MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            quot(0(),s(Y)) -> 0()
            quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
        - Signature:
            {min/2,plus/2,quot/2} / {0/0,Z/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min,plus,quot} and constructors {0,Z,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          min#(X,0()) -> c_1()
          min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
          min#(s(X),s(Y)) -> c_3(min#(X,Y))
          plus#(0(),Y) -> c_4()
          plus#(s(X),Y) -> c_5(plus#(X,Y))
          quot#(0(),s(Y)) -> c_6()
          quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            min#(X,0()) -> c_1()
            min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
            min#(s(X),s(Y)) -> c_3(min#(X,Y))
            plus#(0(),Y) -> c_4()
            plus#(s(X),Y) -> c_5(plus#(X,Y))
            quot#(0(),s(Y)) -> c_6()
            quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
            quot(0(),s(Y)) -> 0()
            quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          min(X,0()) -> X
          min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
          min(s(X),s(Y)) -> min(X,Y)
          plus(0(),Y) -> Y
          plus(s(X),Y) -> s(plus(X,Y))
          min#(X,0()) -> c_1()
          min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
          min#(s(X),s(Y)) -> c_3(min#(X,Y))
          plus#(0(),Y) -> c_4()
          plus#(s(X),Y) -> c_5(plus#(X,Y))
          quot#(0(),s(Y)) -> c_6()
          quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            min#(X,0()) -> c_1()
            min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
            min#(s(X),s(Y)) -> c_3(min#(X,Y))
            plus#(0(),Y) -> c_4()
            plus#(s(X),Y) -> c_5(plus#(X,Y))
            quot#(0(),s(Y)) -> c_6()
            quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4,6}
        by application of
          Pre({1,4,6}) = {2,3,5,7}.
        Here rules are labelled as follows:
          1: min#(X,0()) -> c_1()
          2: min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
          3: min#(s(X),s(Y)) -> c_3(min#(X,Y))
          4: plus#(0(),Y) -> c_4()
          5: plus#(s(X),Y) -> c_5(plus#(X,Y))
          6: quot#(0(),s(Y)) -> c_6()
          7: quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
            min#(s(X),s(Y)) -> c_3(min#(X,Y))
            plus#(s(X),Y) -> c_5(plus#(X,Y))
            quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
        - Weak DPs:
            min#(X,0()) -> c_1()
            plus#(0(),Y) -> c_4()
            quot#(0(),s(Y)) -> c_6()
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
             -->_2 plus#(s(X),Y) -> c_5(plus#(X,Y)):3
             -->_1 min#(s(X),s(Y)) -> c_3(min#(X,Y)):2
             -->_2 plus#(0(),Y) -> c_4():6
             -->_1 min#(X,0()) -> c_1():5
             -->_1 min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z())):1
          
          2:S:min#(s(X),s(Y)) -> c_3(min#(X,Y))
             -->_1 min#(X,0()) -> c_1():5
             -->_1 min#(s(X),s(Y)) -> c_3(min#(X,Y)):2
             -->_1 min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z())):1
          
          3:S:plus#(s(X),Y) -> c_5(plus#(X,Y))
             -->_1 plus#(0(),Y) -> c_4():6
             -->_1 plus#(s(X),Y) -> c_5(plus#(X,Y)):3
          
          4:S:quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
             -->_1 quot#(0(),s(Y)) -> c_6():7
             -->_2 min#(X,0()) -> c_1():5
             -->_1 quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y)):4
             -->_2 min#(s(X),s(Y)) -> c_3(min#(X,Y)):2
             -->_2 min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z())):1
          
          5:W:min#(X,0()) -> c_1()
             
          
          6:W:plus#(0(),Y) -> c_4()
             
          
          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.
          7: quot#(0(),s(Y)) -> c_6()
          5: min#(X,0()) -> c_1()
          6: plus#(0(),Y) -> c_4()
* Step 5: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
            min#(s(X),s(Y)) -> c_3(min#(X,Y))
            plus#(s(X),Y) -> c_5(plus#(X,Y))
            quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,s}
    + 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:
              min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
              min#(s(X),s(Y)) -> c_3(min#(X,Y))
              plus#(s(X),Y) -> c_5(plus#(X,Y))
          - Weak DPs:
              quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
          - Weak TRS:
              min(X,0()) -> X
              min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
              min(s(X),s(Y)) -> min(X,Y)
              plus(0(),Y) -> Y
              plus(s(X),Y) -> s(plus(X,Y))
          - Signature:
              {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,s}
        
        Problem (S)
          - Strict DPs:
              quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
          - Weak DPs:
              min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
              min#(s(X),s(Y)) -> c_3(min#(X,Y))
              plus#(s(X),Y) -> c_5(plus#(X,Y))
          - Weak TRS:
              min(X,0()) -> X
              min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
              min(s(X),s(Y)) -> min(X,Y)
              plus(0(),Y) -> Y
              plus(s(X),Y) -> s(plus(X,Y))
          - Signature:
              {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,s}
** Step 5.a:1: DecomposeDG MAYBE
    + Considered Problem:
        - Strict DPs:
            min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
            min#(s(X),s(Y)) -> c_3(min#(X,Y))
            plus#(s(X),Y) -> c_5(plus#(X,Y))
        - Weak DPs:
            quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
        and a lower component
          min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
          min#(s(X),s(Y)) -> c_3(min#(X,Y))
          plus#(s(X),Y) -> c_5(plus#(X,Y))
        Further, following extension rules are added to the lower component.
          quot#(s(X),s(Y)) -> min#(X,Y)
          quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
*** Step 5.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,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: quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 5.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,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_7) = {1}
        
        Following symbols are considered usable:
          {min,min#,plus#,quot#}
        TcT has computed the following interpretation:
              p(0) = [5]                  
              p(Z) = [8]                  
            p(min) = [1] x1 + [0]         
           p(plus) = [2] x1 + [1] x2 + [6]
           p(quot) = [1] x2 + [1]         
              p(s) = [1] x1 + [2]         
           p(min#) = [1] x2 + [0]         
          p(plus#) = [1] x2 + [1]         
          p(quot#) = [6] x1 + [0]         
            p(c_1) = [1]                  
            p(c_2) = [1]                  
            p(c_3) = [1] x1 + [2]         
            p(c_4) = [1]                  
            p(c_5) = [2]                  
            p(c_6) = [0]                  
            p(c_7) = [1] x1 + [10]        
        
        Following rules are strictly oriented:
        quot#(s(X),s(Y)) = [6] X + [12]                       
                         > [6] X + [10]                       
                         = c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
        
        
        Following rules are (at-least) weakly oriented:
               min(X,0()) =  [1] X + [0]       
                          >= [1] X + [0]       
                          =  X                 
        
        min(min(X,Y),Z()) =  [1] X + [0]       
                          >= [1] X + [0]       
                          =  min(X,plus(Y,Z()))
        
           min(s(X),s(Y)) =  [1] X + [2]       
                          >= [1] X + [0]       
                          =  min(X,Y)          
        
**** Step 5.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

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

*** Step 5.a:1.b:1: DecomposeDG MAYBE
    + Considered Problem:
        - Strict DPs:
            min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
            min#(s(X),s(Y)) -> c_3(min#(X,Y))
            plus#(s(X),Y) -> c_5(plus#(X,Y))
        - Weak DPs:
            quot#(s(X),s(Y)) -> min#(X,Y)
            quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,s}
    + 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
          min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
          min#(s(X),s(Y)) -> c_3(min#(X,Y))
          quot#(s(X),s(Y)) -> min#(X,Y)
          quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
        and a lower component
          plus#(s(X),Y) -> c_5(plus#(X,Y))
        Further, following extension rules are added to the lower component.
          min#(min(X,Y),Z()) -> min#(X,plus(Y,Z()))
          min#(min(X,Y),Z()) -> plus#(Y,Z())
          min#(s(X),s(Y)) -> min#(X,Y)
          quot#(s(X),s(Y)) -> min#(X,Y)
          quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
**** Step 5.a:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
            min#(s(X),s(Y)) -> c_3(min#(X,Y))
        - Weak DPs:
            quot#(s(X),s(Y)) -> min#(X,Y)
            quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,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: min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
          2: min#(s(X),s(Y)) -> c_3(min#(X,Y))
          
        The strictly oriented rules are moved into the weak component.
***** Step 5.a:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
            min#(s(X),s(Y)) -> c_3(min#(X,Y))
        - Weak DPs:
            quot#(s(X),s(Y)) -> min#(X,Y)
            quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,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},
          uargs(c_3) = {1}
        
        Following symbols are considered usable:
          {min,min#,plus#,quot#}
        TcT has computed the following interpretation:
              p(0) = [1]                  
              p(Z) = [6]                  
            p(min) = [1] x1 + [2]         
           p(plus) = [3] x2 + [0]         
           p(quot) = [1] x1 + [2] x2 + [0]
              p(s) = [1] x1 + [8]         
           p(min#) = [2] x1 + [0]         
          p(plus#) = [0]                  
          p(quot#) = [2] x1 + [7]         
            p(c_1) = [0]                  
            p(c_2) = [1] x1 + [8] x2 + [0]
            p(c_3) = [1] x1 + [13]        
            p(c_4) = [1]                  
            p(c_5) = [1] x1 + [2]         
            p(c_6) = [0]                  
            p(c_7) = [2] x1 + [1] x2 + [2]
        
        Following rules are strictly oriented:
        min#(min(X,Y),Z()) = [2] X + [4]                          
                           > [2] X + [0]                          
                           = c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
        
           min#(s(X),s(Y)) = [2] X + [16]                         
                           > [2] X + [13]                         
                           = c_3(min#(X,Y))                       
        
