WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            sqr(0()) -> 0()
            sqr(s(x)) -> +(sqr(x),s(double(x)))
            sqr(s(x)) -> s(+(sqr(x),double(x)))
        - Signature:
            {+/2,double/1,sqr/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          +#(x,0()) -> c_1()
          +#(x,s(y)) -> c_2(+#(x,y))
          double#(0()) -> c_3()
          double#(s(x)) -> c_4(double#(x))
          sqr#(0()) -> c_5()
          sqr#(s(x)) -> c_6(+#(sqr(x),s(double(x))),sqr#(x),double#(x))
          sqr#(s(x)) -> c_7(+#(sqr(x),double(x)),sqr#(x),double#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,0()) -> c_1()
            +#(x,s(y)) -> c_2(+#(x,y))
            double#(0()) -> c_3()
            double#(s(x)) -> c_4(double#(x))
            sqr#(0()) -> c_5()
            sqr#(s(x)) -> c_6(+#(sqr(x),s(double(x))),sqr#(x),double#(x))
            sqr#(s(x)) -> c_7(+#(sqr(x),double(x)),sqr#(x),double#(x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            sqr(0()) -> 0()
            sqr(s(x)) -> +(sqr(x),s(double(x)))
            sqr(s(x)) -> s(+(sqr(x),double(x)))
        - Signature:
            {+/2,double/1,sqr/1,+#/2,double#/1,sqr#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/3,c_7/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,sqr#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,5}
        by application of
          Pre({1,3,5}) = {2,4,6,7}.
        Here rules are labelled as follows:
          1: +#(x,0()) -> c_1()
          2: +#(x,s(y)) -> c_2(+#(x,y))
          3: double#(0()) -> c_3()
          4: double#(s(x)) -> c_4(double#(x))
          5: sqr#(0()) -> c_5()
          6: sqr#(s(x)) -> c_6(+#(sqr(x),s(double(x))),sqr#(x),double#(x))
          7: sqr#(s(x)) -> c_7(+#(sqr(x),double(x)),sqr#(x),double#(x))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
            double#(s(x)) -> c_4(double#(x))
            sqr#(s(x)) -> c_6(+#(sqr(x),s(double(x))),sqr#(x),double#(x))
            sqr#(s(x)) -> c_7(+#(sqr(x),double(x)),sqr#(x),double#(x))
        - Weak DPs:
            +#(x,0()) -> c_1()
            double#(0()) -> c_3()
            sqr#(0()) -> c_5()
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            sqr(0()) -> 0()
            sqr(s(x)) -> +(sqr(x),s(double(x)))
            sqr(s(x)) -> s(+(sqr(x),double(x)))
        - Signature:
            {+/2,double/1,sqr/1,+#/2,double#/1,sqr#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/3,c_7/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,sqr#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:+#(x,s(y)) -> c_2(+#(x,y))
             -->_1 +#(x,0()) -> c_1():5
             -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1
          
          2:S:double#(s(x)) -> c_4(double#(x))
             -->_1 double#(0()) -> c_3():6
             -->_1 double#(s(x)) -> c_4(double#(x)):2
          
          3:S:sqr#(s(x)) -> c_6(+#(sqr(x),s(double(x))),sqr#(x),double#(x))
             -->_2 sqr#(s(x)) -> c_7(+#(sqr(x),double(x)),sqr#(x),double#(x)):4
             -->_2 sqr#(0()) -> c_5():7
             -->_3 double#(0()) -> c_3():6
             -->_2 sqr#(s(x)) -> c_6(+#(sqr(x),s(double(x))),sqr#(x),double#(x)):3
             -->_3 double#(s(x)) -> c_4(double#(x)):2
             -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1
          
