WORST_CASE(?,O(n^3))
* Step 1: DependencyPairs WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
            quot(0(),s(Y)) -> 0()
            quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} and constructors {0,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
          ifMinus#(true(),s(X),Y) -> c_2()
          le#(0(),Y) -> c_3()
          le#(s(X),0()) -> c_4()
          le#(s(X),s(Y)) -> c_5(le#(X,Y))
          minus#(0(),Y) -> c_6()
          minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
          quot#(0(),s(Y)) -> c_8()
          quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
            ifMinus#(true(),s(X),Y) -> c_2()
            le#(0(),Y) -> c_3()
            le#(s(X),0()) -> c_4()
            le#(s(X),s(Y)) -> c_5(le#(X,Y))
            minus#(0(),Y) -> c_6()
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
            quot#(0(),s(Y)) -> c_8()
            quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
            quot(0(),s(Y)) -> 0()
            quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          ifMinus(false(),s(X),Y) -> s(minus(X,Y))
          ifMinus(true(),s(X),Y) -> 0()
          le(0(),Y) -> true()
          le(s(X),0()) -> false()
          le(s(X),s(Y)) -> le(X,Y)
          minus(0(),Y) -> 0()
          minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
          ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
          ifMinus#(true(),s(X),Y) -> c_2()
          le#(0(),Y) -> c_3()
          le#(s(X),0()) -> c_4()
          le#(s(X),s(Y)) -> c_5(le#(X,Y))
          minus#(0(),Y) -> c_6()
          minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
          quot#(0(),s(Y)) -> c_8()
          quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
            ifMinus#(true(),s(X),Y) -> c_2()
            le#(0(),Y) -> c_3()
            le#(s(X),0()) -> c_4()
            le#(s(X),s(Y)) -> c_5(le#(X,Y))
            minus#(0(),Y) -> c_6()
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
            quot#(0(),s(Y)) -> c_8()
            quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,4,6,8}
        by application of
          Pre({2,3,4,6,8}) = {1,5,7,9}.
        Here rules are labelled as follows:
          1: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
          2: ifMinus#(true(),s(X),Y) -> c_2()
          3: le#(0(),Y) -> c_3()
          4: le#(s(X),0()) -> c_4()
          5: le#(s(X),s(Y)) -> c_5(le#(X,Y))
          6: minus#(0(),Y) -> c_6()
          7: minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
          8: quot#(0(),s(Y)) -> c_8()
          9: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
            le#(s(X),s(Y)) -> c_5(le#(X,Y))
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
            quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak DPs:
            ifMinus#(true(),s(X),Y) -> c_2()
            le#(0(),Y) -> c_3()
            le#(s(X),0()) -> c_4()
            minus#(0(),Y) -> c_6()
            quot#(0(),s(Y)) -> c_8()
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
             -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):3
             -->_1 minus#(0(),Y) -> c_6():8
          
          2:S:le#(s(X),s(Y)) -> c_5(le#(X,Y))
             -->_1 le#(s(X),0()) -> c_4():7
             -->_1 le#(0(),Y) -> c_3():6
             -->_1 le#(s(X),s(Y)) -> c_5(le#(X,Y)):2
          
          3:S:minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
             -->_2 le#(s(X),0()) -> c_4():7
             -->_1 ifMinus#(true(),s(X),Y) -> c_2():5
             -->_2 le#(s(X),s(Y)) -> c_5(le#(X,Y)):2
             -->_1 ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)):1
          
          4:S:quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
             -->_1 quot#(0(),s(Y)) -> c_8():9
             -->_2 minus#(0(),Y) -> c_6():8
             -->_1 quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)):4
             -->_2 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):3
          
          5:W:ifMinus#(true(),s(X),Y) -> c_2()
             
          
          6:W:le#(0(),Y) -> c_3()
             
          
          7:W:le#(s(X),0()) -> c_4()
             
          
          8:W:minus#(0(),Y) -> c_6()
             
          
          9:W:quot#(0(),s(Y)) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          9: quot#(0(),s(Y)) -> c_8()
          8: minus#(0(),Y) -> c_6()
          6: le#(0(),Y) -> c_3()
          5: ifMinus#(true(),s(X),Y) -> c_2()
          7: le#(s(X),0()) -> c_4()
* Step 5: Decompose WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
            le#(s(X),s(Y)) -> c_5(le#(X,Y))
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
            quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
              le#(s(X),s(Y)) -> c_5(le#(X,Y))
              minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
          - Weak DPs:
              quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
          - Weak TRS:
              ifMinus(false(),s(X),Y) -> s(minus(X,Y))
              ifMinus(true(),s(X),Y) -> 0()
              le(0(),Y) -> true()
              le(s(X),0()) -> false()
              le(s(X),s(Y)) -> le(X,Y)
              minus(0(),Y) -> 0()
              minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
          - Signature:
              {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
              ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
              ,true}
        
        Problem (S)
          - Strict DPs:
              quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
          - Weak DPs:
              ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
              le#(s(X),s(Y)) -> c_5(le#(X,Y))
              minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
          - Weak TRS:
              ifMinus(false(),s(X),Y) -> s(minus(X,Y))
              ifMinus(true(),s(X),Y) -> 0()
              le(0(),Y) -> true()
              le(s(X),0()) -> false()
              le(s(X),s(Y)) -> le(X,Y)
              minus(0(),Y) -> 0()
              minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
          - Signature:
              {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
              ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
              ,true}
** Step 5.a:1: DecomposeDG WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
            le#(s(X),s(Y)) -> c_5(le#(X,Y))
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
        - Weak DPs:
            quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
        and a lower component
          ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
          le#(s(X),s(Y)) -> c_5(le#(X,Y))
          minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
        Further, following extension rules are added to the lower component.
          quot#(s(X),s(Y)) -> minus#(X,Y)
          quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
*** Step 5.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 5.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, 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:
        The following argument positions are considered usable:
          uargs(c_9) = {1}
        
        Following symbols are considered usable:
          {ifMinus,minus,ifMinus#,le#,minus#,quot#}
        TcT has computed the following interpretation:
                 p(0) = [0]                  
             p(false) = [0]                  
           p(ifMinus) = [1] x2 + [3]         
                p(le) = [0]                  
             p(minus) = [1] x1 + [3]         
              p(quot) = [1] x1 + [1] x2 + [1]
                 p(s) = [1] x1 + [13]        
              p(true) = [0]                  
          p(ifMinus#) = [1] x1 + [2] x2 + [1]
               p(le#) = [1] x1 + [2] x2 + [1]
            p(minus#) = [1]                  
             p(quot#) = [1] x1 + [1]         
               p(c_1) = [1]                  
               p(c_2) = [0]                  
               p(c_3) = [4]                  
               p(c_4) = [2]                  
               p(c_5) = [2] x1 + [8]         
               p(c_6) = [4]                  
               p(c_7) = [4] x1 + [8] x2 + [0]
               p(c_8) = [2]                  
               p(c_9) = [1] x1 + [7] x2 + [2]
        
        Following rules are strictly oriented:
        quot#(s(X),s(Y)) = [1] X + [14]                           
                         > [1] X + [13]                           
                         = c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
        
        
        Following rules are (at-least) weakly oriented:
        ifMinus(false(),s(X),Y) =  [1] X + [16]              
                                >= [1] X + [16]              
                                =  s(minus(X,Y))             
        
         ifMinus(true(),s(X),Y) =  [1] X + [16]              
                                >= [0]                       
                                =  0()                       
        
                   minus(0(),Y) =  [3]                       
                                >= [0]                       
                                =  0()                       
        
                  minus(s(X),Y) =  [1] X + [16]              
                                >= [1] X + [16]              
                                =  ifMinus(le(s(X),Y),s(X),Y)
        
