MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            del(x,cons(y,z)) -> if2(eq(x,y),x,y,z)
            del(x,nil()) -> nil()
            eq(0(),0()) -> true()
            eq(0(),s(y)) -> false()
            eq(s(x),0()) -> false()
            eq(s(x),s(y)) -> eq(x,y)
            if1(false(),x,y,xs) -> min(y,xs)
            if1(true(),x,y,xs) -> min(x,xs)
            if2(false(),x,y,xs) -> cons(y,del(x,xs))
            if2(true(),x,y,xs) -> xs
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            min(x,cons(y,z)) -> if1(le(x,y),x,y,z)
            min(x,nil()) -> x
            minsort(cons(x,y)) -> cons(min(x,y),minsort(del(min(x,y),cons(x,y))))
            minsort(nil()) -> nil()
        - Signature:
            {del/2,eq/2,if1/4,if2/4,le/2,min/2,minsort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {del,eq,if1,if2,le,min,minsort} and constructors {0,cons
            ,false,nil,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          del#(x,cons(y,z)) -> c_1(if2#(eq(x,y),x,y,z),eq#(x,y))
          del#(x,nil()) -> c_2()
          eq#(0(),0()) -> c_3()
          eq#(0(),s(y)) -> c_4()
          eq#(s(x),0()) -> c_5()
          eq#(s(x),s(y)) -> c_6(eq#(x,y))
          if1#(false(),x,y,xs) -> c_7(min#(y,xs))
          if1#(true(),x,y,xs) -> c_8(min#(x,xs))
          if2#(false(),x,y,xs) -> c_9(del#(x,xs))
          if2#(true(),x,y,xs) -> c_10()
          le#(0(),y) -> c_11()
          le#(s(x),0()) -> c_12()
          le#(s(x),s(y)) -> c_13(le#(x,y))
          min#(x,cons(y,z)) -> c_14(if1#(le(x,y),x,y,z),le#(x,y))
          min#(x,nil()) -> c_15()
          minsort#(cons(x,y)) -> c_16(min#(x,y),minsort#(del(min(x,y),cons(x,y))),del#(min(x,y),cons(x,y)),min#(x,y))
          minsort#(nil()) -> c_17()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            del#(x,cons(y,z)) -> c_1(if2#(eq(x,y),x,y,z),eq#(x,y))
            del#(x,nil()) -> c_2()
            eq#(0(),0()) -> c_3()
            eq#(0(),s(y)) -> c_4()
            eq#(s(x),0()) -> c_5()
            eq#(s(x),s(y)) -> c_6(eq#(x,y))
            if1#(false(),x,y,xs) -> c_7(min#(y,xs))
            if1#(true(),x,y,xs) -> c_8(min#(x,xs))
            if2#(false(),x,y,xs) -> c_9(del#(x,xs))
            if2#(true(),x,y,xs) -> c_10()
            le#(0(),y) -> c_11()
            le#(s(x),0()) -> c_12()
            le#(s(x),s(y)) -> c_13(le#(x,y))
            min#(x,cons(y,z)) -> c_14(if1#(le(x,y),x,y,z),le#(x,y))
            min#(x,nil()) -> c_15()
            minsort#(cons(x,y)) -> c_16(min#(x,y),minsort#(del(min(x,y),cons(x,y))),del#(min(x,y),cons(x,y)),min#(x,y))
            minsort#(nil()) -> c_17()
        - Weak TRS:
            del(x,cons(y,z)) -> if2(eq(x,y),x,y,z)
            del(x,nil()) -> nil()
            eq(0(),0()) -> true()
            eq(0(),s(y)) -> false()
            eq(s(x),0()) -> false()
            eq(s(x),s(y)) -> eq(x,y)
            if1(false(),x,y,xs) -> min(y,xs)
            if1(true(),x,y,xs) -> min(x,xs)
            if2(false(),x,y,xs) -> cons(y,del(x,xs))
            if2(true(),x,y,xs) -> xs
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            min(x,cons(y,z)) -> if1(le(x,y),x,y,z)
            min(x,nil()) -> x
            minsort(cons(x,y)) -> cons(min(x,y),minsort(del(min(x,y),cons(x,y))))
            minsort(nil()) -> nil()
        - Signature:
            {del/2,eq/2,if1/4,if2/4,le/2,min/2,minsort/1,del#/2,eq#/2,if1#/4,if2#/4,le#/2,min#/2,minsort#/1} / {0/0
            ,cons/2,false/0,nil/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/1,c_9/1,c_10/0,c_11/0,c_12/0
            ,c_13/1,c_14/2,c_15/0,c_16/4,c_17/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {del#,eq#,if1#,if2#,le#,min#,minsort#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          del(x,cons(y,z)) -> if2(eq(x,y),x,y,z)
          del(x,nil()) -> nil()
          eq(0(),0()) -> true()
          eq(0(),s(y)) -> false()
          eq(s(x),0()) -> false()
          eq(s(x),s(y)) -> eq(x,y)
          if1(false(),x,y,xs) -> min(y,xs)
          if1(true(),x,y,xs) -> min(x,xs)
          if2(false(),x,y,xs) -> cons(y,del(x,xs))
          if2(true(),x,y,xs) -> xs
          le(0(),y) -> true()
          le(s(x),0()) -> false()
          le(s(x),s(y)) -> le(x,y)
          min(x,cons(y,z)) -> if1(le(x,y),x,y,z)
          min(x,nil()) -> x
          del#(x,cons(y,z)) -> c_1(if2#(eq(x,y),x,y,z),eq#(x,y))
          del#(x,nil()) -> c_2()
          eq#(0(),0()) -> c_3()
          eq#(0(),s(y)) -> c_4()
          eq#(s(x),0()) -> c_5()
          eq#(s(x),s(y)) -> c_6(eq#(x,y))
          if1#(false(),x,y,xs) -> c_7(min#(y,xs))
          if1#(true(),x,y,xs) -> c_8(min#(x,xs))
          if2#(false(),x,y,xs) -> c_9(del#(x,xs))
          if2#(true(),x,y,xs) -> c_10()
          le#(0(),y) -> c_11()
          le#(s(x),0()) -> c_12()
          le#(s(x),s(y)) -> c_13(le#(x,y))
          min#(x,cons(y,z)) -> c_14(if1#(le(x,y),x,y,z),le#(x,y))
          min#(x,nil()) -> c_15()
          minsort#(cons(x,y)) -> c_16(min#(x,y),minsort#(del(min(x,y),cons(x,y))),del#(min(x,y),cons(x,y)),min#(x,y))
          minsort#(nil()) -> c_17()
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            del#(x,cons(y,z)) -> c_1(if2#(eq(x,y),x,y,z),eq#(x,y))
            del#(x,nil()) -> c_2()
            eq#(0(),0()) -> c_3()
            eq#(0(),s(y)) -> c_4()
            eq#(s(x),0()) -> c_5()
            eq#(s(x),s(y)) -> c_6(eq#(x,y))
            if1#(false(),x,y,xs) -> c_7(min#(y,xs))
            if1#(true(),x,y,xs) -> c_8(min#(x,xs))
            if2#(false(),x,y,xs) -> c_9(del#(x,xs))
            if2#(true(),x,y,xs) -> c_10()
            le#(0(),y) -> c_11()
            le#(s(x),0()) -> c_12()
            le#(s(x),s(y)) -> c_13(le#(x,y))
            min#(x,cons(y,z)) -> c_14(if1#(le(x,y),x,y,z),le#(x,y))
            min#(x,nil()) -> c_15()
            minsort#(cons(x,y)) -> c_16(min#(x,y),minsort#(del(min(x,y),cons(x,y))),del#(min(x,y),cons(x,y)),min#(x,y))
            minsort#(nil()) -> c_17()
        - Weak TRS:
            del(x,cons(y,z)) -> if2(eq(x,y),x,y,z)
            del(x,nil()) -> nil()
            eq(0(),0()) -> true()
            eq(0(),s(y)) -> false()
            eq(s(x),0()) -> false()
            eq(s(x),s(y)) -> eq(x,y)
            if1(false(),x,y,xs) -> min(y,xs)
            if1(true(),x,y,xs) -> min(x,xs)
            if2(false(),x,y,xs) -> cons(y,del(x,xs))
            if2(true(),x,y,xs) -> xs
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            min(x,cons(y,z)) -> if1(le(x,y),x,y,z)
            min(x,nil()) -> x
        - Signature:
            {del/2,eq/2,if1/4,if2/4,le/2,min/2,minsort/1,del#/2,eq#/2,if1#/4,if2#/4,le#/2,min#/2,minsort#/1} / {0/0
            ,cons/2,false/0,nil/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/1,c_9/1,c_10/0,c_11/0,c_12/0
            ,c_13/1,c_14/2,c_15/0,c_16/4,c_17/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {del#,eq#,if1#,if2#,le#,min#,minsort#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,4,5,10,11,12,15,17}
        by application of
          Pre({2,3,4,5,10,11,12,15,17}) = {1,6,7,8,9,13,14,16}.
        Here rules are labelled as follows:
          1: del#(x,cons(y,z)) -> c_1(if2#(eq(x,y),x,y,z),eq#(x,y))
          2: del#(x,nil()) -> c_2()
          3: eq#(0(),0()) -> c_3()
          4: eq#(0(),s(y)) -> c_4()
          5: eq#(s(x),0()) -> c_5()
          6: eq#(s(x),s(y)) -> c_6(eq#(x,y))
          7: if1#(false(),x,y,xs) -> c_7(min#(y,xs))
          8: if1#(true(),x,y,xs) -> c_8(min#(x,xs))
          9: if2#(false(),x,y,xs) -> c_9(del#(x,xs))
          10: if2#(true(),x,y,xs) -> c_10()
          11: le#(0(),y) -> c_11()
          12: le#(s(x),0()) -> c_12()
          13: le#(s(x),s(y)) -> c_13(le#(x,y))
          14: min#(x,cons(y,z)) -> c_14(if1#(le(x,y),x,y,z),le#(x,y))
          15: min#(x,nil()) -> c_15()
          16: minsort#(cons(x,y)) -> c_16(min#(x,y)
                                         ,minsort#(del(min(x,y),cons(x,y)))
                                         ,del#(min(x,y),cons(x,y))
                                         ,min#(x,y))
          17: minsort#(nil()) -> c_17()
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            del#(x,cons(y,z)) -> c_1(if2#(eq(x,y),x,y,z),eq#(x,y))
            eq#(s(x),s(y)) -> c_6(eq#(x,y))
            if1#(false(),x,y,xs) -> c_7(min#(y,xs))
            if1#(true(),x,y,xs) -> c_8(min#(x,xs))
            if2#(false(),x,y,xs) -> c_9(del#(x,xs))
            le#(s(x),s(y)) -> c_13(le#(x,y))
            min#(x,cons(y,z)) -> c_14(if1#(le(x,y),x,y,z),le#(x,y))
            minsort#(cons(x,y)) -> c_16(min#(x,y),minsort#(del(min(x,y),cons(x,y))),del#(min(x,y),cons(x,y)),min#(x,y))
        - Weak DPs:
            del#(x,nil()) -> c_2()
            eq#(0(),0()) -> c_3()
            eq#(0(),s(y)) -> c_4()
            eq#(s(x),0()) -> c_5()
            if2#(true(),x,y,xs) -> c_10()
            le#(0(),y) -> c_11()
            le#(s(x),0()) -> c_12()
            min#(x,nil()) -> c_15()
            minsort#(nil()) -> c_17()
        - Weak TRS:
            del(x,cons(y,z)) -> if2(eq(x,y),x,y,z)
            del(x,nil()) -> nil()
            eq(0(),0()) -> true()
            eq(0(),s(y)) -> false()
            eq(s(x),0()) -> false()
            eq(s(x),s(y)) -> eq(x,y)
            if1(false(),x,y,xs) -> min(y,xs)
            if1(true(),x,y,xs) -> min(x,xs)
            if2(false(),x,y,xs) -> cons(y,del(x,xs))
            if2(true(),x,y,xs) -> xs
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            min(x,cons(y,z)) -> if1(le(x,y),x,y,z)
            min(x,nil()) -> x
        - Signature:
            {del/2,eq/2,if1/4,if2/4,le/2,min/2,minsort/1,del#/2,eq#/2,if1#/4,if2#/4,le#/2,min#/2,minsort#/1} / {0/0
            ,cons/2,false/0,nil/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/1,c_9/1,c_10/0,c_11/0,c_12/0
            ,c_13/1,c_14/2,c_15/0,c_16/4,c_17/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {del#,eq#,if1#,if2#,le#,min#,minsort#} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:del#(x,cons(y,z)) -> c_1(if2#(eq(x,y),x,y,z),eq#(x,y))
             -->_1 if2#(false(),x,y,xs) -> c_9(del#(x,xs)):5
             -->_2 eq#(s(x),s(y)) -> c_6(eq#(x,y)):2
             -->_1 if2#(true(),x,y,xs) -> c_10():13
             -->_2 eq#(s(x),0()) -> c_5():12
             -->_2 eq#(0(),s(y)) -> c_4():11
             -->_2 eq#(0(),0()) -> c_3():10
          
          2:S:eq#(s(x),s(y)) -> c_6(eq#(x,y))
             -->_1 eq#(s(x),0()) -> c_5():12
             -->_1 eq#(0(),s(y)) -> c_4():11
             -->_1 eq#(0(),0()) -> c_3():10
             -->_1 eq#(s(x),s(y)) -> c_6(eq#(x,y)):2
          
          3:S:if1#(false(),x,y,xs) -> c_7(min#(y,xs))
             -->_1 min#(x,cons(y,z)) -> c_14(if1#(le(x,y),x,y,z),le#(x,y)):7
             -->_1 min#(x,nil()) -> c_15():16
          
          4:S:if1#(true(),x,y,xs) -> c_8(min#(x,xs))
             -->_1 min#(x,cons(y,z)) -> c_14(if1#(le(x,y),x,y,z),le#(x,y)):7
             -->_1 min#(x,nil()) -> c_15():16
          
          5:S:if2#(false(),x,y,xs) -> c_9(del#(x,xs))
             -->_1 del#(x,nil()) -> c_2():9
             -->_1 del#(x,cons(y,z)) -> c_1(if2#(eq(x,y),x,y,z),eq#(x,y)):1
          
          6:S:le#(s(x),s(y)) -> c_13(le#(x,y))
             -->_1 le#(s(x),0()) -> c_12():15
             -->_1 le#(0(),y) -> c_11():14
             -->_1 le#(s(x),s(y)) -> c_13(le#(x,y)):6
          
          7:S:min#(x,cons(y,z)) -> c_14(if1#(le(x,y),x,y,z),le#(x,y))
             -->_2 le#(s(x),0()) -> c_12():15
             -->_2 le#(0(),y) -> c_11():14
             -->_2 le#(s(x),s(y)) -> c_13(le#(x,y)):6
             -->_1 if1#(true(),x,y,xs) -> c_8(min#(x,xs)):4
             -->_1 if1#(false(),x,y,xs) -> c_7(min#(y,xs)):3
          