          4:S:sqr#(s(x)) -> c_7(+#(sqr(x),double(x)),sqr#(x),double#(x))
             -->_2 sqr#(0()) -> c_5():7
             -->_3 double#(0()) -> c_3():6
             -->_1 +#(x,0()) -> c_1():5
             -->_2 sqr#(s(x)) -> c_7(+#(sqr(x),double(x)),sqr#(x),double#(x)):4
             -->_2 sqr#(s(x)) -> c_6(+#(sqr(x),s(double(x))),sqr#(x),double#(x)):3
             -->_3 double#(s(x)) -> c_4(double#(x)):2
             -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1
          
          5:W:+#(x,0()) -> c_1()
             
          
          6:W:double#(0()) -> c_3()
             
          
          7:W:sqr#(0()) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: sqr#(0()) -> c_5()
          6: double#(0()) -> c_3()
          5: +#(x,0()) -> c_1()
* Step 4: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
            double#(s(x)) -> c_4(double#(x))
            sqr#(s(x)) -> c_6(+#(sqr(x),s(double(x))),sqr#(x),double#(x))
            sqr#(s(x)) -> c_7(+#(sqr(x),double(x)),sqr#(x),double#(x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            sqr(0()) -> 0()
            sqr(s(x)) -> +(sqr(x),s(double(x)))
            sqr(s(x)) -> s(+(sqr(x),double(x)))
        - Signature:
            {+/2,double/1,sqr/1,+#/2,double#/1,sqr#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/3,c_7/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,sqr#} and constructors {0,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          sqr#(s(x)) -> c_6(+#(sqr(x),s(double(x))),sqr#(x),double#(x))
          sqr#(s(x)) -> c_7(+#(sqr(x),double(x)),sqr#(x),double#(x))
        and a lower component
          +#(x,s(y)) -> c_2(+#(x,y))
          double#(s(x)) -> c_4(double#(x))
        Further, following extension rules are added to the lower component.
          sqr#(s(x)) -> +#(sqr(x),double(x))
          sqr#(s(x)) -> +#(sqr(x),s(double(x)))
          sqr#(s(x)) -> double#(x)
          sqr#(s(x)) -> sqr#(x)
** Step 4.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sqr#(s(x)) -> c_6(+#(sqr(x),s(double(x))),sqr#(x),double#(x))
            sqr#(s(x)) -> c_7(+#(sqr(x),double(x)),sqr#(x),double#(x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            sqr(0()) -> 0()
            sqr(s(x)) -> +(sqr(x),s(double(x)))
            sqr(s(x)) -> s(+(sqr(x),double(x)))
        - Signature:
            {+/2,double/1,sqr/1,+#/2,double#/1,sqr#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/3,c_7/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,sqr#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:sqr#(s(x)) -> c_6(+#(sqr(x),s(double(x))),sqr#(x),double#(x))
             -->_2 sqr#(s(x)) -> c_7(+#(sqr(x),double(x)),sqr#(x),double#(x)):2
             -->_2 sqr#(s(x)) -> c_6(+#(sqr(x),s(double(x))),sqr#(x),double#(x)):1
          
          2:S:sqr#(s(x)) -> c_7(+#(sqr(x),double(x)),sqr#(x),double#(x))
             -->_2 sqr#(s(x)) -> c_7(+#(sqr(x),double(x)),sqr#(x),double#(x)):2
             -->_2 sqr#(s(x)) -> c_6(+#(sqr(x),s(double(x))),sqr#(x),double#(x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sqr#(s(x)) -> c_6(sqr#(x))
          sqr#(s(x)) -> c_7(sqr#(x))
** Step 4.a:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sqr#(s(x)) -> c_6(sqr#(x))
            sqr#(s(x)) -> c_7(sqr#(x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            sqr(0()) -> 0()
            sqr(s(x)) -> +(sqr(x),s(double(x)))
            sqr(s(x)) -> s(+(sqr(x),double(x)))
        - Signature:
            {+/2,double/1,sqr/1,+#/2,double#/1,sqr#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,sqr#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          sqr#(s(x)) -> c_6(sqr#(x))
          sqr#(s(x)) -> c_7(sqr#(x))
** Step 4.a:3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sqr#(s(x)) -> c_6(sqr#(x))
            sqr#(s(x)) -> c_7(sqr#(x))
        - Signature:
            {+/2,double/1,sqr/1,+#/2,double#/1,sqr#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,sqr#} 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 constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_6) = {1},
            uargs(c_7) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                  p(+) = [0]         
                  p(0) = [0]         
             p(double) = [0]         
                  p(s) = [1] x1 + [1]
                p(sqr) = [0]         
                 p(+#) = [0]         
            p(double#) = [0]         
               p(sqr#) = [1] x1 + [9]
                p(c_1) = [0]         
                p(c_2) = [0]         
                p(c_3) = [0]         
                p(c_4) = [2]         
                p(c_5) = [1]         
                p(c_6) = [1] x1 + [1]
                p(c_7) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          sqr#(s(x)) = [1] x + [10]
                     > [1] x + [9] 
                     = c_7(sqr#(x))
          
          
          Following rules are (at-least) weakly oriented:
          sqr#(s(x)) =  [1] x + [10]
                     >= [1] x + [10]
                     =  c_6(sqr#(x))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 4.a:4: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sqr#(s(x)) -> c_6(sqr#(x))
        - Weak DPs:
            sqr#(s(x)) -> c_7(sqr#(x))
        - Signature:
            {+/2,double/1,sqr/1,+#/2,double#/1,sqr#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,sqr#} 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 constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_6) = {1},
            uargs(c_7) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                  p(+) = [0]         
                  p(0) = [0]         
             p(double) = [0]         
                  p(s) = [1] x1 + [1]
                p(sqr) = [0]         
                 p(+#) = [0]         
            p(double#) = [0]         
               p(sqr#) = [1] x1 + [6]
                p(c_1) = [1]         
                p(c_2) = [1] x1 + [4]
                p(c_3) = [4]         
                p(c_4) = [1]         
                p(c_5) = [1]         
                p(c_6) = [1] x1 + [0]
                p(c_7) = [1] x1 + [1]
          
          Following rules are strictly oriented:
          sqr#(s(x)) = [1] x + [7] 
                     > [1] x + [6] 
                     = c_6(sqr#(x))
          
          
          Following rules are (at-least) weakly oriented:
          sqr#(s(x)) =  [1] x + [7] 
                     >= [1] x + [7] 
                     =  c_7(sqr#(x))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 4.a:5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sqr#(s(x)) -> c_6(sqr#(x))
            sqr#(s(x)) -> c_7(sqr#(x))
        - Signature:
            {+/2,double/1,sqr/1,+#/2,double#/1,sqr#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,sqr#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 4.b:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
            double#(s(x)) -> c_4(double#(x))
        - Weak DPs:
            sqr#(s(x)) -> +#(sqr(x),double(x))
            sqr#(s(x)) -> +#(sqr(x),s(double(x)))
            sqr#(s(x)) -> double#(x)
            sqr#(s(x)) -> sqr#(x)
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            sqr(0()) -> 0()
            sqr(s(x)) -> +(sqr(x),s(double(x)))
            sqr(s(x)) -> s(+(sqr(x),double(x)))
        - Signature:
            {+/2,double/1,sqr/1,+#/2,double#/1,sqr#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/3,c_7/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,sqr#} and constructors {0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_2) = {1},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {+#,double#,sqr#}
        TcT has computed the following interpretation:
                p(+) = [8] x1 + [0]                  
                p(0) = [2]                           
           p(double) = [4] x1 + [0]                  
                p(s) = [1] x1 + [1]                  
              p(sqr) = [3] x1 + [1]                  
               p(+#) = [0]                           
          p(double#) = [4] x1 + [0]                  
             p(sqr#) = [10] x1 + [12]                
              p(c_1) = [1]                           
              p(c_2) = [8] x1 + [0]                  
              p(c_3) = [2]                           
              p(c_4) = [1] x1 + [3]                  
              p(c_5) = [0]                           
              p(c_6) = [1] x1 + [2] x2 + [8] x3 + [1]
              p(c_7) = [1] x1 + [4] x2 + [1] x3 + [2]
        
