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,del#/1,f#/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/2
            ,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {=#,del#,f#} and constructors {.,and,false,nil,true,u,v}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z),=#(x,y))
             -->_1 f#(true(),x,y,z) -> c_7(del#(.(y,z))):3
             -->_1 f#(false(),x,y,z) -> c_6(del#(.(y,z))):2
             -->_2 =#(nil(),nil()) -> c_4():7
             -->_2 =#(nil(),.(y,z)) -> c_3():6
             -->_2 =#(.(x,y),nil()) -> c_2():5
             -->_2 =#(.(x,y),.(u(),v())) -> c_1(=#(x,u()),=#(y,v())):4
          
          2:S:f#(false(),x,y,z) -> c_6(del#(.(y,z)))
             -->_1 del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z),=#(x,y)):1
          
          3:S:f#(true(),x,y,z) -> c_7(del#(.(y,z)))
             -->_1 del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z),=#(x,y)):1
          
          4:W:=#(.(x,y),.(u(),v())) -> c_1(=#(x,u()),=#(y,v()))
             
          
          5:W:=#(.(x,y),nil()) -> c_2()
             
          
          6:W:=#(nil(),.(y,z)) -> c_3()
             
          
          7:W:=#(nil(),nil()) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: =#(.(x,y),.(u(),v())) -> c_1(=#(x,u()),=#(y,v()))
          5: =#(.(x,y),nil()) -> c_2()
          6: =#(nil(),.(y,z)) -> c_3()
          7: =#(nil(),nil()) -> c_4()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z),=#(x,y))
            f#(false(),x,y,z) -> c_6(del#(.(y,z)))
            f#(true(),x,y,z) -> c_7(del#(.(y,z)))
        - 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,del#/1,f#/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/2
            ,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {=#,del#,f#} and constructors {.,and,false,nil,true,u,v}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z),=#(x,y))
             -->_1 f#(true(),x,y,z) -> c_7(del#(.(y,z))):3
             -->_1 f#(false(),x,y,z) -> c_6(del#(.(y,z))):2
          
          2:S:f#(false(),x,y,z) -> c_6(del#(.(y,z)))
             -->_1 del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z),=#(x,y)):1
          
          3:S:f#(true(),x,y,z) -> c_7(del#(.(y,z)))
             -->_1 del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z),=#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z))
* Step 5: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z))
            f#(false(),x,y,z) -> c_6(del#(.(y,z)))
            f#(true(),x,y,z) -> c_7(del#(.(y,z)))
        - 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,del#/1,f#/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/1
            ,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {=#,del#,f#} and constructors {.,and,false,nil,true,u,v}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
          =(.(x,y),nil()) -> false()
          =(nil(),.(y,z)) -> false()
          =(nil(),nil()) -> true()
          del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z))
          f#(false(),x,y,z) -> c_6(del#(.(y,z)))
          f#(true(),x,y,z) -> c_7(del#(.(y,z)))
* Step 6: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z))
            f#(false(),x,y,z) -> c_6(del#(.(y,z)))
            f#(true(),x,y,z) -> c_7(del#(.(y,z)))
        - Weak TRS:
            =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
            =(.(x,y),nil()) -> false()
            =(nil(),.(y,z)) -> false()
            =(nil(),nil()) -> true()
        - Signature:
            {=/2,del/1,f/4,=#/2,del#/1,f#/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/1
            ,c_6/1,c_7/1}
        - 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 constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(f#) = {1},
            uargs(c_5) = {1},
            uargs(c_6) = {1},
            uargs(c_7) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(.) = [1] x1 + [1] x2 + [4]         
                p(=) = [3] x1 + [0]                  
              p(and) = [1] x1 + [1] x2 + [12]        
              p(del) = [2] x1 + [1]                  
                p(f) = [1] x1 + [4] x3 + [1]         
            p(false) = [12]                          
              p(nil) = [4]                           
             p(true) = [12]                          
                p(u) = [2]                           
                p(v) = [12]                          
               p(=#) = [8] x1 + [1] x2 + [1]         
             p(del#) = [3] x1 + [0]                  
               p(f#) = [1] x1 + [3] x3 + [3] x4 + [6]
              p(c_1) = [0]                           
              p(c_2) = [1]                           
              p(c_3) = [0]                           
              p(c_4) = [0]                           
              p(c_5) = [1] x1 + [8]                  
              p(c_6) = [1] x1 + [5]                  
              p(c_7) = [1] x1 + [8]                  
          
          Following rules are strictly oriented:
          del#(.(x,.(y,z))) = [3] x + [3] y + [3] z + [24]
                            > [3] x + [3] y + [3] z + [14]
                            = c_5(f#(=(x,y),x,y,z))       
          
          f#(false(),x,y,z) = [3] y + [3] z + [18]        
                            > [3] y + [3] z + [17]        
                            = c_6(del#(.(y,z)))           
          
          
          Following rules are (at-least) weakly oriented:
              f#(true(),x,y,z) =  [3] y + [3] z + [18]  
                               >= [3] y + [3] z + [20]  
                               =  c_7(del#(.(y,z)))     
          
          =(.(x,y),.(u(),v())) =  [3] x + [3] y + [12]  
                               >= [3] x + [3] y + [12]  
                               =  and(=(x,u()),=(y,v()))
          
               =(.(x,y),nil()) =  [3] x + [3] y + [12]  
                               >= [12]                  
                               =  false()               
          
               =(nil(),.(y,z)) =  [12]                  
                               >= [12]                  
                               =  false()               
          
                =(nil(),nil()) =  [12]                  
                               >= [12]                  
                               =  true()                
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(true(),x,y,z) -> c_7(del#(.(y,z)))
        - Weak DPs:
            del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z))
            f#(false(),x,y,z) -> c_6(del#(.(y,z)))
        - Weak TRS:
            =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
            =(.(x,y),nil()) -> false()
            =(nil(),.(y,z)) -> false()
            =(nil(),nil()) -> true()
        - Signature:
            {=/2,del/1,f/4,=#/2,del#/1,f#/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/1
            ,c_6/1,c_7/1}
        - 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 constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(f#) = {1},
            uargs(c_5) = {1},
            uargs(c_6) = {1},
            uargs(c_7) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(.) = [1] x1 + [1] x2 + [2]                  
                p(=) = [0]                                    
              p(and) = [0]                                    
              p(del) = [0]                                    
                p(f) = [1] x1 + [1] x4 + [4]                  
            p(false) = [0]                                    
              p(nil) = [0]                                    
             p(true) = [0]                                    
                p(u) = [0]                                    
                p(v) = [0]                                    
               p(=#) = [1] x1 + [1] x2 + [4]                  
             p(del#) = [2] x1 + [0]                           
               p(f#) = [1] x1 + [1] x2 + [2] x3 + [2] x4 + [6]
              p(c_1) = [1]                                    
              p(c_2) = [0]                                    
              p(c_3) = [4]                                    
              p(c_4) = [1]                                    
              p(c_5) = [1] x1 + [1]                           
              p(c_6) = [1] x1 + [0]                           
              p(c_7) = [1] x1 + [1]                           
          
          Following rules are strictly oriented:
          f#(true(),x,y,z) = [1] x + [2] y + [2] z + [6]
                           > [2] y + [2] z + [5]        
                           = c_7(del#(.(y,z)))          
          
          
          Following rules are (at-least) weakly oriented:
             del#(.(x,.(y,z))) =  [2] x + [2] y + [2] z + [8]
                               >= [1] x + [2] y + [2] z + [7]
                               =  c_5(f#(=(x,y),x,y,z))      
          
             f#(false(),x,y,z) =  [1] x + [2] y + [2] z + [6]
                               >= [2] y + [2] z + [4]        
                               =  c_6(del#(.(y,z)))          
          
          =(.(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()                     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 8: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            del#(.(x,.(y,z))) -> c_5(f#(=(x,y),x,y,z))
            f#(false(),x,y,z) -> c_6(del#(.(y,z)))
            f#(true(),x,y,z) -> c_7(del#(.(y,z)))
        - Weak TRS:
            =(.(x,y),.(u(),v())) -> and(=(x,u()),=(y,v()))
            =(.(x,y),nil()) -> false()
            =(nil(),.(y,z)) -> false()
            =(nil(),nil()) -> true()
        - Signature:
            {=/2,del/1,f/4,=#/2,del#/1,f#/4} / {./2,and/2,false/0,nil/0,true/0,u/0,v/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/1
            ,c_6/1,c_7/1}
        - 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))