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: 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)
            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:
        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 3: 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)
            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:
        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 4: UsableRules 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)
            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))
          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))
* Step 5: 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))
            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 = Nothing, 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: 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.a:2: WeightGap 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:
        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(ifMinus) = {1},
            uargs(s) = {1},
            uargs(quot#) = {1},
            uargs(c_9) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]                  
               p(false) = [0]                  
             p(ifMinus) = [1] x1 + [1] x2 + [0]
                  p(le) = [0]                  
               p(minus) = [1] x1 + [0]         
                p(quot) = [1] x1 + [1] x2 + [2]
                   p(s) = [1] x1 + [4]         
                p(true) = [0]                  
            p(ifMinus#) = [1] x1 + [2]         
                 p(le#) = [1] x2 + [1]         
              p(minus#) = [8] x1 + [1] x2 + [0]
               p(quot#) = [1] x1 + [11]        
                 p(c_1) = [1] x1 + [2]         
                 p(c_2) = [2]                  
                 p(c_3) = [2]                  
                 p(c_4) = [1]                  
                 p(c_5) = [8]                  
                 p(c_6) = [1]                  
                 p(c_7) = [8] x1 + [8] x2 + [2]
                 p(c_8) = [1]                  
                 p(c_9) = [1] x1 + [1]         
          
          Following rules are strictly oriented:
          quot#(s(X),s(Y)) = [1] X + [15]               
                           > [1] X + [12]               
                           = c_9(quot#(minus(X,Y),s(Y)))
          
          
          Following rules are (at-least) weakly oriented:
          ifMinus(false(),s(X),Y) =  [1] X + [4]               
                                  >= [1] X + [4]               
                                  =  s(minus(X,Y))             
          
           ifMinus(true(),s(X),Y) =  [1] X + [4]               
                                  >= [0]                       
                                  =  0()                       
          
                        le(0(),Y) =  [0]                       
                                  >= [0]                       
                                  =  true()                    
          
                     le(s(X),0()) =  [0]                       
                                  >= [0]                       
                                  =  false()                   
          
                    le(s(X),s(Y)) =  [0]                       
                                  >= [0]                       
                                  =  le(X,Y)                   
          
                     minus(0(),Y) =  [0]                       
                                  >= [0]                       
                                  =  0()                       
          
                    minus(s(X),Y) =  [1] X + [4]               
                                  >= [1] X + [4]               
                                  =  ifMinus(le(s(X),Y),s(X),Y)
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 5.a:3: EmptyProcessor 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:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 5.b:1: DecomposeDG 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:
        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
          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))
          quot#(s(X),s(Y)) -> minus#(X,Y)
          quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
        and a lower component
          le#(s(X),s(Y)) -> c_5(le#(X,Y))
        Further, following extension rules are added to the lower component.
          ifMinus#(false(),s(X),Y) -> minus#(X,Y)
          minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y)
          minus#(s(X),Y) -> le#(s(X),Y)
          quot#(s(X),s(Y)) -> minus#(X,Y)
          quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
*** Step 5.b:1.a:1: 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
          
          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),le#(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
          
        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.b:1.a:2: WeightGap 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:
        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(ifMinus) = {1},
            uargs(s) = {1},
            uargs(ifMinus#) = {1},
            uargs(quot#) = {1},
            uargs(c_1) = {1},
            uargs(c_7) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]                  
               p(false) = [0]                  
             p(ifMinus) = [1] x1 + [0]         
                  p(le) = [0]                  
               p(minus) = [0]                  
                p(quot) = [4] x1 + [2] x2 + [0]
                   p(s) = [1] x1 + [0]         
                p(true) = [0]                  
            p(ifMinus#) = [1] x1 + [1]         
                 p(le#) = [1] x1 + [1]         
              p(minus#) = [0]                  
               p(quot#) = [1] x1 + [4] x2 + [0]
                 p(c_1) = [1] x1 + [0]         
                 p(c_2) = [0]                  
                 p(c_3) = [1]                  
                 p(c_4) = [1]                  
                 p(c_5) = [1] x1 + [0]         
                 p(c_6) = [2]                  
                 p(c_7) = [1] x1 + [3]         
                 p(c_8) = [1]                  
                 p(c_9) = [4] x1 + [2]         
          
          Following rules are strictly oriented:
          ifMinus#(false(),s(X),Y) = [1]             
                                   > [0]             
                                   = c_1(minus#(X,Y))
          
          
          Following rules are (at-least) weakly oriented:
                   minus#(s(X),Y) =  [0]                             
                                  >= [4]                             
                                  =  c_7(ifMinus#(le(s(X),Y),s(X),Y))
          
                 quot#(s(X),s(Y)) =  [1] X + [4] Y + [0]             
                                  >= [0]                             
                                  =  minus#(X,Y)                     
          
                 quot#(s(X),s(Y)) =  [1] X + [4] Y + [0]             
                                  >= [4] Y + [0]                     
                                  =  quot#(minus(X,Y),s(Y))          
          
          ifMinus(false(),s(X),Y) =  [0]                             
                                  >= [0]                             
                                  =  s(minus(X,Y))                   
          
           ifMinus(true(),s(X),Y) =  [0]                             
                                  >= [0]                             
                                  =  0()                             
          
                        le(0(),Y) =  [0]                             
                                  >= [0]                             
                                  =  true()                          
          
                     le(s(X),0()) =  [0]                             
                                  >= [0]                             
                                  =  false()                         
          
                    le(s(X),s(Y)) =  [0]                             
                                  >= [0]                             
                                  =  le(X,Y)                         
          
                     minus(0(),Y) =  [0]                             
                                  >= [0]                             
                                  =  0()                             
          
                    minus(s(X),Y) =  [0]                             
                                  >= [0]                             
                                  =  ifMinus(le(s(X),Y),s(X),Y)      
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 5.b:1.a:3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y))
        - Weak DPs:
            ifMinus#(false(),s(X),Y) -> c_1(minus#(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:
        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(ifMinus) = {1},
            uargs(s) = {1},
            uargs(ifMinus#) = {1},
            uargs(quot#) = {1},
            uargs(c_1) = {1},
            uargs(c_7) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [2]                  
               p(false) = [0]                  
             p(ifMinus) = [1] x1 + [1] x2 + [0]
                  p(le) = [0]                  
               p(minus) = [1] x1 + [0]         
                p(quot) = [1] x2 + [2]         
                   p(s) = [1] x1 + [2]         
                p(true) = [0]                  
            p(ifMinus#) = [1] x1 + [1] x2 + [4]
                 p(le#) = [1] x1 + [1] x2 + [1]
              p(minus#) = [1] x1 + [5]         
               p(quot#) = [1] x1 + [3]         
                 p(c_1) = [1] x1 + [1]         
                 p(c_2) = [0]                  
                 p(c_3) = [1]                  
                 p(c_4) = [0]                  
                 p(c_5) = [2] x1 + [4]         
                 p(c_6) = [0]                  
                 p(c_7) = [1] x1 + [0]         
                 p(c_8) = [4]                  
                 p(c_9) = [4] x2 + [4]         
          
          Following rules are strictly oriented:
          minus#(s(X),Y) = [1] X + [7]                     
                         > [1] X + [6]                     
                         = c_7(ifMinus#(le(s(X),Y),s(X),Y))
          
          
          Following rules are (at-least) weakly oriented:
          ifMinus#(false(),s(X),Y) =  [1] X + [6]               
                                   >= [1] X + [6]               
                                   =  c_1(minus#(X,Y))          
          
                  quot#(s(X),s(Y)) =  [1] X + [5]               
                                   >= [1] X + [5]               
                                   =  minus#(X,Y)               
          
                  quot#(s(X),s(Y)) =  [1] X + [5]               
                                   >= [1] X + [3]               
                                   =  quot#(minus(X,Y),s(Y))    
          
           ifMinus(false(),s(X),Y) =  [1] X + [2]               
                                   >= [1] X + [2]               
                                   =  s(minus(X,Y))             
          
            ifMinus(true(),s(X),Y) =  [1] X + [2]               
                                   >= [2]                       
                                   =  0()                       
          