**** Step 5.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 5.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
             -->_1 quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
**** Step 5.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 5.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
            le#(s(X),s(Y)) -> c_5(le#(X,Y))
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
        - Weak DPs:
            quot#(s(X),s(Y)) -> minus#(X,Y)
            quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + 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:
          2: le#(s(X),s(Y)) -> c_5(le#(X,Y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 5.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
            le#(s(X),s(Y)) -> c_5(le#(X,Y))
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
        - Weak DPs:
            quot#(s(X),s(Y)) -> minus#(X,Y)
            quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + 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_1) = {1},
          uargs(c_5) = {1},
          uargs(c_7) = {1,2}
        
        Following symbols are considered usable:
          {ifMinus,le,minus,ifMinus#,le#,minus#,quot#}
        TcT has computed the following interpretation:
                 p(0) = [0]                          
                        [0]                          
                        [0]                          
             p(false) = [0]                          
                        [0]                          
                        [1]                          
           p(ifMinus) = [0 0 0]      [0 1 0]      [0]
                        [0 0 1] x1 + [1 0 0] x2 + [0]
                        [0 0 0]      [0 0 1]      [0]
                p(le) = [0 0 0]      [0]             
                        [0 1 1] x1 + [0]             
                        [0 0 0]      [1]             
             p(minus) = [0 1 0]      [0]             
                        [0 1 0] x1 + [0]             
                        [0 0 1]      [0]             
              p(quot) = [0]                          
                        [0]                          
                        [0]                          
                 p(s) = [0 1 1]      [0]             
                        [0 1 1] x1 + [1]             
                        [0 0 1]      [1]             
              p(true) = [0]                          
                        [0]                          
                        [0]                          
          p(ifMinus#) = [1 0 0]      [0 0 0]      [1]
                        [1 0 1] x2 + [0 1 1] x3 + [0]
                        [1 0 0]      [0 1 1]      [1]
               p(le#) = [0 0 1]      [0 0 0]      [1]
                        [1 0 0] x1 + [0 0 1] x2 + [1]
                        [0 1 1]      [0 1 0]      [0]
            p(minus#) = [0 1 1]      [0 0 0]      [1]
                        [0 1 0] x1 + [0 1 1] x2 + [1]
                        [0 0 0]      [0 0 0]      [0]
             p(quot#) = [0 1 0]      [0 0 0]      [0]
                        [1 0 1] x1 + [1 0 1] x2 + [0]
                        [0 1 1]      [0 0 0]      [0]
               p(c_1) = [1 0 0]      [0]             
                        [0 1 0] x1 + [0]             
                        [0 0 0]      [0]             
               p(c_2) = [0]                          
                        [0]                          
                        [0]                          
               p(c_3) = [0]                          
                        [0]                          
                        [0]                          
               p(c_4) = [0]                          
                        [0]                          
                        [0]                          
               p(c_5) = [1 0 0]      [0]             
                        [0 0 0] x1 + [1]             
                        [0 0 1]      [0]             
               p(c_6) = [0]                          
                        [0]                          
                        [0]                          
               p(c_7) = [1 0 0]      [1 0 0]      [0]
                        [0 0 1] x1 + [0 0 0] x2 + [1]
                        [0 0 0]      [0 0 0]      [0]
               p(c_8) = [0]                          
                        [0]                          
                        [0]                          
               p(c_9) = [0]                          
                        [0]                          
                        [0]                          
        
        Following rules are strictly oriented:
        le#(s(X),s(Y)) = [0 0 1]     [0 0 0]     [2]
                         [0 1 1] X + [0 0 1] Y + [2]
                         [0 1 2]     [0 1 1]     [3]
                       > [0 0 1]     [0 0 0]     [1]
                         [0 0 0] X + [0 0 0] Y + [1]
                         [0 1 1]     [0 1 0]     [0]
                       = c_5(le#(X,Y))              
        
        
        Following rules are (at-least) weakly oriented:
        ifMinus#(false(),s(X),Y) =  [0 1 1]     [0 0 0]     [1]                 
                                    [0 1 2] X + [0 1 1] Y + [1]                 
                                    [0 1 1]     [0 1 1]     [1]                 
                                 >= [0 1 1]     [0 0 0]     [1]                 
                                    [0 1 0] X + [0 1 1] Y + [1]                 
                                    [0 0 0]     [0 0 0]     [0]                 
                                 =  c_1(minus#(X,Y))                            
        
                  minus#(s(X),Y) =  [0 1 2]     [0 0 0]     [3]                 
                                    [0 1 1] X + [0 1 1] Y + [2]                 
                                    [0 0 0]     [0 0 0]     [0]                 
                                 >= [0 1 2]     [0 0 0]     [3]                 
                                    [0 1 1] X + [0 1 1] Y + [2]                 
                                    [0 0 0]     [0 0 0]     [0]                 
                                 =  c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
        
                quot#(s(X),s(Y)) =  [0 1 1]     [0 0 0]     [1]                 
                                    [0 1 2] X + [0 1 2] Y + [2]                 
                                    [0 1 2]     [0 0 0]     [2]                 
                                 >= [0 1 1]     [0 0 0]     [1]                 
                                    [0 1 0] X + [0 1 1] Y + [1]                 
                                    [0 0 0]     [0 0 0]     [0]                 
                                 =  minus#(X,Y)                                 
        
                quot#(s(X),s(Y)) =  [0 1 1]     [0 0 0]     [1]                 
                                    [0 1 2] X + [0 1 2] Y + [2]                 
                                    [0 1 2]     [0 0 0]     [2]                 
                                 >= [0 1 0]     [0 0 0]     [0]                 
                                    [0 1 1] X + [0 1 2] Y + [1]                 
                                    [0 1 1]     [0 0 0]     [0]                 
                                 =  quot#(minus(X,Y),s(Y))                      
        
         ifMinus(false(),s(X),Y) =  [0 1 1]     [1]                             
                                    [0 1 1] X + [1]                             
                                    [0 0 1]     [1]                             
                                 >= [0 1 1]     [0]                             
                                    [0 1 1] X + [1]                             
                                    [0 0 1]     [1]                             
                                 =  s(minus(X,Y))                               
        
          ifMinus(true(),s(X),Y) =  [0 1 1]     [1]                             
                                    [0 1 1] X + [0]                             
                                    [0 0 1]     [1]                             
                                 >= [0]                                         
                                    [0]                                         
                                    [0]                                         
                                 =  0()                                         
        
                       le(0(),Y) =  [0]                                         
                                    [0]                                         
                                    [1]                                         
                                 >= [0]                                         
                                    [0]                                         
                                    [0]                                         
                                 =  true()                                      
        
                    le(s(X),0()) =  [0 0 0]     [0]                             
                                    [0 1 2] X + [2]                             
                                    [0 0 0]     [1]                             
                                 >= [0]                                         
                                    [0]                                         
                                    [1]                                         
                                 =  false()                                     
        
                   le(s(X),s(Y)) =  [0 0 0]     [0]                             
                                    [0 1 2] X + [2]                             
                                    [0 0 0]     [1]                             
                                 >= [0 0 0]     [0]                             
                                    [0 1 1] X + [0]                             
                                    [0 0 0]     [1]                             
                                 =  le(X,Y)                                     
        
                    minus(0(),Y) =  [0]                                         
                                    [0]                                         
                                    [0]                                         
                                 >= [0]                                         
                                    [0]                                         
                                    [0]                                         
                                 =  0()                                         
        
                   minus(s(X),Y) =  [0 1 1]     [1]                             
                                    [0 1 1] X + [1]                             
                                    [0 0 1]     [1]                             
                                 >= [0 1 1]     [1]                             
                                    [0 1 1] X + [1]                             
                                    [0 0 1]     [1]                             
                                 =  ifMinus(le(s(X),Y),s(X),Y)                  
        
**** Step 5.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
        - Weak DPs:
            le#(s(X),s(Y)) -> c_5(le#(X,Y))
            quot#(s(X),s(Y)) -> minus#(X,Y)
            quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 5.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
        - Weak DPs:
            le#(s(X),s(Y)) -> c_5(le#(X,Y))
            quot#(s(X),s(Y)) -> minus#(X,Y)
            quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
             -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):2
          
          2:S:minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
             -->_2 le#(s(X),s(Y)) -> c_5(le#(X,Y)):3
             -->_1 ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)):1
          
          3:W:le#(s(X),s(Y)) -> c_5(le#(X,Y))
             -->_1 le#(s(X),s(Y)) -> c_5(le#(X,Y)):3
          
          4:W:quot#(s(X),s(Y)) -> minus#(X,Y)
             -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):2
          
          5:W:quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
             -->_1 quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)):5
             -->_1 quot#(s(X),s(Y)) -> minus#(X,Y):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: le#(s(X),s(Y)) -> c_5(le#(X,Y))
**** Step 5.a:1.b:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
        - Weak DPs:
            quot#(s(X),s(Y)) -> minus#(X,Y)
            quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
             -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):2
          
          2:S:minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
             -->_1 ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)):1
          
          4:W:quot#(s(X),s(Y)) -> minus#(X,Y)
             -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):2
          
          5:W:quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
             -->_1 quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)):5
             -->_1 quot#(s(X),s(Y)) -> minus#(X,Y):4
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y))
**** Step 5.a:1.b:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y))
        - Weak DPs:
            quot#(s(X),s(Y)) -> minus#(X,Y)
            quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
          2: minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y))
          
        The strictly oriented rules are moved into the weak component.
***** Step 5.a:1.b:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y))
        - Weak DPs:
            quot#(s(X),s(Y)) -> minus#(X,Y)
            quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, 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:
        The following argument positions are considered usable:
          uargs(c_1) = {1},
          uargs(c_7) = {1}
        
        Following symbols are considered usable:
          {ifMinus,minus,ifMinus#,le#,minus#,quot#}
        TcT has computed the following interpretation:
                 p(0) = [11]         
             p(false) = [0]          
           p(ifMinus) = [1] x2 + [5] 
                p(le) = [0]          
             p(minus) = [1] x1 + [5] 
              p(quot) = [8] x1 + [0] 
                 p(s) = [1] x1 + [8] 
              p(true) = [2]          
          p(ifMinus#) = [2] x2 + [0] 
               p(le#) = [0]          
            p(minus#) = [2] x1 + [11]
             p(quot#) = [3] x1 + [2] 
               p(c_1) = [1] x1 + [1] 
               p(c_2) = [1]          
               p(c_3) = [8]          
               p(c_4) = [8]          
               p(c_5) = [2] x1 + [0] 
               p(c_6) = [1]          
               p(c_7) = [1] x1 + [0] 
               p(c_8) = [2]          
               p(c_9) = [1] x1 + [8] 
        
        Following rules are strictly oriented:
        ifMinus#(false(),s(X),Y) = [2] X + [16]                    
                                 > [2] X + [12]                    
                                 = c_1(minus#(X,Y))                
        
                  minus#(s(X),Y) = [2] X + [27]                    
                                 > [2] X + [16]                    
                                 = c_7(ifMinus#(le(s(X),Y),s(X),Y))
        
        
        Following rules are (at-least) weakly oriented:
               quot#(s(X),s(Y)) =  [3] X + [26]              
                                >= [2] X + [11]              
                                =  minus#(X,Y)               
        
               quot#(s(X),s(Y)) =  [3] X + [26]              
                                >= [3] X + [17]              
                                =  quot#(minus(X,Y),s(Y))    
        
        ifMinus(false(),s(X),Y) =  [1] X + [13]              
                                >= [1] X + [13]              
                                =  s(minus(X,Y))             
        
         ifMinus(true(),s(X),Y) =  [1] X + [13]              
                                >= [11]                      
                                =  0()                       
        
                   minus(0(),Y) =  [16]                      
                                >= [11]                      
                                =  0()                       
        
                  minus(s(X),Y) =  [1] X + [13]              
                                >= [1] X + [13]              
                                =  ifMinus(le(s(X),Y),s(X),Y)
        
***** Step 5.a:1.b:1.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y))
            quot#(s(X),s(Y)) -> minus#(X,Y)
            quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 5.a:1.b:1.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y))
            quot#(s(X),s(Y)) -> minus#(X,Y)
            quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
             -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y)):2
          
          2:W:minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y))
             -->_1 ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)):1
          
          3:W:quot#(s(X),s(Y)) -> minus#(X,Y)
             -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y)):2
          
          4:W:quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
             -->_1 quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)):4
             -->_1 quot#(s(X),s(Y)) -> minus#(X,Y):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
          3: quot#(s(X),s(Y)) -> minus#(X,Y)
          1: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
          2: minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y))
***** Step 5.a:1.b:1.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
            le#(s(X),s(Y)) -> c_5(le#(X,Y))
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
             -->_2 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):4
             -->_1 quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)):1
          
          2:W:ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
             -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):4
          
          3:W:le#(s(X),s(Y)) -> c_5(le#(X,Y))
             -->_1 le#(s(X),s(Y)) -> c_5(le#(X,Y)):3
          
          4:W:minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
             -->_2 le#(s(X),s(Y)) -> c_5(le#(X,Y)):3
             -->_1 ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y))
          2: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y))
          3: le#(s(X),s(Y)) -> c_5(le#(X,Y))
** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y))
             -->_1 quot#(s(X),s(Y)) -> c_9(quot#(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:
          quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)))
** Step 5.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)))
          
        The strictly oriented rules are moved into the weak component.
*** Step 5.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, 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:
        The following argument positions are considered usable:
          uargs(c_9) = {1}
        
        Following symbols are considered usable:
          {ifMinus,minus,ifMinus#,le#,minus#,quot#}
        TcT has computed the following interpretation:
                 p(0) = [0]                  
             p(false) = [0]                  
           p(ifMinus) = [1] x2 + [11]        
                p(le) = [0]                  
             p(minus) = [1] x1 + [11]        
              p(quot) = [1] x1 + [0]         
                 p(s) = [1] x1 + [13]        
              p(true) = [0]                  
          p(ifMinus#) = [8] x2 + [4] x3 + [2]
               p(le#) = [2] x2 + [2]         
            p(minus#) = [1] x2 + [1]         
             p(quot#) = [2] x1 + [0]         
               p(c_1) = [1] x1 + [0]         
               p(c_2) = [1]                  
               p(c_3) = [1]                  
               p(c_4) = [1]                  
               p(c_5) = [2] x1 + [0]         
               p(c_6) = [2]                  
               p(c_7) = [1] x1 + [2] x2 + [8]
               p(c_8) = [2]                  
               p(c_9) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        quot#(s(X),s(Y)) = [2] X + [26]               
                         > [2] X + [22]               
                         = c_9(quot#(minus(X,Y),s(Y)))
        
        
        Following rules are (at-least) weakly oriented:
        ifMinus(false(),s(X),Y) =  [1] X + [24]              
                                >= [1] X + [24]              
                                =  s(minus(X,Y))             
        
         ifMinus(true(),s(X),Y) =  [1] X + [24]              
                                >= [0]                       
                                =  0()                       
        
                   minus(0(),Y) =  [11]                      
                                >= [0]                       
                                =  0()                       
        
                  minus(s(X),Y) =  [1] X + [24]              
                                >= [1] X + [24]              
                                =  ifMinus(le(s(X),Y),s(X),Y)
        
*** Step 5.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)))
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)))
             -->_1 quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)))
*** Step 5.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            ifMinus(false(),s(X),Y) -> s(minus(X,Y))
            ifMinus(true(),s(X),Y) -> 0()
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(0(),Y) -> 0()
            minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
        - Signature:
            {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/1,c_2/0
            ,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^3))