        Following rules are strictly oriented:
        double#(s(x)) = [4] x + [4]    
                      > [4] x + [3]    
                      = c_4(double#(x))
        
        
        Following rules are (at-least) weakly oriented:
        +#(x,s(y)) =  [0]                    
                   >= [0]                    
                   =  c_2(+#(x,y))           
        
        sqr#(s(x)) =  [10] x + [22]          
                   >= [0]                    
                   =  +#(sqr(x),double(x))   
        
        sqr#(s(x)) =  [10] x + [22]          
                   >= [0]                    
                   =  +#(sqr(x),s(double(x)))
        
        sqr#(s(x)) =  [10] x + [22]          
                   >= [4] x + [0]            
                   =  double#(x)             
        
        sqr#(s(x)) =  [10] x + [22]          
                   >= [10] x + [12]          
                   =  sqr#(x)                
        
** Step 4.b:2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
        - Weak DPs:
            double#(s(x)) -> c_4(double#(x))
            sqr#(s(x)) -> +#(sqr(x),double(x))
            sqr#(s(x)) -> +#(sqr(x),s(double(x)))
            sqr#(s(x)) -> double#(x)
            sqr#(s(x)) -> sqr#(x)
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            sqr(0()) -> 0()
            sqr(s(x)) -> +(sqr(x),s(double(x)))
            sqr(s(x)) -> s(+(sqr(x),double(x)))
        - Signature:
            {+/2,double/1,sqr/1,+#/2,double#/1,sqr#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/3,c_7/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,sqr#} and constructors {0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_2) = {1},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {double,+#,double#,sqr#}
        TcT has computed the following interpretation:
                p(+) = [1] x2 + [6]         
                p(0) = [7]                  
           p(double) = [3] x1 + [2]         
                p(s) = [1] x1 + [2]         
              p(sqr) = [0]                  
               p(+#) = [4] x2 + [0]         
          p(double#) = [0]                  
             p(sqr#) = [12] x1 + [0]        
              p(c_1) = [1]                  
              p(c_2) = [1] x1 + [4]         
              p(c_3) = [0]                  
              p(c_4) = [1] x1 + [0]         
              p(c_5) = [1]                  
              p(c_6) = [1] x1 + [4] x2 + [4]
              p(c_7) = [2] x2 + [1] x3 + [2]
        
        Following rules are strictly oriented:
        +#(x,s(y)) = [4] y + [8] 
                   > [4] y + [4] 
                   = c_2(+#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        double#(s(x)) =  [0]                    
                      >= [0]                    
                      =  c_4(double#(x))        
        
           sqr#(s(x)) =  [12] x + [24]          
                      >= [12] x + [8]           
                      =  +#(sqr(x),double(x))   
        
           sqr#(s(x)) =  [12] x + [24]          
                      >= [12] x + [16]          
                      =  +#(sqr(x),s(double(x)))
        
           sqr#(s(x)) =  [12] x + [24]          
                      >= [0]                    
                      =  double#(x)             
        
           sqr#(s(x)) =  [12] x + [24]          
                      >= [12] x + [0]           
                      =  sqr#(x)                
        
          double(0()) =  [23]                   
                      >= [7]                    
                      =  0()                    
        
         double(s(x)) =  [3] x + [8]            
                      >= [3] x + [6]            
                      =  s(s(double(x)))        
        
** Step 4.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
            double#(s(x)) -> c_4(double#(x))
            sqr#(s(x)) -> +#(sqr(x),double(x))
            sqr#(s(x)) -> +#(sqr(x),s(double(x)))
            sqr#(s(x)) -> double#(x)
            sqr#(s(x)) -> sqr#(x)
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            sqr(0()) -> 0()
            sqr(s(x)) -> +(sqr(x),s(double(x)))
            sqr(s(x)) -> s(+(sqr(x),double(x)))
        - Signature:
            {+/2,double/1,sqr/1,+#/2,double#/1,sqr#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/3,c_7/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,sqr#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))