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: Failure 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:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE