WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            div(0(),s(Y)) -> 0()
            div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
            minus(X,0()) -> X
            minus(s(X),s(Y)) -> p(minus(X,Y))
            p(s(X)) -> X
        - Signature:
            {div/2,minus/2,p/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          div#(0(),s(Y)) -> c_1()
          div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
          minus#(X,0()) -> c_3()
          minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y)),minus#(X,Y))
          p#(s(X)) -> c_5()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            div#(0(),s(Y)) -> c_1()
            div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
            minus#(X,0()) -> c_3()
            minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y)),minus#(X,Y))
            p#(s(X)) -> c_5()
        - Weak TRS:
            div(0(),s(Y)) -> 0()
            div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
            minus(X,0()) -> X
            minus(s(X),s(Y)) -> p(minus(X,Y))
            p(s(X)) -> X
        - Signature:
            {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,minus#,p#} 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}.
        Here rules are labelled as follows:
          1: div#(0(),s(Y)) -> c_1()
          2: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
          3: minus#(X,0()) -> c_3()
          4: minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y)),minus#(X,Y))
          5: p#(s(X)) -> c_5()
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
            minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y)),minus#(X,Y))
        - Weak DPs:
            div#(0(),s(Y)) -> c_1()
            minus#(X,0()) -> c_3()
            p#(s(X)) -> c_5()
        - Weak TRS:
            div(0(),s(Y)) -> 0()
            div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
            minus(X,0()) -> X
            minus(s(X),s(Y)) -> p(minus(X,Y))
            p(s(X)) -> X
        - Signature:
            {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,minus#,p#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
             -->_2 minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y)),minus#(X,Y)):2
             -->_2 minus#(X,0()) -> c_3():4
             -->_1 div#(0(),s(Y)) -> c_1():3
             -->_1 div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y)):1
          
          2:S:minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y)),minus#(X,Y))
             -->_1 p#(s(X)) -> c_5():5
             -->_2 minus#(X,0()) -> c_3():4
             -->_2 minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y)),minus#(X,Y)):2
          
          3:W:div#(0(),s(Y)) -> c_1()
             
          
          4:W:minus#(X,0()) -> c_3()
             
          
          5:W:p#(s(X)) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: div#(0(),s(Y)) -> c_1()
          4: minus#(X,0()) -> c_3()
          5: p#(s(X)) -> c_5()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
            minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y)),minus#(X,Y))
        - Weak TRS:
            div(0(),s(Y)) -> 0()
            div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
            minus(X,0()) -> X
            minus(s(X),s(Y)) -> p(minus(X,Y))
            p(s(X)) -> X
        - Signature:
            {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,minus#,p#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
             -->_2 minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y)),minus#(X,Y)):2
             -->_1 div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y)):1
          
          2:S:minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y)),minus#(X,Y))
             -->_2 minus#(s(X),s(Y)) -> c_4(p#(minus(X,Y)),minus#(X,Y)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          minus#(s(X),s(Y)) -> c_4(minus#(X,Y))
* Step 5: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
            minus#(s(X),s(Y)) -> c_4(minus#(X,Y))
        - Weak TRS:
            div(0(),s(Y)) -> 0()
            div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
            minus(X,0()) -> X
            minus(s(X),s(Y)) -> p(minus(X,Y))
            p(s(X)) -> X
        - Signature:
            {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,minus#,p#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          minus(X,0()) -> X
          minus(s(X),s(Y)) -> p(minus(X,Y))
          p(s(X)) -> X
          div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
          minus#(s(X),s(Y)) -> c_4(minus#(X,Y))
* Step 6: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
            minus#(s(X),s(Y)) -> c_4(minus#(X,Y))
        - Weak TRS:
            minus(X,0()) -> X
            minus(s(X),s(Y)) -> p(minus(X,Y))
            p(s(X)) -> X
        - Signature:
            {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,minus#,p#} 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
          div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
        and a lower component
          minus#(s(X),s(Y)) -> c_4(minus#(X,Y))
        Further, following extension rules are added to the lower component.
          div#(s(X),s(Y)) -> div#(minus(X,Y),s(Y))
          div#(s(X),s(Y)) -> minus#(X,Y)
** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak TRS:
            minus(X,0()) -> X
            minus(s(X),s(Y)) -> p(minus(X,Y))
            p(s(X)) -> X
        - Signature:
            {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,minus#,p#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
             -->_1 div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)))
** Step 6.a:2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)))
        - Weak TRS:
            minus(X,0()) -> X
            minus(s(X),s(Y)) -> p(minus(X,Y))
            p(s(X)) -> X
        - Signature:
            {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,minus#,p#} 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(p) = {1},
            uargs(div#) = {1},
            uargs(c_2) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [2]                  
               p(div) = [1] x1 + [0]         
             p(minus) = [1] x1 + [4]         
                 p(p) = [1] x1 + [0]         
                 p(s) = [1] x1 + [7]         
              p(div#) = [1] x1 + [1]         
            p(minus#) = [1] x1 + [1] x2 + [1]
                p(p#) = [1] x1 + [1]         
               p(c_1) = [8]                  
               p(c_2) = [1] x1 + [2]         
               p(c_3) = [1]                  
               p(c_4) = [2] x1 + [1]         
               p(c_5) = [8]                  
          
          Following rules are strictly oriented:
          div#(s(X),s(Y)) = [1] X + [8]               
                          > [1] X + [7]               
                          = c_2(div#(minus(X,Y),s(Y)))
          
          
          Following rules are (at-least) weakly oriented:
              minus(X,0()) =  [1] X + [4]  
                           >= [1] X + [0]  
                           =  X            
          
          minus(s(X),s(Y)) =  [1] X + [11] 
                           >= [1] X + [4]  
                           =  p(minus(X,Y))
          
                   p(s(X)) =  [1] X + [7]  
                           >= [1] X + [0]  
                           =  X            
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 6.a:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)))
        - Weak TRS:
            minus(X,0()) -> X
            minus(s(X),s(Y)) -> p(minus(X,Y))
            p(s(X)) -> X
        - Signature:
            {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,minus#,p#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 6.b:1: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            minus#(s(X),s(Y)) -> c_4(minus#(X,Y))
        - Weak DPs:
            div#(s(X),s(Y)) -> div#(minus(X,Y),s(Y))
            div#(s(X),s(Y)) -> minus#(X,Y)
        - Weak TRS:
            minus(X,0()) -> X
            minus(s(X),s(Y)) -> p(minus(X,Y))
            p(s(X)) -> X
        - Signature:
            {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,minus#,p#} 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(p) = {1},
            uargs(div#) = {1},
            uargs(c_4) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [0]                  
               p(div) = [1] x1 + [1] x2 + [0]
             p(minus) = [1] x1 + [0]         
                 p(p) = [1] x1 + [1]         
                 p(s) = [1] x1 + [1]         
              p(div#) = [1] x1 + [1] x2 + [0]
            p(minus#) = [1] x2 + [0]         
                p(p#) = [1] x1 + [2]         
               p(c_1) = [1]                  
               p(c_2) = [1] x1 + [1] x2 + [0]
               p(c_3) = [1]                  
               p(c_4) = [1] x1 + [0]         
               p(c_5) = [2]                  
          
          Following rules are strictly oriented:
          minus#(s(X),s(Y)) = [1] Y + [1]     
                            > [1] Y + [0]     
                            = c_4(minus#(X,Y))
          
          
          Following rules are (at-least) weakly oriented:
           div#(s(X),s(Y)) =  [1] X + [1] Y + [2]  
                           >= [1] X + [1] Y + [1]  
                           =  div#(minus(X,Y),s(Y))
          
           div#(s(X),s(Y)) =  [1] X + [1] Y + [2]  
                           >= [1] Y + [0]          
                           =  minus#(X,Y)          
          
              minus(X,0()) =  [1] X + [0]          
                           >= [1] X + [0]          
                           =  X                    
          
          minus(s(X),s(Y)) =  [1] X + [1]          
                           >= [1] X + [1]          
                           =  p(minus(X,Y))        
          
                   p(s(X)) =  [1] X + [2]          
                           >= [1] X + [0]          
                           =  X                    
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            div#(s(X),s(Y)) -> div#(minus(X,Y),s(Y))
            div#(s(X),s(Y)) -> minus#(X,Y)
            minus#(s(X),s(Y)) -> c_4(minus#(X,Y))
        - Weak TRS:
            minus(X,0()) -> X
            minus(s(X),s(Y)) -> p(minus(X,Y))
            p(s(X)) -> X
        - Signature:
            {div/2,minus/2,p/1,div#/2,minus#/2,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,minus#,p#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))