* Step 1: Sum WORST_CASE(Omega(n^1),O(n^4))
    + Considered Problem:
        - Strict TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            log(s(0())) -> 0()
            log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus,le,log,minus,quot} and constructors {0,false,s
            ,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            log(s(0())) -> 0()
            log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus,le,log,minus,quot} and constructors {0,false,s
            ,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          le(x,y){x -> s(x),y -> s(y)} =
            le(s(x),s(y)) ->^+ le(x,y)
              = C[le(x,y) = le(x,y){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^4))
    + Considered Problem:
        - Strict TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            log(s(0())) -> 0()
            log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus,le,log,minus,quot} and constructors {0,false,s
            ,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          if_minus#(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))
          log#(s(0())) -> c_6()
          log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
          minus#(0(),y) -> c_8()
          minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
          quot#(0(),s(y)) -> c_10()
          quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^4))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            if_minus#(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))
            log#(s(0())) -> c_6()
            log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            minus#(0(),y) -> c_8()
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
            quot#(0(),s(y)) -> c_10()
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            log(s(0())) -> 0()
            log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          if_minus(false(),s(x),y) -> s(minus(x,y))
          if_minus(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) -> if_minus(le(s(x),y),s(x),y)
          quot(0(),s(y)) -> 0()
          quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
          if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          if_minus#(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))
          log#(s(0())) -> c_6()
          log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
          minus#(0(),y) -> c_8()
          minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
          quot#(0(),s(y)) -> c_10()
          quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^4))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            if_minus#(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))
            log#(s(0())) -> c_6()
            log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            minus#(0(),y) -> c_8()
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
            quot#(0(),s(y)) -> c_10()
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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
          {2,3,4,6,8,10}
        by application of
          Pre({2,3,4,6,8,10}) = {1,5,7,9,11}.
        Here rules are labelled as follows:
          1: if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          2: if_minus#(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: log#(s(0())) -> c_6()
          7: log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
          8: minus#(0(),y) -> c_8()
          9: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
          10: quot#(0(),s(y)) -> c_10()
          11: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^4))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            le#(s(x),s(y)) -> c_5(le#(x,y))
            log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak DPs:
            if_minus#(true(),s(x),y) -> c_2()
            le#(0(),y) -> c_3()
            le#(s(x),0()) -> c_4()
            log#(s(0())) -> c_6()
            minus#(0(),y) -> c_8()
            quot#(0(),s(y)) -> c_10()
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):4
             -->_1 minus#(0(),y) -> c_8():10
          
          2:S:le#(s(x),s(y)) -> c_5(le#(x,y))
             -->_1 le#(s(x),0()) -> c_4():8
             -->_1 le#(0(),y) -> c_3():7
             -->_1 le#(s(x),s(y)) -> c_5(le#(x,y)):2
          
          3:S:log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
             -->_2 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):5
             -->_2 quot#(0(),s(y)) -> c_10():11
             -->_1 log#(s(0())) -> c_6():9
             -->_1 log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):3
          
          4:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
             -->_2 le#(s(x),0()) -> c_4():8
             -->_1 if_minus#(true(),s(x),y) -> c_2():6
             -->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):2
             -->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1
          
          5:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(0(),s(y)) -> c_10():11
             -->_2 minus#(0(),y) -> c_8():10
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):5
             -->_2 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):4
          
          6:W:if_minus#(true(),s(x),y) -> c_2()
             
          
          7:W:le#(0(),y) -> c_3()
             
          
          8:W:le#(s(x),0()) -> c_4()
             
          
          9:W:log#(s(0())) -> c_6()
             
          
          10:W:minus#(0(),y) -> c_8()
             
          
          11:W:quot#(0(),s(y)) -> c_10()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          9: log#(s(0())) -> c_6()
          11: quot#(0(),s(y)) -> c_10()
          10: minus#(0(),y) -> c_8()
          7: le#(0(),y) -> c_3()
          6: if_minus#(true(),s(x),y) -> c_2()
          8: le#(s(x),0()) -> c_4()
** Step 1.b:5: Decompose WORST_CASE(?,O(n^4))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            le#(s(x),s(y)) -> c_5(le#(x,y))
            log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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:
              if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
              le#(s(x),s(y)) -> c_5(le#(x,y))
              minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
          - Weak DPs:
              log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
              quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
          - Weak TRS:
              if_minus(false(),s(x),y) -> s(minus(x,y))
              if_minus(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) -> if_minus(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:
              {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
              ,false,s,true}
        