        
        Following rules are (at-least) weakly oriented:
         quot#(s(X),s(Y)) =  [2] X + [23]        
                          >= [2] X + [0]         
                          =  min#(X,Y)           
        
         quot#(s(X),s(Y)) =  [2] X + [23]        
                          >= [2] X + [11]        
                          =  quot#(min(X,Y),s(Y))
        
               min(X,0()) =  [1] X + [2]         
                          >= [1] X + [0]         
                          =  X                   
        
        min(min(X,Y),Z()) =  [1] X + [4]         
                          >= [1] X + [2]         
                          =  min(X,plus(Y,Z()))  
        
           min(s(X),s(Y)) =  [1] X + [10]        
                          >= [1] X + [2]         
                          =  min(X,Y)            
        
***** Step 5.a:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
            min#(s(X),s(Y)) -> c_3(min#(X,Y))
            quot#(s(X),s(Y)) -> min#(X,Y)
            quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

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

**** Step 5.a:1.b:1.b:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          plus#(s(X),Y) -> c_5(plus#(X,Y))
      - Weak DPs:
          min#(min(X,Y),Z()) -> min#(X,plus(Y,Z()))
          min#(min(X,Y),Z()) -> plus#(Y,Z())
          min#(s(X),s(Y)) -> min#(X,Y)
          quot#(s(X),s(Y)) -> min#(X,Y)
          quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
      - Weak TRS:
          min(X,0()) -> X
          min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
          min(s(X),s(Y)) -> min(X,Y)
          plus(0(),Y) -> Y
          plus(s(X),Y) -> s(plus(X,Y))
      - Signature:
          {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
        - Weak DPs:
            min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
            min#(s(X),s(Y)) -> c_3(min#(X,Y))
            plus#(s(X),Y) -> c_5(plus#(X,Y))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
             -->_2 min#(s(X),s(Y)) -> c_3(min#(X,Y)):3
             -->_2 min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z())):2
             -->_1 quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y)):1
          
          2:W:min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
             -->_2 plus#(s(X),Y) -> c_5(plus#(X,Y)):4
             -->_1 min#(s(X),s(Y)) -> c_3(min#(X,Y)):3
             -->_1 min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z())):2
          
          3:W:min#(s(X),s(Y)) -> c_3(min#(X,Y))
             -->_1 min#(s(X),s(Y)) -> c_3(min#(X,Y)):3
             -->_1 min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z())):2
          
          4:W:plus#(s(X),Y) -> c_5(plus#(X,Y))
             -->_1 plus#(s(X),Y) -> c_5(plus#(X,Y)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: min#(s(X),s(Y)) -> c_3(min#(X,Y))
          2: min#(min(X,Y),Z()) -> c_2(min#(X,plus(Y,Z())),plus#(Y,Z()))
          4: plus#(s(X),Y) -> c_5(plus#(X,Y))
** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y))
             -->_1 quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)),min#(X,Y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)))
** Step 5.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,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: quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)))
          
        The strictly oriented rules are moved into the weak component.
*** Step 5.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,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_7) = {1}
        
        Following symbols are considered usable:
          {min,min#,plus#,quot#}
        TcT has computed the following interpretation:
              p(0) = [0]                  
              p(Z) = [3]                  
            p(min) = [1] x1 + [0]         
           p(plus) = [3] x2 + [12]        
           p(quot) = [1] x1 + [1] x2 + [1]
              p(s) = [1] x1 + [8]         
           p(min#) = [1] x1 + [1] x2 + [2]
          p(plus#) = [1] x1 + [1]         
          p(quot#) = [1] x1 + [2] x2 + [0]
            p(c_1) = [1]                  
            p(c_2) = [8] x1 + [1] x2 + [8]
            p(c_3) = [1] x1 + [8]         
            p(c_4) = [1]                  
            p(c_5) = [2] x1 + [2]         
            p(c_6) = [1]                  
            p(c_7) = [1] x1 + [7]         
        
        Following rules are strictly oriented:
        quot#(s(X),s(Y)) = [1] X + [2] Y + [24]     
                         > [1] X + [2] Y + [23]     
                         = c_7(quot#(min(X,Y),s(Y)))
        
        
        Following rules are (at-least) weakly oriented:
               min(X,0()) =  [1] X + [0]       
                          >= [1] X + [0]       
                          =  X                 
        
        min(min(X,Y),Z()) =  [1] X + [0]       
                          >= [1] X + [0]       
                          =  min(X,plus(Y,Z()))
        
           min(s(X),s(Y)) =  [1] X + [8]       
                          >= [1] X + [0]       
                          =  min(X,Y)          
        
*** Step 5.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            quot#(s(X),s(Y)) -> c_7(quot#(min(X,Y),s(Y)))
        - Weak TRS:
            min(X,0()) -> X
            min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
            min(s(X),s(Y)) -> min(X,Y)
            plus(0(),Y) -> Y
            plus(s(X),Y) -> s(plus(X,Y))
        - Signature:
            {min/2,plus/2,quot/2,min#/2,plus#/2,quot#/2} / {0/0,Z/0,s/1,c_1/0,c_2/2,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {min#,plus#,quot#} and constructors {0,Z,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

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

MAYBE