* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            log(s(0())) -> 0()
            log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
            minus(x,0()) -> x
            minus(x,s(y)) -> pred(minus(x,y))
            pred(s(x)) -> x
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {log/1,minus/2,pred/1,quot/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log,minus,pred,quot} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          log#(s(0())) -> c_1()
          log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
          minus#(x,0()) -> c_3()
          minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
          pred#(s(x)) -> c_5()
          quot#(0(),s(y)) -> c_6()
          quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            log#(s(0())) -> c_1()
            log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            minus#(x,0()) -> c_3()
            minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
            pred#(s(x)) -> c_5()
            quot#(0(),s(y)) -> c_6()
            quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            log(s(0())) -> 0()
            log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
            minus(x,0()) -> x
            minus(x,s(y)) -> pred(minus(x,y))
            pred(s(x)) -> x
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0
            ,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          minus(x,0()) -> x
          minus(x,s(y)) -> pred(minus(x,y))
          pred(s(x)) -> x
          quot(0(),s(y)) -> 0()
          quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
          log#(s(0())) -> c_1()
          log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
          minus#(x,0()) -> c_3()
          minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
          pred#(s(x)) -> c_5()
          quot#(0(),s(y)) -> c_6()
          quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            log#(s(0())) -> c_1()
            log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            minus#(x,0()) -> c_3()
            minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
            pred#(s(x)) -> c_5()
            quot#(0(),s(y)) -> c_6()
            quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            minus(x,0()) -> x
            minus(x,s(y)) -> pred(minus(x,y))
            pred(s(x)) -> x
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0
            ,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,5,6}
        by application of
          Pre({1,3,5,6}) = {2,4,7}.
        Here rules are labelled as follows:
          1: log#(s(0())) -> c_1()
          2: log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
          3: minus#(x,0()) -> c_3()
          4: minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
          5: pred#(s(x)) -> c_5()
          6: quot#(0(),s(y)) -> c_6()
          7: quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
            quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak DPs:
            log#(s(0())) -> c_1()
            minus#(x,0()) -> c_3()
            pred#(s(x)) -> c_5()
            quot#(0(),s(y)) -> c_6()
        - Weak TRS:
            minus(x,0()) -> x
            minus(x,s(y)) -> pred(minus(x,y))
            pred(s(x)) -> x
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0
            ,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
             -->_2 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
             -->_2 quot#(0(),s(y)) -> c_6():7
             -->_1 log#(s(0())) -> c_1():4
             -->_1 log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
          
          2:S:minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
             -->_1 pred#(s(x)) -> c_5():6
             -->_2 minus#(x,0()) -> c_3():5
             -->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2
          
          3:S:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(0(),s(y)) -> c_6():7
             -->_2 minus#(x,0()) -> c_3():5
             -->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
             -->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2
          
          4:W:log#(s(0())) -> c_1()
             
          
          5:W:minus#(x,0()) -> c_3()
             
          
          6:W:pred#(s(x)) -> c_5()
             
          
          7:W:quot#(0(),s(y)) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: log#(s(0())) -> c_1()
          6: pred#(s(x)) -> c_5()
          5: minus#(x,0()) -> c_3()
          7: quot#(0(),s(y)) -> c_6()
* Step 5: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
            quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            minus(x,0()) -> x
            minus(x,s(y)) -> pred(minus(x,y))
            pred(s(x)) -> x
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0
            ,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
             -->_2 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
             -->_1 log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
          
          2:S:minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
             -->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2
          
          3:S:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
             -->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          minus#(x,s(y)) -> c_4(minus#(x,y))
* Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            minus#(x,s(y)) -> c_4(minus#(x,y))
            quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            minus(x,0()) -> x
            minus(x,s(y)) -> pred(minus(x,y))
            pred(s(x)) -> x
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0
            ,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
    + 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:
          2: minus#(x,s(y)) -> c_4(minus#(x,y))
          3: quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
          
        The strictly oriented rules are moved into the weak component.
** Step 6.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
            minus#(x,s(y)) -> c_4(minus#(x,y))
            quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            minus(x,0()) -> x
            minus(x,s(y)) -> pred(minus(x,y))
            pred(s(x)) -> x
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0
            ,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
    + 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_2) = {1,2},
          uargs(c_4) = {1},
          uargs(c_7) = {1,2}
        
        Following symbols are considered usable:
          {minus,pred,quot,log#,minus#,pred#,quot#}
        TcT has computed the following interpretation:
               p(0) = 0           
             p(log) = 2 + 2*x1^2  
           p(minus) = x1          
            p(pred) = x1          
            p(quot) = x1          
               p(s) = 1 + x1      
            p(log#) = x1 + x1^2   
          p(minus#) = x2          
           p(pred#) = 1           
           p(quot#) = x1*x2 + 2*x2
             p(c_1) = 0           
             p(c_2) = x1 + x2     
             p(c_3) = 0           
             p(c_4) = x1          
             p(c_5) = 0           
             p(c_6) = 0           
             p(c_7) = x1 + x2     
        
        Following rules are strictly oriented:
          minus#(x,s(y)) = 1 + y                                  
                         > y                                      
                         = c_4(minus#(x,y))                       
        
        quot#(s(x),s(y)) = 3 + x + x*y + 3*y                      
                         > 2 + x + x*y + 3*y                      
                         = c_7(quot#(minus(x,y),s(y)),minus#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
          log#(s(s(x))) =  6 + 5*x + x^2                                     
                        >= 6 + 5*x + x^2                                     
                        =  c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        
           minus(x,0()) =  x                                                 
                        >= x                                                 
                        =  x                                                 
        
          minus(x,s(y)) =  x                                                 
                        >= x                                                 
                        =  pred(minus(x,y))                                  
        
             pred(s(x)) =  1 + x                                             
                        >= x                                                 
                        =  x                                                 
        
         quot(0(),s(y)) =  0                                                 
                        >= 0                                                 
                        =  0()                                               
        
        quot(s(x),s(y)) =  1 + x                                             
                        >= 1 + x                                             
                        =  s(quot(minus(x,y),s(y)))                          
        
** Step 6.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        - Weak DPs:
            minus#(x,s(y)) -> c_4(minus#(x,y))
            quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            minus(x,0()) -> x
            minus(x,s(y)) -> pred(minus(x,y))
            pred(s(x)) -> x
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0
            ,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        - Weak DPs:
            minus#(x,s(y)) -> c_4(minus#(x,y))
            quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            minus(x,0()) -> x
            minus(x,s(y)) -> pred(minus(x,y))
            pred(s(x)) -> x
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0
            ,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
             -->_2 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
             -->_1 log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
          
          2:W:minus#(x,s(y)) -> c_4(minus#(x,y))
             -->_1 minus#(x,s(y)) -> c_4(minus#(x,y)):2
          
          3:W:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
             -->_2 minus#(x,s(y)) -> c_4(minus#(x,y)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
          2: minus#(x,s(y)) -> c_4(minus#(x,y))
** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
        - Weak TRS:
            minus(x,0()) -> x
            minus(x,s(y)) -> pred(minus(x,y))
            pred(s(x)) -> x
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0
            ,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
             -->_1 log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))))
** Step 6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))))
        - Weak TRS:
            minus(x,0()) -> x
            minus(x,s(y)) -> pred(minus(x,y))
            pred(s(x)) -> x
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0
            ,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
    + 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_2(log#(s(quot(x,s(s(0()))))))
          
        The strictly oriented rules are moved into the weak component.
*** Step 6.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))))
        - Weak TRS:
            minus(x,0()) -> x
            minus(x,s(y)) -> pred(minus(x,y))
            pred(s(x)) -> x
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0
            ,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
    + 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_2) = {1}
        
        Following symbols are considered usable:
          {minus,pred,quot,log#,minus#,pred#,quot#}
        TcT has computed the following interpretation:
               p(0) = [0]                  
             p(log) = [0]                  
           p(minus) = [1] x1 + [0]         
            p(pred) = [1] x1 + [0]         
            p(quot) = [1] x1 + [0]         
               p(s) = [1] x1 + [8]         
            p(log#) = [1] x1 + [8]         
          p(minus#) = [1] x1 + [0]         
           p(pred#) = [2] x1 + [1]         
           p(quot#) = [1] x1 + [1] x2 + [0]
             p(c_1) = [1]                  
             p(c_2) = [1] x1 + [7]         
             p(c_3) = [0]                  
             p(c_4) = [1] x1 + [0]         
             p(c_5) = [1]                  
             p(c_6) = [0]                  
             p(c_7) = [1] x1 + [1] x2 + [1]
        
        Following rules are strictly oriented:
        log#(s(s(x))) = [1] x + [24]                   
                      > [1] x + [23]                   
                      = c_2(log#(s(quot(x,s(s(0()))))))
        
        
        Following rules are (at-least) weakly oriented:
           minus(x,0()) =  [1] x + [0]             
                        >= [1] x + [0]             
                        =  x                       
        
          minus(x,s(y)) =  [1] x + [0]             
                        >= [1] x + [0]             
                        =  pred(minus(x,y))        
        
             pred(s(x)) =  [1] x + [8]             
                        >= [1] x + [0]             
                        =  x                       
        
         quot(0(),s(y)) =  [0]                     
                        >= [0]                     
                        =  0()                     
        
        quot(s(x),s(y)) =  [1] x + [8]             
                        >= [1] x + [8]             
                        =  s(quot(minus(x,y),s(y)))
        
*** Step 6.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))))
        - Weak TRS:
            minus(x,0()) -> x
            minus(x,s(y)) -> pred(minus(x,y))
            pred(s(x)) -> x
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0
            ,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))))
        - Weak TRS:
            minus(x,0()) -> x
            minus(x,s(y)) -> pred(minus(x,y))
            pred(s(x)) -> x
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0
            ,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))))
             -->_1 log#(s(s(x))) -> c_2(log#(s(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_2(log#(s(quot(x,s(s(0()))))))
*** Step 6.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            minus(x,0()) -> x
            minus(x,s(y)) -> pred(minus(x,y))
            pred(s(x)) -> x
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0
            ,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))