* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            -(x,0()) -> x
            -(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0())
            -(0(),y) -> 0()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {-/2,p/1} / {0/0,greater/2,if/3,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-,p} and constructors {0,greater,if,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          -#(x,0()) -> c_1()
          -#(x,s(y)) -> c_2(-#(x,p(s(y))))
          -#(0(),y) -> c_3()
          p#(0()) -> c_4()
          p#(s(x)) -> c_5()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(x,0()) -> c_1()
            -#(x,s(y)) -> c_2(-#(x,p(s(y))))
            -#(0(),y) -> c_3()
            p#(0()) -> c_4()
            p#(s(x)) -> c_5()
        - Strict TRS:
            -(x,0()) -> x
            -(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0())
            -(0(),y) -> 0()
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          p(s(x)) -> x
          -#(x,0()) -> c_1()
          -#(x,s(y)) -> c_2(-#(x,p(s(y))))
          -#(0(),y) -> c_3()
          p#(0()) -> c_4()
          p#(s(x)) -> c_5()
* Step 3: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(x,0()) -> c_1()
            -#(x,s(y)) -> c_2(-#(x,p(s(y))))
            -#(0(),y) -> c_3()
            p#(0()) -> c_4()
            p#(s(x)) -> c_5()
        - Strict TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
    + 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(-#) = {2},
            uargs(c_2) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                  p(-) = [0]                           
                  p(0) = [0]                           
            p(greater) = [1] x1 + [1] x2 + [0]         
                 p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
                  p(p) = [1] x1 + [0]                  
                  p(s) = [1] x1 + [11]                 
                 p(-#) = [1] x2 + [0]                  
                 p(p#) = [0]                           
                p(c_1) = [0]                           
                p(c_2) = [1] x1 + [0]                  
                p(c_3) = [0]                           
                p(c_4) = [0]                           
                p(c_5) = [0]                           
          
          Following rules are strictly oriented:
          p(s(x)) = [1] x + [11]
                  > [1] x + [0] 
                  = x           
          
          
          Following rules are (at-least) weakly oriented:
           -#(x,0()) =  [0]               
                     >= [0]               
                     =  c_1()             
          
          -#(x,s(y)) =  [1] y + [11]      
                     >= [1] y + [11]      
                     =  c_2(-#(x,p(s(y))))
          
           -#(0(),y) =  [1] y + [0]       
                     >= [0]               
                     =  c_3()             
          
             p#(0()) =  [0]               
                     >= [0]               
                     =  c_4()             
          
            p#(s(x)) =  [0]               
                     >= [0]               
                     =  c_5()             
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(x,0()) -> c_1()
            -#(x,s(y)) -> c_2(-#(x,p(s(y))))
            -#(0(),y) -> c_3()
            p#(0()) -> c_4()
            p#(s(x)) -> c_5()
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,4,5}
        by application of
          Pre({1,3,4,5}) = {2}.
        Here rules are labelled as follows:
          1: -#(x,0()) -> c_1()
          2: -#(x,s(y)) -> c_2(-#(x,p(s(y))))
          3: -#(0(),y) -> c_3()
          4: p#(0()) -> c_4()
          5: p#(s(x)) -> c_5()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(x,s(y)) -> c_2(-#(x,p(s(y))))
        - Weak DPs:
            -#(x,0()) -> c_1()
            -#(0(),y) -> c_3()
            p#(0()) -> c_4()
            p#(s(x)) -> c_5()
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:-#(x,s(y)) -> c_2(-#(x,p(s(y))))
             -->_1 -#(0(),y) -> c_3():3
             -->_1 -#(x,0()) -> c_1():2
             -->_1 -#(x,s(y)) -> c_2(-#(x,p(s(y)))):1
          
          2:W:-#(x,0()) -> c_1()
             
          
          3:W:-#(0(),y) -> c_3()
             
          
          4:W:p#(0()) -> c_4()
             
          
          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.
          5: p#(s(x)) -> c_5()
          4: p#(0()) -> c_4()
          2: -#(x,0()) -> c_1()
          3: -#(0(),y) -> c_3()
* Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(x,s(y)) -> c_2(-#(x,p(s(y))))
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: -#(x,s(y)) -> c_2(-#(x,p(s(y))))
          
        The strictly oriented rules are moved into the weak component.
** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(x,s(y)) -> c_2(-#(x,p(s(y))))
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {p,-#,p#}
        TcT has computed the following interpretation:
                p(-) = [4 0 4]      [0 0 0]      [0]             
                       [0 2 1] x1 + [0 0 0] x2 + [0]             
                       [0 0 2]      [1 4 4]      [0]             
                p(0) = [0]                                       
                       [1]                                       
                       [0]                                       
          p(greater) = [0 0 1]      [0 1 1]      [0]             
                       [0 0 2] x1 + [0 0 2] x2 + [1]             
                       [0 0 0]      [0 0 0]      [1]             
               p(if) = [1 1 0]      [0 0 0]      [0 1 1]      [1]
                       [0 0 4] x1 + [0 0 4] x2 + [0 0 2] x3 + [0]
                       [0 0 0]      [0 0 0]      [0 0 0]      [0]
                p(p) = [4 1 4]      [0]                          
                       [2 4 0] x1 + [1]                          
                       [0 1 0]      [0]                          
                p(s) = [1 2 0]      [0]                          
                       [0 0 1] x1 + [2]                          
                       [0 0 1]      [3]                          
               p(-#) = [0 1 0]      [0 0 4]      [1]             
                       [4 4 0] x1 + [1 0 0] x2 + [0]             
                       [0 0 1]      [1 0 0]      [0]             
               p(p#) = [0]                                       
                       [1]                                       
                       [0]                                       
              p(c_1) = [0]                                       
                       [0]                                       
                       [2]                                       
              p(c_2) = [1 0 0]      [1]                          
                       [0 0 0] x1 + [0]                          
                       [0 0 0]      [0]                          
              p(c_3) = [1]                                       
                       [0]                                       
                       [2]                                       
              p(c_4) = [0]                                       
                       [1]                                       
                       [1]                                       
              p(c_5) = [1]                                       
                       [0]                                       
                       [1]                                       
        
        Following rules are strictly oriented:
        -#(x,s(y)) = [0 1 0]     [0 0 4]     [13]
                     [4 4 0] x + [1 2 0] y + [0] 
                     [0 0 1]     [1 2 0]     [0] 
                   > [0 1 0]     [0 0 4]     [10]
                     [0 0 0] x + [0 0 0] y + [0] 
                     [0 0 0]     [0 0 0]     [0] 
                   = c_2(-#(x,p(s(y))))          
        
        
        Following rules are (at-least) weakly oriented:
        p(s(x)) =  [4 8 5]     [14]
                   [2 4 4] x + [9] 
                   [0 0 1]     [2] 
                >= [1 0 0]     [0] 
                   [0 1 0] x + [0] 
                   [0 0 1]     [0] 
                =  x               
        
** Step 6.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            -#(x,s(y)) -> c_2(-#(x,p(s(y))))
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            -#(x,s(y)) -> c_2(-#(x,p(s(y))))
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:-#(x,s(y)) -> c_2(-#(x,p(s(y))))
             -->_1 -#(x,s(y)) -> c_2(-#(x,p(s(y)))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: -#(x,s(y)) -> c_2(-#(x,p(s(y))))
** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            p(s(x)) -> x
        - Signature:
            {-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-#,p#} and constructors {0,greater,if,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))