        Problem (S)
          - Strict DPs:
              log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
              quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
          - Weak DPs:
              if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
              le#(s(x),s(y)) -> c_5(le#(x,y))
              minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
          - Weak TRS:
              if_minus(false(),s(x),y) -> s(minus(x,y))
              if_minus(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) -> if_minus(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:
              {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
              ,false,s,true}
*** Step 1.b:5.a:1: DecomposeDG WORST_CASE(?,O(n^4))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        - Weak DPs:
            log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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
          log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        and a lower component
          if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          le#(s(x),s(y)) -> c_5(le#(x,y))
          minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
          quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        Further, following extension rules are added to the lower component.
          log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
          log#(s(s(x))) -> quot#(x,s(s(0())))
**** Step 1.b:5.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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: log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
          
        The strictly oriented rules are moved into the weak component.
***** Step 1.b:5.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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_7) = {1}
        
        Following symbols are considered usable:
          {if_minus,minus,quot,if_minus#,le#,log#,minus#,quot#}
        TcT has computed the following interpretation:
                  p(0) = [0]                           
              p(false) = [0]                           
           p(if_minus) = [1] x2 + [0]                  
                 p(le) = [8] x2 + [0]                  
                p(log) = [2] x1 + [1]                  
              p(minus) = [1] x1 + [0]                  
               p(quot) = [1] x1 + [0]                  
                  p(s) = [1] x1 + [2]                  
               p(true) = [0]                           
          p(if_minus#) = [1] x1 + [1] x2 + [1] x3 + [8]
                p(le#) = [1] x2 + [1]                  
               p(log#) = [4] x1 + [0]                  
             p(minus#) = [1] x1 + [2] x2 + [1]         
              p(quot#) = [1] x1 + [1] x2 + [2]         
                p(c_1) = [1] x1 + [8]                  
                p(c_2) = [8]                           
                p(c_3) = [0]                           
                p(c_4) = [1]                           
                p(c_5) = [1] x1 + [0]                  
                p(c_6) = [1]                           
                p(c_7) = [1] x1 + [6]                  
                p(c_8) = [1]                           
                p(c_9) = [1]                           
               p(c_10) = [1]                           
               p(c_11) = [1] x1 + [1]                  
        
        Following rules are strictly oriented:
        log#(s(s(x))) = [4] x + [16]                                      
                      > [4] x + [14]                                      
                      = c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        
        
        Following rules are (at-least) weakly oriented:
        if_minus(false(),s(x),y) =  [1] x + [2]                
                                 >= [1] x + [2]                
                                 =  s(minus(x,y))              
        
         if_minus(true(),s(x),y) =  [1] x + [2]                
                                 >= [0]                        
                                 =  0()                        
        
                    minus(0(),y) =  [0]                        
                                 >= [0]                        
                                 =  0()                        
        
                   minus(s(x),y) =  [1] x + [2]                
                                 >= [1] x + [2]                
                                 =  if_minus(le(s(x),y),s(x),y)
        
                  quot(0(),s(y)) =  [0]                        
                                 >= [0]                        
                                 =  0()                        
        
                 quot(s(x),s(y)) =  [1] x + [2]                
                                 >= [1] x + [2]                
                                 =  s(quot(minus(x,y),s(y)))   
        
***** Step 1.b:5.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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 1.b:5.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
             -->_1 log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
***** Step 1.b:5.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 1.b:5.a:1.b:1: DecomposeDG WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        - Weak DPs:
            log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
            log#(s(s(x))) -> quot#(x,s(s(0())))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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
          log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
          log#(s(s(x))) -> quot#(x,s(s(0())))
          quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        and a lower component
          if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          le#(s(x),s(y)) -> c_5(le#(x,y))
          minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        Further, following extension rules are added to the lower component.
          log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
          log#(s(s(x))) -> quot#(x,s(s(0())))
          quot#(s(x),s(y)) -> minus#(x,y)
          quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
***** Step 1.b:5.a:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak DPs:
            log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
            log#(s(s(x))) -> quot#(x,s(s(0())))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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 = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
          
        The strictly oriented rules are moved into the weak component.
****** Step 1.b:5.a:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak DPs:
            log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
            log#(s(s(x))) -> quot#(x,s(s(0())))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        NaturalMI {miDimension = 3, 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 (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_11) = {1}
        
        Following symbols are considered usable:
          {if_minus,minus,quot,if_minus#,le#,log#,minus#,quot#}
        TcT has computed the following interpretation:
                  p(0) = [0]                          
                         [0]                          
                         [0]                          
              p(false) = [0]                          
                         [0]                          
                         [0]                          
           p(if_minus) = [1 0 0]      [0]             
                         [0 0 0] x2 + [0]             
                         [0 0 0]      [0]             
                 p(le) = [0]                          
                         [0]                          
                         [0]                          
                p(log) = [0]                          
                         [0]                          
                         [0]                          
              p(minus) = [1 0 0]      [0]             
                         [0 0 1] x1 + [0]             
                         [1 0 0]      [0]             
               p(quot) = [1 0 0]      [0 0 0]      [0]
                         [0 1 0] x1 + [0 0 0] x2 + [1]
                         [0 0 0]      [1 0 0]      [1]
                  p(s) = [1 0 0]      [1]             
                         [0 0 0] x1 + [0]             
                         [0 0 0]      [0]             
               p(true) = [0]                          
                         [0]                          
                         [0]                          
          p(if_minus#) = [0]                          
                         [0]                          
                         [0]                          
                p(le#) = [0]                          
                         [0]                          
                         [0]                          
               p(log#) = [1 0 0]      [0]             
                         [0 0 0] x1 + [0]             
                         [1 0 0]      [1]             
             p(minus#) = [0]                          
                         [0]                          
                         [0]                          
              p(quot#) = [1 0 0]      [0]             
                         [0 0 0] x1 + [0]             
                         [0 0 0]      [0]             
                p(c_1) = [0]                          
                         [0]                          
                         [0]                          
                p(c_2) = [0]                          
                         [0]                          
                         [0]                          
                p(c_3) = [0]                          
                         [0]                          
                         [0]                          
                p(c_4) = [0]                          
                         [0]                          
                         [0]                          
                p(c_5) = [0]                          
                         [0]                          
                         [0]                          
                p(c_6) = [0]                          
                         [0]                          
                         [0]                          
                p(c_7) = [0]                          
                         [0]                          
                         [0]                          
                p(c_8) = [0]                          
                         [0]                          
                         [0]                          
                p(c_9) = [0]                          
                         [0]                          
                         [0]                          
               p(c_10) = [0]                          
                         [0]                          
                         [0]                          
               p(c_11) = [1 0 0]      [0]             
                         [0 0 0] x1 + [0]             
                         [0 0 0]      [0]             
        
        Following rules are strictly oriented:
        quot#(s(x),s(y)) = [1 0 0]     [1]                         
                           [0 0 0] x + [0]                         
                           [0 0 0]     [0]                         
                         > [1 0 0]     [0]                         
                           [0 0 0] x + [0]                         
                           [0 0 0]     [0]                         
                         = c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
                   log#(s(s(x))) =  [1 0 0]     [2]            
                                    [0 0 0] x + [0]            
                                    [1 0 0]     [3]            
                                 >= [1 0 0]     [1]            
                                    [0 0 0] x + [0]            
                                    [1 0 0]     [2]            
                                 =  log#(s(quot(x,s(s(0()))))) 
        
                   log#(s(s(x))) =  [1 0 0]     [2]            
                                    [0 0 0] x + [0]            
                                    [1 0 0]     [3]            
                                 >= [1 0 0]     [0]            
                                    [0 0 0] x + [0]            
                                    [0 0 0]     [0]            
                                 =  quot#(x,s(s(0())))         
        
        if_minus(false(),s(x),y) =  [1 0 0]     [1]            
                                    [0 0 0] x + [0]            
                                    [0 0 0]     [0]            
                                 >= [1 0 0]     [1]            
                                    [0 0 0] x + [0]            
                                    [0 0 0]     [0]            
                                 =  s(minus(x,y))              
        
         if_minus(true(),s(x),y) =  [1 0 0]     [1]            
                                    [0 0 0] x + [0]            
                                    [0 0 0]     [0]            
                                 >= [0]                        
                                    [0]                        
                                    [0]                        
                                 =  0()                        
        
                    minus(0(),y) =  [0]                        
                                    [0]                        
                                    [0]                        
                                 >= [0]                        
                                    [0]                        
                                    [0]                        
                                 =  0()                        
        
                   minus(s(x),y) =  [1 0 0]     [1]            
                                    [0 0 0] x + [0]            
                                    [1 0 0]     [1]            
                                 >= [1 0 0]     [1]            
                                    [0 0 0] x + [0]            
                                    [0 0 0]     [0]            
                                 =  if_minus(le(s(x),y),s(x),y)
        
                  quot(0(),s(y)) =  [0 0 0]     [0]            
                                    [0 0 0] y + [1]            
                                    [1 0 0]     [2]            
                                 >= [0]                        
                                    [0]                        
                                    [0]                        
                                 =  0()                        
        
                 quot(s(x),s(y)) =  [1 0 0]     [0 0 0]     [1]
                                    [0 0 0] x + [0 0 0] y + [1]
                                    [0 0 0]     [1 0 0]     [2]
                                 >= [1 0 0]     [1]            
                                    [0 0 0] x + [0]            
                                    [0 0 0]     [0]            
                                 =  s(quot(minus(x,y),s(y)))   
        
****** Step 1.b:5.a:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
            log#(s(s(x))) -> quot#(x,s(s(0())))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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 1.b:5.a:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
            log#(s(s(x))) -> quot#(x,s(s(0())))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
             -->_1 log#(s(s(x))) -> quot#(x,s(s(0()))):2
             -->_1 log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))):1
          
          2:W:log#(s(s(x))) -> quot#(x,s(s(0())))
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):3
          
          3:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),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.
          1: log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
          2: log#(s(s(x))) -> quot#(x,s(s(0())))
          3: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
****** Step 1.b:5.a:1.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

***** Step 1.b:5.a:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        - Weak DPs:
            log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
            log#(s(s(x))) -> quot#(x,s(s(0())))
            quot#(s(x),s(y)) -> minus#(x,y)
            quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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 = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 2, 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 1.b:5.a:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        - Weak DPs:
            log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
            log#(s(s(x))) -> quot#(x,s(s(0())))
            quot#(s(x),s(y)) -> minus#(x,y)
            quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        NaturalMI {miDimension = 2, 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:
        The following argument positions are considered usable:
          uargs(c_1) = {1},
          uargs(c_5) = {1},
          uargs(c_9) = {1,2}
        
        Following symbols are considered usable:
          {if_minus,minus,quot,if_minus#,le#,log#,minus#,quot#}
        TcT has computed the following interpretation:
                  p(0) = [0]                      
                         [0]                      
              p(false) = [0]                      
                         [0]                      
           p(if_minus) = [1 0] x2 + [0]           
                         [0 1]      [0]           
                 p(le) = [0]                      
                         [0]                      
                p(log) = [0 2] x1 + [0]           
                         [2 2]      [0]           
              p(minus) = [1 0] x1 + [0]           
                         [0 1]      [0]           
               p(quot) = [1 0] x1 + [0]           
                         [0 1]      [0]           
                  p(s) = [1 2] x1 + [0]           
                         [0 1]      [1]           
               p(true) = [0]                      
                         [0]                      
          p(if_minus#) = [1 0] x2 + [0 0] x3 + [0]
                         [0 3]      [0 3]      [0]
                p(le#) = [0 1] x1 + [0]           
                         [0 1]      [0]           
               p(log#) = [1 0] x1 + [3]           
                         [1 0]      [3]           
             p(minus#) = [1 2] x1 + [0]           
                         [0 1]      [2]           
              p(quot#) = [1 0] x1 + [0]           
                         [1 2]      [1]           
                p(c_1) = [1 0] x1 + [0]           
                         [0 0]      [3]           
                p(c_2) = [0]                      
                         [0]                      
                p(c_3) = [0]                      
                         [0]                      
                p(c_4) = [0]                      
                         [0]                      
                p(c_5) = [1 0] x1 + [0]           
                         [0 0]      [0]           
                p(c_6) = [0]                      
                         [0]                      
                p(c_7) = [2 0] x1 + [0 0] x2 + [0]
                         [0 2]      [0 2]      [1]
                p(c_8) = [0]                      
                         [0]                      
                p(c_9) = [1 0] x1 + [2 0] x2 + [0]
                         [0 0]      [0 1]      [2]
               p(c_10) = [0]                      
                         [1]                      
               p(c_11) = [0 0] x2 + [0]           
                         [0 1]      [0]           
        
        Following rules are strictly oriented:
        le#(s(x),s(y)) = [0 1] x + [1]
                         [0 1]     [1]
                       > [0 1] x + [0]
                         [0 0]     [0]
                       = c_5(le#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        if_minus#(false(),s(x),y) =  [1 2] x + [0 0] y + [0]                      
                                     [0 3]     [0 3]     [3]                      
                                  >= [1 2] x + [0]                                
                                     [0 0]     [3]                                
                                  =  c_1(minus#(x,y))                             
        
                    log#(s(s(x))) =  [1 4] x + [5]                                
                                     [1 4]     [5]                                
                                  >= [1 2] x + [3]                                
                                     [1 2]     [3]                                
                                  =  log#(s(quot(x,s(s(0())))))                   
        
                    log#(s(s(x))) =  [1 4] x + [5]                                
                                     [1 4]     [5]                                
                                  >= [1 0] x + [0]                                
                                     [1 2]     [1]                                
                                  =  quot#(x,s(s(0())))                           
        
                   minus#(s(x),y) =  [1 4] x + [2]                                
                                     [0 1]     [3]                                
                                  >= [1 4] x + [2]                                
                                     [0 1]     [3]                                
                                  =  c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        
                 quot#(s(x),s(y)) =  [1 2] x + [0]                                
                                     [1 4]     [3]                                
                                  >= [1 2] x + [0]                                
                                     [0 1]     [2]                                
                                  =  minus#(x,y)                                  
        
                 quot#(s(x),s(y)) =  [1 2] x + [0]                                
                                     [1 4]     [3]                                
                                  >= [1 0] x + [0]                                
                                     [1 2]     [1]                                
                                  =  quot#(minus(x,y),s(y))                       
        
         if_minus(false(),s(x),y) =  [1 2] x + [0]                                
                                     [0 1]     [1]                                
                                  >= [1 2] x + [0]                                
                                     [0 1]     [1]                                
                                  =  s(minus(x,y))                                
        
          if_minus(true(),s(x),y) =  [1 2] x + [0]                                
                                     [0 1]     [1]                                
                                  >= [0]                                          
                                     [0]                                          
                                  =  0()                                          
        
                     minus(0(),y) =  [0]                                          
                                     [0]                                          
                                  >= [0]                                          
                                     [0]                                          
                                  =  0()                                          
        
                    minus(s(x),y) =  [1 2] x + [0]                                
                                     [0 1]     [1]                                
                                  >= [1 2] x + [0]                                
                                     [0 1]     [1]                                
                                  =  if_minus(le(s(x),y),s(x),y)                  
        
                   quot(0(),s(y)) =  [0]                                          
                                     [0]                                          
                                  >= [0]                                          
                                     [0]                                          
                                  =  0()                                          
        
                  quot(s(x),s(y)) =  [1 2] x + [0]                                
                                     [0 1]     [1]                                
                                  >= [1 2] x + [0]                                
                                     [0 1]     [1]                                
                                  =  s(quot(minus(x,y),s(y)))                     
        
****** Step 1.b:5.a:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        - Weak DPs:
            le#(s(x),s(y)) -> c_5(le#(x,y))
            log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
            log#(s(s(x))) -> quot#(x,s(s(0())))
            quot#(s(x),s(y)) -> minus#(x,y)
            quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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 1.b:5.a:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        - Weak DPs:
            le#(s(x),s(y)) -> c_5(le#(x,y))
            log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
            log#(s(s(x))) -> quot#(x,s(s(0())))
            quot#(s(x),s(y)) -> minus#(x,y)
            quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
          
          2:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
             -->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):3
             -->_1 if_minus#(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:log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
             -->_1 log#(s(s(x))) -> quot#(x,s(s(0()))):5
             -->_1 log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))):4
          
          5:W:log#(s(s(x))) -> quot#(x,s(s(0())))
             -->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):7
             -->_1 quot#(s(x),s(y)) -> minus#(x,y):6
          
          6:W:quot#(s(x),s(y)) -> minus#(x,y)
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
          
          7:W:quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
             -->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):7
             -->_1 quot#(s(x),s(y)) -> minus#(x,y):6
          
        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 1.b:5.a:1.b:1.b:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        - Weak DPs:
            log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
            log#(s(s(x))) -> quot#(x,s(s(0())))
            quot#(s(x),s(y)) -> minus#(x,y)
            quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
          
          2:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
             -->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1
          
          4:W:log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
             -->_1 log#(s(s(x))) -> quot#(x,s(s(0()))):5
             -->_1 log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))):4
          
          5:W:log#(s(s(x))) -> quot#(x,s(s(0())))
             -->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):7
             -->_1 quot#(s(x),s(y)) -> minus#(x,y):6
          
          6:W:quot#(s(x),s(y)) -> minus#(x,y)
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
          
          7:W:quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
             -->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):7
             -->_1 quot#(s(x),s(y)) -> minus#(x,y):6
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
****** Step 1.b:5.a:1.b:1.b:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
        - Weak DPs:
            log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
            log#(s(s(x))) -> quot#(x,s(s(0())))
            quot#(s(x),s(y)) -> minus#(x,y)
            quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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: if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          2: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
          
        The strictly oriented rules are moved into the weak component.
******* Step 1.b:5.a:1.b:1.b:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
        - Weak DPs:
            log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
            log#(s(s(x))) -> quot#(x,s(s(0())))
            quot#(s(x),s(y)) -> minus#(x,y)
            quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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_9) = {1}
        
        Following symbols are considered usable:
          {if_minus,minus,quot,if_minus#,le#,log#,minus#,quot#}
        TcT has computed the following interpretation:
                  p(0) = [0]                  
              p(false) = [0]                  
           p(if_minus) = [1] x2 + [0]         
                 p(le) = [0]                  
                p(log) = [1] x1 + [0]         
              p(minus) = [1] x1 + [0]         
               p(quot) = [1] x1 + [0]         
                  p(s) = [1] x1 + [4]         
               p(true) = [0]                  
          p(if_minus#) = [2] x2 + [2]         
                p(le#) = [1] x1 + [1]         
               p(log#) = [2] x1 + [9]         
             p(minus#) = [2] x1 + [6]         
              p(quot#) = [2] x1 + [8]         
                p(c_1) = [1] x1 + [3]         
                p(c_2) = [1]                  
                p(c_3) = [0]                  
                p(c_4) = [8]                  
                p(c_5) = [0]                  
                p(c_6) = [0]                  
                p(c_7) = [1] x1 + [1] x2 + [0]
                p(c_8) = [0]                  
                p(c_9) = [1] x1 + [2]         
               p(c_10) = [2]                  
               p(c_11) = [1] x1 + [1] x2 + [1]
        
        Following rules are strictly oriented:
        if_minus#(false(),s(x),y) = [2] x + [10]                     
                                  > [2] x + [9]                      
                                  = c_1(minus#(x,y))                 
        
                   minus#(s(x),y) = [2] x + [14]                     
                                  > [2] x + [12]                     
                                  = c_9(if_minus#(le(s(x),y),s(x),y))
        
        
        Following rules are (at-least) weakly oriented:
                   log#(s(s(x))) =  [2] x + [25]               
                                 >= [2] x + [17]               
                                 =  log#(s(quot(x,s(s(0()))))) 
        
                   log#(s(s(x))) =  [2] x + [25]               
                                 >= [2] x + [8]                
                                 =  quot#(x,s(s(0())))         
        
                quot#(s(x),s(y)) =  [2] x + [16]               
                                 >= [2] x + [6]                
                                 =  minus#(x,y)                
        
                quot#(s(x),s(y)) =  [2] x + [16]               
                                 >= [2] x + [8]                
                                 =  quot#(minus(x,y),s(y))     
        
        if_minus(false(),s(x),y) =  [1] x + [4]                
                                 >= [1] x + [4]                
                                 =  s(minus(x,y))              
        
         if_minus(true(),s(x),y) =  [1] x + [4]                
                                 >= [0]                        
                                 =  0()                        
        
                    minus(0(),y) =  [0]                        
                                 >= [0]                        
                                 =  0()                        
        
                   minus(s(x),y) =  [1] x + [4]                
                                 >= [1] x + [4]                
                                 =  if_minus(le(s(x),y),s(x),y)
        
                  quot(0(),s(y)) =  [0]                        
                                 >= [0]                        
                                 =  0()                        
        
                 quot(s(x),s(y)) =  [1] x + [4]                
                                 >= [1] x + [4]                
                                 =  s(quot(minus(x,y),s(y)))   
        
******* Step 1.b:5.a:1.b:1.b:1.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
            log#(s(s(x))) -> quot#(x,s(s(0())))
            minus#(s(x),y) -> c_9(if_minus#(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:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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 1.b:5.a:1.b:1.b:1.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
            log#(s(s(x))) -> quot#(x,s(s(0())))
            minus#(s(x),y) -> c_9(if_minus#(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:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)):4
          
          2:W:log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
             -->_1 log#(s(s(x))) -> quot#(x,s(s(0()))):3
             -->_1 log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))):2
          
          3:W:log#(s(s(x))) -> quot#(x,s(s(0())))
             -->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):6
             -->_1 quot#(s(x),s(y)) -> minus#(x,y):5
          
          4:W:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
             -->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1
          
          5:W:quot#(s(x),s(y)) -> minus#(x,y)
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)):4
          
          6:W:quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
             -->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):6
             -->_1 quot#(s(x),s(y)) -> minus#(x,y):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
          3: log#(s(s(x))) -> quot#(x,s(s(0())))
          6: quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
          5: quot#(s(x),s(y)) -> minus#(x,y)
          1: if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          4: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
******* Step 1.b:5.a:1.b:1.b:1.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
             -->_2 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):2
             -->_1 log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
          
          2:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_2 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):5
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):2
          
          3:W:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):5
          
          4:W:le#(s(x),s(y)) -> c_5(le#(x,y))
             -->_1 le#(s(x),s(y)) -> c_5(le#(x,y)):4
          
          5:W:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
             -->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):4
             -->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
          3: if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          4: le#(s(x),s(y)) -> c_5(le#(x,y))
*** Step 1.b:5.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
             -->_2 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):2
             -->_1 log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
          
          2:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
*** Step 1.b:5.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
          2: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
          
        The strictly oriented rules are moved into the weak component.
**** Step 1.b:5.b:3.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_7) = {1,2},
          uargs(c_11) = {1}
        
        Following symbols are considered usable:
          {if_minus,minus,quot,if_minus#,le#,log#,minus#,quot#}
        TcT has computed the following interpretation:
                  p(0) = 0                   
              p(false) = 0                   
           p(if_minus) = x2                  
                 p(le) = 2                   
                p(log) = 1 + x1^2            
              p(minus) = x1                  
               p(quot) = x1                  
                  p(s) = 1 + x1              
               p(true) = 0                   
          p(if_minus#) = 2*x2 + x2^2 + 2*x3^2
                p(le#) = 0                   
               p(log#) = 3 + x1^2            
             p(minus#) = 2*x1*x2 + 2*x2^2    
              p(quot#) = x1                  
                p(c_1) = 1                   
                p(c_2) = 0                   
                p(c_3) = 0                   
                p(c_4) = 0                   
                p(c_5) = 0                   
                p(c_6) = 0                   
                p(c_7) = x1 + x2             
                p(c_8) = 1                   
                p(c_9) = 0                   
               p(c_10) = 0                   
               p(c_11) = x1                  
        
        Following rules are strictly oriented:
           log#(s(s(x))) = 7 + 4*x + x^2                                     
                         > 4 + 3*x + x^2                                     
                         = c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        
        quot#(s(x),s(y)) = 1 + x                                             
                         > x                                                 
                         = c_11(quot#(minus(x,y),s(y)))                      
        
        
        Following rules are (at-least) weakly oriented:
        if_minus(false(),s(x),y) =  1 + x                      
                                 >= 1 + x                      
                                 =  s(minus(x,y))              
        
         if_minus(true(),s(x),y) =  1 + x                      
                                 >= 0                          
                                 =  0()                        
        
                    minus(0(),y) =  0                          
                                 >= 0                          
                                 =  0()                        
        
                   minus(s(x),y) =  1 + x                      
                                 >= 1 + x                      
                                 =  if_minus(le(s(x),y),s(x),y)
        
                  quot(0(),s(y)) =  0                          
                                 >= 0                          
                                 =  0()                        
        
                 quot(s(x),s(y)) =  1 + x                      
                                 >= 1 + x                      
                                 =  s(quot(minus(x,y),s(y)))   
        
**** Step 1.b:5.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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 1.b:5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
             -->_2 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y))):2
             -->_1 log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
          
          2:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y))):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
          2: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
**** Step 1.b:5.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(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) -> if_minus(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:
            {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,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,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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(Omega(n^1),O(n^4))