          8:S:minsort#(cons(x,y)) -> c_16(min#(x,y)
                                         ,minsort#(del(min(x,y),cons(x,y)))
                                         ,del#(min(x,y),cons(x,y))
                                         ,min#(x,y))
             -->_2 minsort#(nil()) -> c_17():17
             -->_4 min#(x,nil()) -> c_15():16
             -->_1 min#(x,nil()) -> c_15():16
             -->_2 minsort#(cons(x,y)) -> c_16(min#(x,y)
                                              ,minsort#(del(min(x,y),cons(x,y)))
                                              ,del#(min(x,y),cons(x,y))
                                              ,min#(x,y)):8
             -->_4 min#(x,cons(y,z)) -> c_14(if1#(le(x,y),x,y,z),le#(x,y)):7
             -->_1 min#(x,cons(y,z)) -> c_14(if1#(le(x,y),x,y,z),le#(x,y)):7
             -->_3 del#(x,cons(y,z)) -> c_1(if2#(eq(x,y),x,y,z),eq#(x,y)):1
          
          9:W:del#(x,nil()) -> c_2()
             
          
          10:W:eq#(0(),0()) -> c_3()
             
          
          11:W:eq#(0(),s(y)) -> c_4()
             
          
          12:W:eq#(s(x),0()) -> c_5()
             
          
          13:W:if2#(true(),x,y,xs) -> c_10()
             
          
          14:W:le#(0(),y) -> c_11()
             
          
          15:W:le#(s(x),0()) -> c_12()
             
          
          16:W:min#(x,nil()) -> c_15()
             
          
          17:W:minsort#(nil()) -> c_17()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          17: minsort#(nil()) -> c_17()
          16: min#(x,nil()) -> c_15()
          14: le#(0(),y) -> c_11()
          15: le#(s(x),0()) -> c_12()
          13: if2#(true(),x,y,xs) -> c_10()
          10: eq#(0(),0()) -> c_3()
          11: eq#(0(),s(y)) -> c_4()
          12: eq#(s(x),0()) -> c_5()
          9: del#(x,nil()) -> c_2()
* Step 5: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          del#(x,cons(y,z)) -> c_1(if2#(eq(x,y),x,y,z),eq#(x,y))
          eq#(s(x),s(y)) -> c_6(eq#(x,y))
          if1#(false(),x,y,xs) -> c_7(min#(y,xs))
          if1#(true(),x,y,xs) -> c_8(min#(x,xs))
          if2#(false(),x,y,xs) -> c_9(del#(x,xs))
          le#(s(x),s(y)) -> c_13(le#(x,y))
          min#(x,cons(y,z)) -> c_14(if1#(le(x,y),x,y,z),le#(x,y))
          minsort#(cons(x,y)) -> c_16(min#(x,y),minsort#(del(min(x,y),cons(x,y))),del#(min(x,y),cons(x,y)),min#(x,y))
      - Weak TRS:
          del(x,cons(y,z)) -> if2(eq(x,y),x,y,z)
          del(x,nil()) -> nil()
          eq(0(),0()) -> true()
          eq(0(),s(y)) -> false()
          eq(s(x),0()) -> false()
          eq(s(x),s(y)) -> eq(x,y)
          if1(false(),x,y,xs) -> min(y,xs)
          if1(true(),x,y,xs) -> min(x,xs)
          if2(false(),x,y,xs) -> cons(y,del(x,xs))
          if2(true(),x,y,xs) -> xs
          le(0(),y) -> true()
          le(s(x),0()) -> false()
          le(s(x),s(y)) -> le(x,y)
          min(x,cons(y,z)) -> if1(le(x,y),x,y,z)
          min(x,nil()) -> x
      - Signature:
          {del/2,eq/2,if1/4,if2/4,le/2,min/2,minsort/1,del#/2,eq#/2,if1#/4,if2#/4,le#/2,min#/2,minsort#/1} / {0/0
          ,cons/2,false/0,nil/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/0,c_6/1,c_7/1,c_8/1,c_9/1,c_10/0,c_11/0,c_12/0
          ,c_13/1,c_14/2,c_15/0,c_16/4,c_17/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {del#,eq#,if1#,if2#,le#,min#,minsort#} and constructors {0
          ,cons,false,nil,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE