WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            div(x,y) -> quot(x,y,y)
            div(0(),y) -> 0()
            quot(x,0(),s(z)) -> s(div(x,s(z)))
            quot(0(),s(y),z) -> 0()
            quot(s(x),s(y),z) -> quot(x,y,z)
        - Signature:
            {div/2,quot/3} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div,quot} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          div#(x,y) -> c_1(quot#(x,y,y))
          div#(0(),y) -> c_2()
          quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
          quot#(0(),s(y),z) -> c_4()
          quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            div#(0(),y) -> c_2()
            quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
            quot#(0(),s(y),z) -> c_4()
            quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
        - Strict TRS:
            div(x,y) -> quot(x,y,y)
            div(0(),y) -> 0()
            quot(x,0(),s(z)) -> s(div(x,s(z)))
            quot(0(),s(y),z) -> 0()
            quot(s(x),s(y),z) -> quot(x,y,z)
        - Signature:
            {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          div#(x,y) -> c_1(quot#(x,y,y))
          div#(0(),y) -> c_2()
          quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
          quot#(0(),s(y),z) -> c_4()
          quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            div#(0(),y) -> c_2()
            quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
            quot#(0(),s(y),z) -> c_4()
            quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
        - Signature:
            {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4}
        by application of
          Pre({2,4}) = {1,3,5}.
        Here rules are labelled as follows:
          1: div#(x,y) -> c_1(quot#(x,y,y))
          2: div#(0(),y) -> c_2()
          3: quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
          4: quot#(0(),s(y),z) -> c_4()
          5: quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
            quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
        - Weak DPs:
            div#(0(),y) -> c_2()
            quot#(0(),s(y),z) -> c_4()
        - Signature:
            {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:div#(x,y) -> c_1(quot#(x,y,y))
             -->_1 quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)):3
             -->_1 quot#(0(),s(y),z) -> c_4():5
          
          2:S:quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
             -->_1 div#(0(),y) -> c_2():4
             -->_1 div#(x,y) -> c_1(quot#(x,y,y)):1
          
          3:S:quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
             -->_1 quot#(0(),s(y),z) -> c_4():5
             -->_1 quot#(s(x),s(y),z) -> c_5(quot#(x,y,z)):3
             -->_1 quot#(x,0(),s(z)) -> c_3(div#(x,s(z))):2
          
          4:W:div#(0(),y) -> c_2()
             
          
          5:W:quot#(0(),s(y),z) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: div#(0(),y) -> c_2()
          5: quot#(0(),s(y),z) -> c_4()
* Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
            quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
        - Signature:
            {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,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:
          3: quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
          
        Consider the set of all dependency pairs
          1: div#(x,y) -> c_1(quot#(x,y,y))
          2: quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
          3: quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {3}
        These cover all (indirect) predecessors of dependency pairs
          {1,2,3}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
            quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
        - Signature:
            {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,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_1) = {1},
          uargs(c_3) = {1},
          uargs(c_5) = {1}
        
        Following symbols are considered usable:
          {div#,quot#}
        TcT has computed the following interpretation:
              p(0) = [0]          
            p(div) = [1] x1 + [1] 
           p(quot) = [1] x1 + [1] 
              p(s) = [1] x1 + [8] 
           p(div#) = [2] x1 + [0] 
          p(quot#) = [2] x1 + [0] 
            p(c_1) = [1] x1 + [0] 
            p(c_2) = [8]          
            p(c_3) = [1] x1 + [0] 
            p(c_4) = [1]          
            p(c_5) = [1] x1 + [12]
        
        Following rules are strictly oriented:
        quot#(s(x),s(y),z) = [2] x + [16]     
                           > [2] x + [12]     
                           = c_5(quot#(x,y,z))
        
        
        Following rules are (at-least) weakly oriented:
                div#(x,y) =  [2] x + [0]      
                          >= [2] x + [0]      
                          =  c_1(quot#(x,y,y))
        
        quot#(x,0(),s(z)) =  [2] x + [0]      
                          >= [2] x + [0]      
                          =  c_3(div#(x,s(z)))
        
** Step 5.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            quot#(x,0(),s(z)) -> c_3(div#(x,s(z)))
        - Weak DPs:
            quot#(s(x),s(y),z) -> c_5(quot#(x,y,z))
        - Signature:
            {div/2,quot/3,div#/2,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,quot#} and constructors {0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

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

WORST_CASE(?,O(n^1))