                         le(0(),Y) =  [0]                       
                                   >= [0]                       
                                   =  true()                    
          
                      le(s(X),0()) =  [0]                       
                                   >= [0]                       
                                   =  false()                   
          
                     le(s(X),s(Y)) =  [0]                       
                                   >= [0]                       
                                   =  le(X,Y)                   
          
                      minus(0(),Y) =  [2]                       
                                   >= [2]                       
                                   =  0()                       
          
                     minus(s(X),Y) =  [1] X + [2]               
                                   >= [1] X + [2]               
                                   =  ifMinus(le(s(X),Y),s(X),Y)
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 5.b:1.a:4: EmptyProcessor 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:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 5.b:1.b:1: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            le#(s(X),s(Y)) -> c_5(le#(X,Y))
        - Weak DPs:
            ifMinus#(false(),s(X),Y) -> minus#(X,Y)
            minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y)
            minus#(s(X),Y) -> le#(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_7/2,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:
        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(ifMinus) = {1},
            uargs(s) = {1},
            uargs(ifMinus#) = {1},
            uargs(quot#) = {1},
            uargs(c_5) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                   p(0) = [0]                  
               p(false) = [0]                  
             p(ifMinus) = [1] x1 + [1] x2 + [0]
                  p(le) = [0]                  
               p(minus) = [1] x1 + [0]         
                p(quot) = [1] x1 + [1] x2 + [2]
                   p(s) = [1] x1 + [2]         
                p(true) = [0]                  
            p(ifMinus#) = [1] x1 + [1] x2 + [1]
                 p(le#) = [1] x1 + [2]         
              p(minus#) = [1] x1 + [2]         
               p(quot#) = [1] x1 + [0]         
                 p(c_1) = [4]                  
                 p(c_2) = [4]                  
                 p(c_3) = [1]                  
                 p(c_4) = [0]                  
                 p(c_5) = [1] x1 + [0]         
                 p(c_6) = [0]                  
                 p(c_7) = [4] x1 + [4]         
                 p(c_8) = [0]                  
                 p(c_9) = [2] x2 + [1]         
          
          Following rules are strictly oriented:
          le#(s(X),s(Y)) = [1] X + [4]  
                         > [1] X + [2]  
                         = c_5(le#(X,Y))
          
          
          Following rules are (at-least) weakly oriented:
          ifMinus#(false(),s(X),Y) =  [1] X + [3]                
                                   >= [1] X + [2]                
                                   =  minus#(X,Y)                
          
                    minus#(s(X),Y) =  [1] X + [4]                
                                   >= [1] X + [3]                
                                   =  ifMinus#(le(s(X),Y),s(X),Y)
          
                    minus#(s(X),Y) =  [1] X + [4]                
                                   >= [1] X + [4]                
                                   =  le#(s(X),Y)                
          
                  quot#(s(X),s(Y)) =  [1] X + [2]                
                                   >= [1] X + [2]                
                                   =  minus#(X,Y)                
          
                  quot#(s(X),s(Y)) =  [1] X + [2]                
                                   >= [1] X + [0]                
                                   =  quot#(minus(X,Y),s(Y))     
          
           ifMinus(false(),s(X),Y) =  [1] X + [2]                
                                   >= [1] X + [2]                
                                   =  s(minus(X,Y))              
          
            ifMinus(true(),s(X),Y) =  [1] X + [2]                
                                   >= [0]                        
                                   =  0()                        
          
                         le(0(),Y) =  [0]                        
                                   >= [0]                        
                                   =  true()                     
          
                      le(s(X),0()) =  [0]                        
                                   >= [0]                        
                                   =  false()                    
          
                     le(s(X),s(Y)) =  [0]                        
                                   >= [0]                        
                                   =  le(X,Y)                    
          
                      minus(0(),Y) =  [0]                        
                                   >= [0]                        
                                   =  0()                        
          
                     minus(s(X),Y) =  [1] X + [2]                
                                   >= [1] X + [2]                
                                   =  ifMinus(le(s(X),Y),s(X),Y) 
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 5.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            ifMinus#(false(),s(X),Y) -> minus#(X,Y)
            le#(s(X),s(Y)) -> c_5(le#(X,Y))
            minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y)
            minus#(s(X),Y) -> le#(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_7/2,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).

WORST_CASE(?,O(n^3))