* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
    + Considered Problem:
        - Strict TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
            sort(cons(n,x)) -> cons(min(cons(n,x)),sort(replace(min(cons(n,x)),n,x)))
            sort(nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq,if_min,if_replace,le,min,replace
            ,sort} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
            sort(cons(n,x)) -> cons(min(cons(n,x)),sort(replace(min(cons(n,x)),n,x)))
            sort(nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq,if_min,if_replace,le,min,replace
            ,sort} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          eq(x,y){x -> s(x),y -> s(y)} =
            eq(s(x),s(y)) ->^+ eq(x,y)
              = C[eq(x,y) = eq(x,y){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
            sort(cons(n,x)) -> cons(min(cons(n,x)),sort(replace(min(cons(n,x)),n,x)))
            sort(nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq,if_min,if_replace,le,min,replace
            ,sort} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          eq#(0(),0()) -> c_1()
          eq#(0(),s(m)) -> c_2()
          eq#(s(n),0()) -> c_3()
          eq#(s(n),s(m)) -> c_4(eq#(n,m))
          if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
          if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
          if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
          if_replace#(true(),n,m,cons(k,x)) -> c_8()
          le#(0(),m) -> c_9()
          le#(s(n),0()) -> c_10()
          le#(s(n),s(m)) -> c_11(le#(n,m))
          min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
          min#(cons(0(),nil())) -> c_13()
          min#(cons(s(n),nil())) -> c_14()
          replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
          replace#(n,m,nil()) -> c_16()
          sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                  ,sort#(replace(min(cons(n,x)),n,x))
                                  ,replace#(min(cons(n,x)),n,x)
                                  ,min#(cons(n,x)))
          sort#(nil()) -> c_18()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            eq#(0(),0()) -> c_1()
            eq#(0(),s(m)) -> c_2()
            eq#(s(n),0()) -> c_3()
            eq#(s(n),s(m)) -> c_4(eq#(n,m))
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            if_replace#(true(),n,m,cons(k,x)) -> c_8()
            le#(0(),m) -> c_9()
            le#(s(n),0()) -> c_10()
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
            min#(cons(0(),nil())) -> c_13()
            min#(cons(s(n),nil())) -> c_14()
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
            replace#(n,m,nil()) -> c_16()
            sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                    ,sort#(replace(min(cons(n,x)),n,x))
                                    ,replace#(min(cons(n,x)),n,x)
                                    ,min#(cons(n,x)))
            sort#(nil()) -> c_18()
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
            sort(cons(n,x)) -> cons(min(cons(n,x)),sort(replace(min(cons(n,x)),n,x)))
            sort(nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          eq(0(),0()) -> true()
          eq(0(),s(m)) -> false()
          eq(s(n),0()) -> false()
          eq(s(n),s(m)) -> eq(n,m)
          if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
          if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
          if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
          if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
          le(0(),m) -> true()
          le(s(n),0()) -> false()
          le(s(n),s(m)) -> le(n,m)
          min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
          min(cons(0(),nil())) -> 0()
          min(cons(s(n),nil())) -> s(n)
          replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
          replace(n,m,nil()) -> nil()
          eq#(0(),0()) -> c_1()
          eq#(0(),s(m)) -> c_2()
          eq#(s(n),0()) -> c_3()
          eq#(s(n),s(m)) -> c_4(eq#(n,m))
          if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
          if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
          if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
          if_replace#(true(),n,m,cons(k,x)) -> c_8()
          le#(0(),m) -> c_9()
          le#(s(n),0()) -> c_10()
          le#(s(n),s(m)) -> c_11(le#(n,m))
          min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
          min#(cons(0(),nil())) -> c_13()
          min#(cons(s(n),nil())) -> c_14()
          replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
          replace#(n,m,nil()) -> c_16()
          sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                  ,sort#(replace(min(cons(n,x)),n,x))
                                  ,replace#(min(cons(n,x)),n,x)
                                  ,min#(cons(n,x)))
          sort#(nil()) -> c_18()
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            eq#(0(),0()) -> c_1()
            eq#(0(),s(m)) -> c_2()
            eq#(s(n),0()) -> c_3()
            eq#(s(n),s(m)) -> c_4(eq#(n,m))
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            if_replace#(true(),n,m,cons(k,x)) -> c_8()
            le#(0(),m) -> c_9()
            le#(s(n),0()) -> c_10()
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
            min#(cons(0(),nil())) -> c_13()
            min#(cons(s(n),nil())) -> c_14()
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
            replace#(n,m,nil()) -> c_16()
            sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                    ,sort#(replace(min(cons(n,x)),n,x))
                                    ,replace#(min(cons(n,x)),n,x)
                                    ,min#(cons(n,x)))
            sort#(nil()) -> c_18()
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} 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
          {1,2,3,8,9,10,13,14,16,18}
        by application of
          Pre({1,2,3,8,9,10,13,14,16,18}) = {4,5,6,7,11,12,15,17}.
        Here rules are labelled as follows:
          1: eq#(0(),0()) -> c_1()
          2: eq#(0(),s(m)) -> c_2()
          3: eq#(s(n),0()) -> c_3()
          4: eq#(s(n),s(m)) -> c_4(eq#(n,m))
          5: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
          6: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
          7: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
          8: if_replace#(true(),n,m,cons(k,x)) -> c_8()
          9: le#(0(),m) -> c_9()
          10: le#(s(n),0()) -> c_10()
          11: le#(s(n),s(m)) -> c_11(le#(n,m))
          12: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
          13: min#(cons(0(),nil())) -> c_13()
          14: min#(cons(s(n),nil())) -> c_14()
          15: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
          16: replace#(n,m,nil()) -> c_16()
          17: sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                      ,sort#(replace(min(cons(n,x)),n,x))
                                      ,replace#(min(cons(n,x)),n,x)
                                      ,min#(cons(n,x)))
          18: sort#(nil()) -> c_18()
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            eq#(s(n),s(m)) -> c_4(eq#(n,m))
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
            sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                    ,sort#(replace(min(cons(n,x)),n,x))
                                    ,replace#(min(cons(n,x)),n,x)
                                    ,min#(cons(n,x)))
        - Weak DPs:
            eq#(0(),0()) -> c_1()
            eq#(0(),s(m)) -> c_2()
            eq#(s(n),0()) -> c_3()
            if_replace#(true(),n,m,cons(k,x)) -> c_8()
            le#(0(),m) -> c_9()
            le#(s(n),0()) -> c_10()
            min#(cons(0(),nil())) -> c_13()
            min#(cons(s(n),nil())) -> c_14()
            replace#(n,m,nil()) -> c_16()
            sort#(nil()) -> c_18()
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:eq#(s(n),s(m)) -> c_4(eq#(n,m))
             -->_1 eq#(s(n),0()) -> c_3():11
             -->_1 eq#(0(),s(m)) -> c_2():10
             -->_1 eq#(0(),0()) -> c_1():9
             -->_1 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1
          
          2:S:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6
             -->_1 min#(cons(s(n),nil())) -> c_14():16
             -->_1 min#(cons(0(),nil())) -> c_13():15
          
          3:S:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6
             -->_1 min#(cons(s(n),nil())) -> c_14():16
             -->_1 min#(cons(0(),nil())) -> c_13():15
          
          4:S:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
             -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):7
             -->_1 replace#(n,m,nil()) -> c_16():17
          
          5:S:le#(s(n),s(m)) -> c_11(le#(n,m))
             -->_1 le#(s(n),0()) -> c_10():14
             -->_1 le#(0(),m) -> c_9():13
             -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):5
          
          6:S:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
             -->_2 le#(s(n),0()) -> c_10():14
             -->_2 le#(0(),m) -> c_9():13
             -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):5
             -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):3
             -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):2
          
          7:S:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
             -->_1 if_replace#(true(),n,m,cons(k,x)) -> c_8():12
             -->_2 eq#(s(n),0()) -> c_3():11
             -->_2 eq#(0(),s(m)) -> c_2():10
             -->_2 eq#(0(),0()) -> c_1():9
             -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):4
             -->_2 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1
          
          8:S:sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                      ,sort#(replace(min(cons(n,x)),n,x))
                                      ,replace#(min(cons(n,x)),n,x)
                                      ,min#(cons(n,x)))
             -->_2 sort#(nil()) -> c_18():18
             -->_3 replace#(n,m,nil()) -> c_16():17
             -->_4 min#(cons(s(n),nil())) -> c_14():16
             -->_1 min#(cons(s(n),nil())) -> c_14():16
             -->_4 min#(cons(0(),nil())) -> c_13():15
             -->_1 min#(cons(0(),nil())) -> c_13():15
             -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                           ,sort#(replace(min(cons(n,x)),n,x))
                                           ,replace#(min(cons(n,x)),n,x)
                                           ,min#(cons(n,x))):8
             -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):7
             -->_4 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6
          
          9:W:eq#(0(),0()) -> c_1()
             
          
          10:W:eq#(0(),s(m)) -> c_2()
             
          
          11:W:eq#(s(n),0()) -> c_3()
             
          
          12:W:if_replace#(true(),n,m,cons(k,x)) -> c_8()
             
          
          13:W:le#(0(),m) -> c_9()
             
          
          14:W:le#(s(n),0()) -> c_10()
             
          
          15:W:min#(cons(0(),nil())) -> c_13()
             
          
          16:W:min#(cons(s(n),nil())) -> c_14()
             
          
          17:W:replace#(n,m,nil()) -> c_16()
             
          
          18:W:sort#(nil()) -> c_18()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          18: sort#(nil()) -> c_18()
          17: replace#(n,m,nil()) -> c_16()
          12: if_replace#(true(),n,m,cons(k,x)) -> c_8()
          15: min#(cons(0(),nil())) -> c_13()
          16: min#(cons(s(n),nil())) -> c_14()
          13: le#(0(),m) -> c_9()
          14: le#(s(n),0()) -> c_10()
          9: eq#(0(),0()) -> c_1()
          10: eq#(0(),s(m)) -> c_2()
          11: eq#(s(n),0()) -> c_3()
** Step 1.b:5: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            eq#(s(n),s(m)) -> c_4(eq#(n,m))
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
            sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                    ,sort#(replace(min(cons(n,x)),n,x))
                                    ,replace#(min(cons(n,x)),n,x)
                                    ,min#(cons(n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + 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:
              eq#(s(n),s(m)) -> c_4(eq#(n,m))
          - Weak DPs:
              if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
              if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
              if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
              le#(s(n),s(m)) -> c_11(le#(n,m))
              min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
              replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
              sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                      ,sort#(replace(min(cons(n,x)),n,x))
                                      ,replace#(min(cons(n,x)),n,x)
                                      ,min#(cons(n,x)))
          - Weak TRS:
              eq(0(),0()) -> true()
              eq(0(),s(m)) -> false()
              eq(s(n),0()) -> false()
              eq(s(n),s(m)) -> eq(n,m)
              if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
              if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
              if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
              if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
              le(0(),m) -> true()
              le(s(n),0()) -> false()
              le(s(n),s(m)) -> le(n,m)
              min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
              min(cons(0(),nil())) -> 0()
              min(cons(s(n),nil())) -> s(n)
              replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
              replace(n,m,nil()) -> nil()
          - Signature:
              {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
              ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1
              ,c_8/0,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
              ,sort#} and constructors {0,cons,false,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
              if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
              if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
              le#(s(n),s(m)) -> c_11(le#(n,m))
              min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
              replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
              sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                      ,sort#(replace(min(cons(n,x)),n,x))
                                      ,replace#(min(cons(n,x)),n,x)
                                      ,min#(cons(n,x)))
          - Weak DPs:
              eq#(s(n),s(m)) -> c_4(eq#(n,m))
          - Weak TRS:
              eq(0(),0()) -> true()
              eq(0(),s(m)) -> false()
              eq(s(n),0()) -> false()
              eq(s(n),s(m)) -> eq(n,m)
              if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
              if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
              if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
              if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
              le(0(),m) -> true()
              le(s(n),0()) -> false()
              le(s(n),s(m)) -> le(n,m)
              min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
              min(cons(0(),nil())) -> 0()
              min(cons(s(n),nil())) -> s(n)
              replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
              replace(n,m,nil()) -> nil()
          - Signature:
              {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
              ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1
              ,c_8/0,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
              ,sort#} and constructors {0,cons,false,nil,s,true}
*** Step 1.b:5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            eq#(s(n),s(m)) -> c_4(eq#(n,m))
        - Weak DPs:
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
            sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                    ,sort#(replace(min(cons(n,x)),n,x))
                                    ,replace#(min(cons(n,x)),n,x)
                                    ,min#(cons(n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:eq#(s(n),s(m)) -> c_4(eq#(n,m))
             -->_1 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1
          
          2:W:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6
          
          3:W:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6
          
          4:W:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
             -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):7
          
          5:W:le#(s(n),s(m)) -> c_11(le#(n,m))
             -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):5
          
          6:W:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
             -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):2
             -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):3
             -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):5
          
          7:W:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
             -->_2 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1
             -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):4
          
          8:W:sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                      ,sort#(replace(min(cons(n,x)),n,x))
                                      ,replace#(min(cons(n,x)),n,x)
                                      ,min#(cons(n,x)))
             -->_4 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6
             -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):7
             -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                           ,sort#(replace(min(cons(n,x)),n,x))
                                           ,replace#(min(cons(n,x)),n,x)
                                           ,min#(cons(n,x))):8
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
          6: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
          3: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
          5: le#(s(n),s(m)) -> c_11(le#(n,m))
*** Step 1.b:5.a:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            eq#(s(n),s(m)) -> c_4(eq#(n,m))
        - Weak DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
            sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                    ,sort#(replace(min(cons(n,x)),n,x))
                                    ,replace#(min(cons(n,x)),n,x)
                                    ,min#(cons(n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:eq#(s(n),s(m)) -> c_4(eq#(n,m))
             -->_1 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1
          
          4:W:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
             -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):7
          
          7:W:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
             -->_2 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1
             -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):4
          
          8:W:sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                      ,sort#(replace(min(cons(n,x)),n,x))
                                      ,replace#(min(cons(n,x)),n,x)
                                      ,min#(cons(n,x)))
             -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):7
             -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                           ,sort#(replace(min(cons(n,x)),n,x))
                                           ,replace#(min(cons(n,x)),n,x)
                                           ,min#(cons(n,x))):8
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
*** Step 1.b:5.a:3: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            eq#(s(n),s(m)) -> c_4(eq#(n,m))
        - Weak DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + 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
          sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        and a lower component
          eq#(s(n),s(m)) -> c_4(eq#(n,m))
          if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
          replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
        Further, following extension rules are added to the lower component.
          sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x)
          sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x))
**** Step 1.b:5.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + 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: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
          
        The strictly oriented rules are moved into the weak component.
***** Step 1.b:5.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + 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_17) = {1,2}
        
        Following symbols are considered usable:
          {if_replace,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#}
        TcT has computed the following interpretation:
                    p(0) = [1]                  
                 p(cons) = [1] x1 + [1] x2 + [2]
                   p(eq) = [0]                  
                p(false) = [0]                  
               p(if_min) = [4] x1 + [2]         
           p(if_replace) = [1] x3 + [1] x4 + [1]
                   p(le) = [2] x1 + [2]         
                  p(min) = [2]                  
                  p(nil) = [5]                  
              p(replace) = [1] x2 + [1] x3 + [1]
                    p(s) = [4]                  
                 p(sort) = [4] x1 + [4]         
                 p(true) = [0]                  
                  p(eq#) = [4] x2 + [0]         
              p(if_min#) = [1] x1 + [4]         
          p(if_replace#) = [2] x1 + [1] x4 + [0]
                  p(le#) = [0]                  
                 p(min#) = [0]                  
             p(replace#) = [0]                  
                p(sort#) = [4] x1 + [4]         
                  p(c_1) = [0]                  
                  p(c_2) = [1]                  
                  p(c_3) = [0]                  
                  p(c_4) = [1] x1 + [0]         
                  p(c_5) = [0]                  
                  p(c_6) = [2]                  
                  p(c_7) = [1] x1 + [2]         
                  p(c_8) = [4]                  
                  p(c_9) = [1]                  
                 p(c_10) = [0]                  
                 p(c_11) = [1] x1 + [2]         
                 p(c_12) = [4]                  
                 p(c_13) = [0]                  
                 p(c_14) = [0]                  
                 p(c_15) = [0]                  
                 p(c_16) = [0]                  
                 p(c_17) = [1] x1 + [1] x2 + [0]
                 p(c_18) = [4]                  
        
        Following rules are strictly oriented:
        sort#(cons(n,x)) = [4] n + [4] x + [12]                                                 
                         > [4] n + [4] x + [8]                                                  
                         = c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        
        
        Following rules are (at-least) weakly oriented:
        if_replace(false(),n,m,cons(k,x)) =  [1] k + [1] m + [1] x + [3]      
                                          >= [1] k + [1] m + [1] x + [3]      
                                          =  cons(k,replace(n,m,x))           
        
         if_replace(true(),n,m,cons(k,x)) =  [1] k + [1] m + [1] x + [3]      
                                          >= [1] m + [1] x + [2]              
                                          =  cons(m,x)                        
        
                   replace(n,m,cons(k,x)) =  [1] k + [1] m + [1] x + [3]      
                                          >= [1] k + [1] m + [1] x + [3]      
                                          =  if_replace(eq(n,k),n,m,cons(k,x))
        
                       replace(n,m,nil()) =  [1] m + [6]                      
                                          >= [5]                              
                                          =  nil()                            
        
***** Step 1.b:5.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 1.b:5.a:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
             -->_1 sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
***** Step 1.b:5.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 1.b:5.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            eq#(s(n),s(m)) -> c_4(eq#(n,m))
        - Weak DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
            sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x)
            sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: eq#(s(n),s(m)) -> c_4(eq#(n,m))
          
        The strictly oriented rules are moved into the weak component.
***** Step 1.b:5.a:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            eq#(s(n),s(m)) -> c_4(eq#(n,m))
        - Weak DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
            sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x)
            sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        NaturalMI {miDimension = 2, 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 (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_4) = {1},
          uargs(c_7) = {1},
          uargs(c_15) = {1,2}
        
        Following symbols are considered usable:
          {if_replace,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#}
        TcT has computed the following interpretation:
                    p(0) = [0]                      
                           [0]                      
                 p(cons) = [0 0] x1 + [0 1] x2 + [0]
                           [0 1]      [0 1]      [2]
                   p(eq) = [0]                      
                           [0]                      
                p(false) = [0]                      
                           [0]                      
               p(if_min) = [0 1] x1 + [1]           
                           [1 1]      [2]           
           p(if_replace) = [0 1] x3 + [1 0] x4 + [2]
                           [0 1]      [0 1]      [2]
                   p(le) = [3 0] x1 + [2]           
                           [3 1]      [1]           
                  p(min) = [2]                      
                           [0]                      
                  p(nil) = [0]                      
                           [1]                      
              p(replace) = [0 3] x2 + [1 0] x3 + [2]
                           [0 1]      [0 1]      [2]
                    p(s) = [0 0] x1 + [1]           
                           [0 1]      [2]           
                 p(sort) = [0 2] x1 + [0]           
                           [1 0]      [1]           
                 p(true) = [0]                      
                           [0]                      
                  p(eq#) = [0 1] x2 + [0]           
                           [0 3]      [0]           
              p(if_min#) = [0]                      
                           [0]                      
          p(if_replace#) = [1 0] x4 + [0]           
                           [0 0]      [1]           
                  p(le#) = [0 0] x1 + [0 1] x2 + [2]
                           [1 2]      [1 2]      [0]
                 p(min#) = [2 2] x1 + [0]           
                           [0 0]      [0]           
             p(replace#) = [0 0] x2 + [0 1] x3 + [0]
                           [0 1]      [0 1]      [1]
                p(sort#) = [0 1] x1 + [3]           
                           [0 1]      [2]           
                  p(c_1) = [0]                      
                           [1]                      
                  p(c_2) = [1]                      
                           [0]                      
                  p(c_3) = [1]                      
                           [0]                      
                  p(c_4) = [1 0] x1 + [0]           
                           [0 0]      [0]           
                  p(c_5) = [2 1] x1 + [0]           
                           [1 0]      [0]           
                  p(c_6) = [0 2] x1 + [2]           
                           [2 0]      [2]           
                  p(c_7) = [1 0] x1 + [0]           
                           [0 0]      [0]           
                  p(c_8) = [0]                      
                           [0]                      
                  p(c_9) = [0]                      
                           [0]                      
                 p(c_10) = [0]                      
                           [0]                      
                 p(c_11) = [0 0] x1 + [1]           
                           [0 2]      [1]           
                 p(c_12) = [0 0] x2 + [0]           
                           [0 2]      [0]           
                 p(c_13) = [0]                      
                           [0]                      
                 p(c_14) = [2]                      
                           [2]                      
                 p(c_15) = [1 2] x1 + [1 0] x2 + [0]
                           [0 0]      [1 0]      [3]
                 p(c_16) = [1]                      
                           [0]                      
                 p(c_17) = [0 0] x1 + [0 1] x2 + [1]
                           [0 2]      [0 0]      [0]
                 p(c_18) = [1]                      
                           [0]                      
        
        Following rules are strictly oriented:
        eq#(s(n),s(m)) = [0 1] m + [2]
                         [0 3]     [6]
                       > [0 1] m + [0]
                         [0 0]     [0]
                       = c_4(eq#(n,m))
        
        
        Following rules are (at-least) weakly oriented:
        if_replace#(false(),n,m,cons(k,x)) =  [0 1] x + [0]                                    
                                              [0 0]     [1]                                    
                                           >= [0 1] x + [0]                                    
                                              [0 0]     [0]                                    
                                           =  c_7(replace#(n,m,x))                             
        
                   replace#(n,m,cons(k,x)) =  [0 1] k + [0 0] m + [0 1] x + [2]                
                                              [0 1]     [0 1]     [0 1]     [3]                
                                           >= [0 1] k + [0 1] x + [2]                          
                                              [0 1]     [0 0]     [3]                          
                                           =  c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
        
                          sort#(cons(n,x)) =  [0 1] n + [0 1] x + [5]                          
                                              [0 1]     [0 1]     [4]                          
                                           >= [0 0] n + [0 1] x + [0]                          
                                              [0 1]     [0 1]     [1]                          
                                           =  replace#(min(cons(n,x)),n,x)                     
        
                          sort#(cons(n,x)) =  [0 1] n + [0 1] x + [5]                          
                                              [0 1]     [0 1]     [4]                          
                                           >= [0 1] n + [0 1] x + [5]                          
                                              [0 1]     [0 1]     [4]                          
                                           =  sort#(replace(min(cons(n,x)),n,x))               
        
         if_replace(false(),n,m,cons(k,x)) =  [0 0] k + [0 1] m + [0 1] x + [2]                
                                              [0 1]     [0 1]     [0 1]     [4]                
                                           >= [0 0] k + [0 1] m + [0 1] x + [2]                
                                              [0 1]     [0 1]     [0 1]     [4]                
                                           =  cons(k,replace(n,m,x))                           
        
          if_replace(true(),n,m,cons(k,x)) =  [0 0] k + [0 1] m + [0 1] x + [2]                
                                              [0 1]     [0 1]     [0 1]     [4]                
                                           >= [0 0] m + [0 1] x + [0]                          
                                              [0 1]     [0 1]     [2]                          
                                           =  cons(m,x)                                        
        
                    replace(n,m,cons(k,x)) =  [0 0] k + [0 3] m + [0 1] x + [2]                
                                              [0 1]     [0 1]     [0 1]     [4]                
                                           >= [0 0] k + [0 1] m + [0 1] x + [2]                
                                              [0 1]     [0 1]     [0 1]     [4]                
                                           =  if_replace(eq(n,k),n,m,cons(k,x))                
        
                        replace(n,m,nil()) =  [0 3] m + [2]                                    
                                              [0 1]     [3]                                    
                                           >= [0]                                              
                                              [1]                                              
                                           =  nil()                                            
        
***** Step 1.b:5.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            eq#(s(n),s(m)) -> c_4(eq#(n,m))
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
            sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x)
            sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 1.b:5.a:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            eq#(s(n),s(m)) -> c_4(eq#(n,m))
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
            sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x)
            sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:eq#(s(n),s(m)) -> c_4(eq#(n,m))
             -->_1 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1
          
          2:W:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
             -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):3
          
          3:W:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
             -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):2
             -->_2 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1
          
          4:W:sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x)
             -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):3
          
          5:W:sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x))
             -->_1 sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)):5
             -->_1 sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x))
          4: sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x)
          2: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
          3: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
          1: eq#(s(n),s(m)) -> c_4(eq#(n,m))
***** Step 1.b:5.a:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
            sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                    ,sort#(replace(min(cons(n,x)),n,x))
                                    ,replace#(min(cons(n,x)),n,x)
                                    ,min#(cons(n,x)))
        - Weak DPs:
            eq#(s(n),s(m)) -> c_4(eq#(n,m))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
          
          2:S:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
          
          3:S:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
             -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):6
          
          4:S:le#(s(n),s(m)) -> c_11(le#(n,m))
             -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):4
          
          5:S:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
             -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):4
             -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):2
             -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):1
          
          6:S:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
             -->_2 eq#(s(n),s(m)) -> c_4(eq#(n,m)):8
             -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):3
          
          7:S:sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                      ,sort#(replace(min(cons(n,x)),n,x))
                                      ,replace#(min(cons(n,x)),n,x)
                                      ,min#(cons(n,x)))
             -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                           ,sort#(replace(min(cons(n,x)),n,x))
                                           ,replace#(min(cons(n,x)),n,x)
                                           ,min#(cons(n,x))):7
             -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):6
             -->_4 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
          
          8:W:eq#(s(n),s(m)) -> c_4(eq#(n,m))
             -->_1 eq#(s(n),s(m)) -> c_4(eq#(n,m)):8
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: eq#(s(n),s(m)) -> c_4(eq#(n,m))
*** Step 1.b:5.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
            sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                    ,sort#(replace(min(cons(n,x)),n,x))
                                    ,replace#(min(cons(n,x)),n,x)
                                    ,min#(cons(n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
          
          2:S:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
          
          3:S:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
             -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):6
          
          4:S:le#(s(n),s(m)) -> c_11(le#(n,m))
             -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):4
          
          5:S:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
             -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):4
             -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):2
             -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):1
          
          6:S:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k))
             -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):3
          
          7:S:sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                      ,sort#(replace(min(cons(n,x)),n,x))
                                      ,replace#(min(cons(n,x)),n,x)
                                      ,min#(cons(n,x)))
             -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                           ,sort#(replace(min(cons(n,x)),n,x))
                                           ,replace#(min(cons(n,x)),n,x)
                                           ,min#(cons(n,x))):7
             -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):6
             -->_4 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
*** Step 1.b:5.b:3: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
            sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                    ,sort#(replace(min(cons(n,x)),n,x))
                                    ,replace#(min(cons(n,x)),n,x)
                                    ,min#(cons(n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/4,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + 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:
              if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
              if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
              le#(s(n),s(m)) -> c_11(le#(n,m))
              min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
          - Weak DPs:
              if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
              replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
              sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                      ,sort#(replace(min(cons(n,x)),n,x))
                                      ,replace#(min(cons(n,x)),n,x)
                                      ,min#(cons(n,x)))
          - Weak TRS:
              eq(0(),0()) -> true()
              eq(0(),s(m)) -> false()
              eq(s(n),0()) -> false()
              eq(s(n),s(m)) -> eq(n,m)
              if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
              if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
              if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
              if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
              le(0(),m) -> true()
              le(s(n),0()) -> false()
              le(s(n),s(m)) -> le(n,m)
              min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
              min(cons(0(),nil())) -> 0()
              min(cons(s(n),nil())) -> s(n)
              replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
              replace(n,m,nil()) -> nil()
          - Signature:
              {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
              ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1
              ,c_8/0,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/4,c_18/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
              ,sort#} and constructors {0,cons,false,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
              replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
              sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                      ,sort#(replace(min(cons(n,x)),n,x))
                                      ,replace#(min(cons(n,x)),n,x)
                                      ,min#(cons(n,x)))
          - Weak DPs:
              if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
              if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
              le#(s(n),s(m)) -> c_11(le#(n,m))
              min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
          - Weak TRS:
              eq(0(),0()) -> true()
              eq(0(),s(m)) -> false()
              eq(s(n),0()) -> false()
              eq(s(n),s(m)) -> eq(n,m)
              if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
              if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
              if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
              if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
              le(0(),m) -> true()
              le(s(n),0()) -> false()
              le(s(n),s(m)) -> le(n,m)
              min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
              min(cons(0(),nil())) -> 0()
              min(cons(s(n),nil())) -> s(n)
              replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
              replace(n,m,nil()) -> nil()
          - Signature:
              {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
              ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1
              ,c_8/0,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/4,c_18/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
              ,sort#} and constructors {0,cons,false,nil,s,true}
**** Step 1.b:5.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
        - Weak DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
            sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                    ,sort#(replace(min(cons(n,x)),n,x))
                                    ,replace#(min(cons(n,x)),n,x)
                                    ,min#(cons(n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/4,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
          
          2:S:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
          
          3:W:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
             -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):6
          
          4:S:le#(s(n),s(m)) -> c_11(le#(n,m))
             -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):4
          
          5:S:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
             -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):1
             -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):2
             -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):4
          
          6:W:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
             -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):3
          
          7:W:sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                      ,sort#(replace(min(cons(n,x)),n,x))
                                      ,replace#(min(cons(n,x)),n,x)
                                      ,min#(cons(n,x)))
             -->_4 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
             -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):6
             -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                           ,sort#(replace(min(cons(n,x)),n,x))
                                           ,replace#(min(cons(n,x)),n,x)
                                           ,min#(cons(n,x))):7
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
          6: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
**** Step 1.b:5.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
        - Weak DPs:
            sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                    ,sort#(replace(min(cons(n,x)),n,x))
                                    ,replace#(min(cons(n,x)),n,x)
                                    ,min#(cons(n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/4,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
          
          2:S:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
          
          4:S:le#(s(n),s(m)) -> c_11(le#(n,m))
             -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):4
          
          5:S:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
             -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):1
             -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):2
             -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):4
          
          7:W:sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                      ,sort#(replace(min(cons(n,x)),n,x))
                                      ,replace#(min(cons(n,x)),n,x)
                                      ,min#(cons(n,x)))
             -->_4 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5
             -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                           ,sort#(replace(min(cons(n,x)),n,x))
                                           ,replace#(min(cons(n,x)),n,x)
                                           ,min#(cons(n,x))):7
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x)))
**** Step 1.b:5.b:3.a:3: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
        - Weak DPs:
            sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + 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
          sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x)))
        and a lower component
          if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
          if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
          le#(s(n),s(m)) -> c_11(le#(n,m))
          min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
        Further, following extension rules are added to the lower component.
          sort#(cons(n,x)) -> min#(cons(n,x))
          sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x))
***** Step 1.b:5.b:3.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + 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: sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x)))
          
        The strictly oriented rules are moved into the weak component.
****** Step 1.b:5.b:3.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + 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_17) = {2}
        
        Following symbols are considered usable:
          {if_replace,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#}
        TcT has computed the following interpretation:
                    p(0) = [2]                  
                 p(cons) = [1] x1 + [1] x2 + [2]
                   p(eq) = [0]                  
                p(false) = [0]                  
               p(if_min) = [1] x1 + [0]         
           p(if_replace) = [1] x3 + [1] x4 + [0]
                   p(le) = [5] x1 + [0]         
                  p(min) = [2]                  
                  p(nil) = [1]                  
              p(replace) = [1] x2 + [1] x3 + [0]
                    p(s) = [1]                  
                 p(sort) = [2] x1 + [1]         
                 p(true) = [0]                  
                  p(eq#) = [2] x1 + [2]         
              p(if_min#) = [4] x1 + [1]         
          p(if_replace#) = [1] x1 + [4] x3 + [0]
                  p(le#) = [0]                  
                 p(min#) = [2]                  
             p(replace#) = [1] x3 + [4]         
                p(sort#) = [4] x1 + [0]         
                  p(c_1) = [1]                  
                  p(c_2) = [0]                  
                  p(c_3) = [2]                  
                  p(c_4) = [0]                  
                  p(c_5) = [1] x1 + [0]         
                  p(c_6) = [4] x1 + [1]         
                  p(c_7) = [1]                  
                  p(c_8) = [0]                  
                  p(c_9) = [0]                  
                 p(c_10) = [0]                  
                 p(c_11) = [4]                  
                 p(c_12) = [1] x2 + [0]         
                 p(c_13) = [0]                  
                 p(c_14) = [1]                  
                 p(c_15) = [0]                  
                 p(c_16) = [1]                  
                 p(c_17) = [2] x1 + [1] x2 + [2]
                 p(c_18) = [2]                  
        
        Following rules are strictly oriented:
        sort#(cons(n,x)) = [4] n + [4] x + [8]                                                     
                         > [4] n + [4] x + [6]                                                     
                         = c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x)))
        
        
        Following rules are (at-least) weakly oriented:
        if_replace(false(),n,m,cons(k,x)) =  [1] k + [1] m + [1] x + [2]      
                                          >= [1] k + [1] m + [1] x + [2]      
                                          =  cons(k,replace(n,m,x))           
        
         if_replace(true(),n,m,cons(k,x)) =  [1] k + [1] m + [1] x + [2]      
                                          >= [1] m + [1] x + [2]              
                                          =  cons(m,x)                        
        
                   replace(n,m,cons(k,x)) =  [1] k + [1] m + [1] x + [2]      
                                          >= [1] k + [1] m + [1] x + [2]      
                                          =  if_replace(eq(n,k),n,m,cons(k,x))
        
                       replace(n,m,nil()) =  [1] m + [1]                      
                                          >= [1]                              
                                          =  nil()                            
        
****** Step 1.b:5.b:3.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 1.b:5.b:3.a:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x)))
             -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x)))
****** Step 1.b:5.b:3.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

***** Step 1.b:5.b:3.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
        - Weak DPs:
            sort#(cons(n,x)) -> min#(cons(n,x))
            sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          3: le#(s(n),s(m)) -> c_11(le#(n,m))
          4: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
          
        Consider the set of all dependency pairs
          1: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
          2: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
          3: le#(s(n),s(m)) -> c_11(le#(n,m))
          4: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
          5: sort#(cons(n,x)) -> min#(cons(n,x))
          6: sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x))
        Processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {3,4}
        These cover all (indirect) predecessors of dependency pairs
          {1,2,3,4}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
****** Step 1.b:5.b:3.a:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
        - Weak DPs:
            sort#(cons(n,x)) -> min#(cons(n,x))
            sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        NaturalMI {miDimension = 3, 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 (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_5) = {1},
          uargs(c_6) = {1},
          uargs(c_11) = {1},
          uargs(c_12) = {1,2}
        
        Following symbols are considered usable:
          {if_replace,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#}
        TcT has computed the following interpretation:
                    p(0) = [1]                          
                           [0]                          
                           [1]                          
                 p(cons) = [0 0 0]      [0 1 0]      [0]
                           [0 0 0] x1 + [0 0 1] x2 + [0]
                           [0 0 1]      [0 0 1]      [1]
                   p(eq) = [0 0 1]      [1 1 0]      [0]
                           [0 1 0] x1 + [0 1 0] x2 + [1]
                           [1 0 0]      [1 0 0]      [0]
                p(false) = [1]                          
                           [0]                          
                           [1]                          
               p(if_min) = [0 0 0]      [0 0 0]      [1]
                           [0 0 1] x1 + [0 1 0] x2 + [1]
                           [1 1 1]      [0 0 0]      [0]
           p(if_replace) = [1 1 1]      [1 0 1]      [1]
                           [0 1 1] x3 + [0 0 1] x4 + [0]
                           [0 0 1]      [0 0 1]      [0]
                   p(le) = [0 0 0]      [0 0 0]      [0]
                           [0 1 0] x1 + [0 1 0] x2 + [0]
                           [0 0 0]      [0 1 1]      [0]
                  p(min) = [0 0 0]      [0]             
                           [0 0 1] x1 + [0]             
                           [0 0 0]      [1]             
                  p(nil) = [1]                          
                           [1]                          
                           [1]                          
              p(replace) = [1 1 1]      [1 0 1]      [1]
                           [1 1 1] x2 + [0 0 1] x3 + [0]
                           [0 0 1]      [0 0 1]      [0]
                    p(s) = [0 0 1]      [0]             
                           [0 0 0] x1 + [1]             
                           [0 0 1]      [1]             
                 p(sort) = [0]                          
                           [0]                          
                           [0]                          
                 p(true) = [1]                          
                           [1]                          
                           [0]                          
                  p(eq#) = [0]                          
                           [0]                          
                           [0]                          
              p(if_min#) = [1 0 0]      [0]             
                           [1 0 0] x2 + [1]             
                           [0 0 0]      [0]             
          p(if_replace#) = [0]                          
                           [0]                          
                           [0]                          
                  p(le#) = [0 0 0]      [0 0 1]      [0]
                           [0 0 0] x1 + [0 1 0] x2 + [0]
                           [0 0 1]      [0 0 0]      [0]
                 p(min#) = [0 1 0]      [0]             
                           [0 0 1] x1 + [0]             
                           [0 0 0]      [0]             
             p(replace#) = [0]                          
                           [0]                          
                           [0]                          
                p(sort#) = [0 0 1]      [0]             
                           [0 0 1] x1 + [0]             
                           [0 0 0]      [0]             
                  p(c_1) = [0]                          
                           [0]                          
                           [0]                          
                  p(c_2) = [0]                          
                           [0]                          
                           [0]                          
                  p(c_3) = [0]                          
                           [0]                          
                           [0]                          
                  p(c_4) = [0]                          
                           [0]                          
                           [0]                          
                  p(c_5) = [1 0 0]      [0]             
                           [0 0 0] x1 + [1]             
                           [0 0 0]      [0]             
                  p(c_6) = [1 0 0]      [0]             
                           [0 0 0] x1 + [1]             
                           [0 0 0]      [0]             
                  p(c_7) = [0]                          
                           [0]                          
                           [0]                          
                  p(c_8) = [0]                          
                           [0]                          
                           [0]                          
                  p(c_9) = [0]                          
                           [0]                          
                           [0]                          
                 p(c_10) = [0]                          
                           [0]                          
                           [0]                          
                 p(c_11) = [1 0 0]      [0]             
                           [0 0 0] x1 + [1]             
                           [0 0 0]      [0]             
                 p(c_12) = [1 0 0]      [1 0 0]      [0]
                           [0 1 0] x1 + [0 0 1] x2 + [1]
                           [0 0 0]      [0 0 0]      [0]
                 p(c_13) = [0]                          
                           [0]                          
                           [0]                          
                 p(c_14) = [0]                          
                           [0]                          
                           [0]                          
                 p(c_15) = [0]                          
                           [0]                          
                           [0]                          
                 p(c_16) = [0]                          
                           [0]                          
                           [0]                          
                 p(c_17) = [0]                          
                           [0]                          
                           [0]                          
                 p(c_18) = [0]                          
                           [0]                          
                           [0]                          
        
        Following rules are strictly oriented:
                 le#(s(n),s(m)) = [0 0 1]     [0 0 0]     [1]                      
                                  [0 0 0] m + [0 0 0] n + [1]                      
                                  [0 0 0]     [0 0 1]     [1]                      
                                > [0 0 1]     [0]                                  
                                  [0 0 0] m + [1]                                  
                                  [0 0 0]     [0]                                  
                                = c_11(le#(n,m))                                   
        
        min#(cons(n,cons(m,x))) = [0 0 1]     [0 0 0]     [0 0 1]     [1]          
                                  [0 0 1] m + [0 0 1] n + [0 0 1] x + [2]          
                                  [0 0 0]     [0 0 0]     [0 0 0]     [0]          
                                > [0 0 1]     [0 0 0]     [0 0 1]     [0]          
                                  [0 0 0] m + [0 0 1] n + [0 0 1] x + [2]          
                                  [0 0 0]     [0 0 0]     [0 0 0]     [0]          
                                = c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
        
        
        Following rules are (at-least) weakly oriented:
        if_min#(false(),cons(n,cons(m,x))) =  [0 0 1]     [0]                        
                                              [0 0 1] x + [1]                        
                                              [0 0 0]     [0]                        
                                           >= [0 0 1]     [0]                        
                                              [0 0 0] x + [1]                        
                                              [0 0 0]     [0]                        
                                           =  c_5(min#(cons(m,x)))                   
        
         if_min#(true(),cons(n,cons(m,x))) =  [0 0 1]     [0]                        
                                              [0 0 1] x + [1]                        
                                              [0 0 0]     [0]                        
                                           >= [0 0 1]     [0]                        
                                              [0 0 0] x + [1]                        
                                              [0 0 0]     [0]                        
                                           =  c_6(min#(cons(n,x)))                   
        
                          sort#(cons(n,x)) =  [0 0 1]     [0 0 1]     [1]            
                                              [0 0 1] n + [0 0 1] x + [1]            
                                              [0 0 0]     [0 0 0]     [0]            
                                           >= [0 0 0]     [0 0 1]     [0]            
                                              [0 0 1] n + [0 0 1] x + [1]            
                                              [0 0 0]     [0 0 0]     [0]            
                                           =  min#(cons(n,x))                        
        
                          sort#(cons(n,x)) =  [0 0 1]     [0 0 1]     [1]            
                                              [0 0 1] n + [0 0 1] x + [1]            
                                              [0 0 0]     [0 0 0]     [0]            
                                           >= [0 0 1]     [0 0 1]     [0]            
                                              [0 0 1] n + [0 0 1] x + [0]            
                                              [0 0 0]     [0 0 0]     [0]            
                                           =  sort#(replace(min(cons(n,x)),n,x))     
        
         if_replace(false(),n,m,cons(k,x)) =  [0 0 1]     [1 1 1]     [0 1 1]     [2]
                                              [0 0 1] k + [0 1 1] m + [0 0 1] x + [1]
                                              [0 0 1]     [0 0 1]     [0 0 1]     [1]
                                           >= [0 0 0]     [1 1 1]     [0 0 1]     [0]
                                              [0 0 0] k + [0 0 1] m + [0 0 1] x + [0]
                                              [0 0 1]     [0 0 1]     [0 0 1]     [1]
                                           =  cons(k,replace(n,m,x))                 
        
          if_replace(true(),n,m,cons(k,x)) =  [0 0 1]     [1 1 1]     [0 1 1]     [2]
                                              [0 0 1] k + [0 1 1] m + [0 0 1] x + [1]
                                              [0 0 1]     [0 0 1]     [0 0 1]     [1]
                                           >= [0 0 0]     [0 1 0]     [0]            
                                              [0 0 0] m + [0 0 1] x + [0]            
                                              [0 0 1]     [0 0 1]     [1]            
                                           =  cons(m,x)                              
        
                    replace(n,m,cons(k,x)) =  [0 0 1]     [1 1 1]     [0 1 1]     [2]
                                              [0 0 1] k + [1 1 1] m + [0 0 1] x + [1]
                                              [0 0 1]     [0 0 1]     [0 0 1]     [1]
                                           >= [0 0 1]     [1 1 1]     [0 1 1]     [2]
                                              [0 0 1] k + [0 1 1] m + [0 0 1] x + [1]
                                              [0 0 1]     [0 0 1]     [0 0 1]     [1]
                                           =  if_replace(eq(n,k),n,m,cons(k,x))      
        
                        replace(n,m,nil()) =  [1 1 1]     [3]                        
                                              [1 1 1] m + [1]                        
                                              [0 0 1]     [1]                        
                                           >= [1]                                    
                                              [1]                                    
                                              [1]                                    
                                           =  nil()                                  
        
****** Step 1.b:5.b:3.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
        - Weak DPs:
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
            sort#(cons(n,x)) -> min#(cons(n,x))
            sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 1.b:5.b:3.a:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
            sort#(cons(n,x)) -> min#(cons(n,x))
            sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):4
          
          2:W:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):4
          
          3:W:le#(s(n),s(m)) -> c_11(le#(n,m))
             -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):3
          
          4:W:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
             -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):3
             -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):2
             -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):1
          
          5:W:sort#(cons(n,x)) -> min#(cons(n,x))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):4
          
          6:W:sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x))
             -->_1 sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)):6
             -->_1 sort#(cons(n,x)) -> min#(cons(n,x)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x))
          5: sort#(cons(n,x)) -> min#(cons(n,x))
          1: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
          4: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
          2: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
          3: le#(s(n),s(m)) -> c_11(le#(n,m))
****** Step 1.b:5.b:3.a:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 1.b:5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
            sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                    ,sort#(replace(min(cons(n,x)),n,x))
                                    ,replace#(min(cons(n,x)),n,x)
                                    ,min#(cons(n,x)))
        - Weak DPs:
            if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
            if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
            le#(s(n),s(m)) -> c_11(le#(n,m))
            min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/4,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
             -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):2
          
          2:S:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
             -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):1
          
          3:S:sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                      ,sort#(replace(min(cons(n,x)),n,x))
                                      ,replace#(min(cons(n,x)),n,x)
                                      ,min#(cons(n,x)))
             -->_4 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):7
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):7
             -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                           ,sort#(replace(min(cons(n,x)),n,x))
                                           ,replace#(min(cons(n,x)),n,x)
                                           ,min#(cons(n,x))):3
             -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):2
          
          4:W:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):7
          
          5:W:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
             -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):7
          
          6:W:le#(s(n),s(m)) -> c_11(le#(n,m))
             -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):6
          
          7:W:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
             -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):6
             -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):5
             -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m))
          5: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x)))
          4: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x)))
          6: le#(s(n),s(m)) -> c_11(le#(n,m))
**** Step 1.b:5.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
            sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                    ,sort#(replace(min(cons(n,x)),n,x))
                                    ,replace#(min(cons(n,x)),n,x)
                                    ,min#(cons(n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/4,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
             -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):2
          
          2:S:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
             -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):1
          
          3:S:sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                      ,sort#(replace(min(cons(n,x)),n,x))
                                      ,replace#(min(cons(n,x)),n,x)
                                      ,min#(cons(n,x)))
             -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x))
                                           ,sort#(replace(min(cons(n,x)),n,x))
                                           ,replace#(min(cons(n,x)),n,x)
                                           ,min#(cons(n,x))):3
             -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
**** Step 1.b:5.b:3.b:3: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + 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:
              if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
              replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
          - Weak DPs:
              sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
          - Weak TRS:
              eq(0(),0()) -> true()
              eq(0(),s(m)) -> false()
              eq(s(n),0()) -> false()
              eq(s(n),s(m)) -> eq(n,m)
              if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
              if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
              if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
              if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
              le(0(),m) -> true()
              le(s(n),0()) -> false()
              le(s(n),s(m)) -> le(n,m)
              min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
              min(cons(0(),nil())) -> 0()
              min(cons(s(n),nil())) -> s(n)
              replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
              replace(n,m,nil()) -> nil()
          - Signature:
              {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
              ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1
              ,c_8/0,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
              ,sort#} and constructors {0,cons,false,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
          - Weak DPs:
              if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
              replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
          - Weak TRS:
              eq(0(),0()) -> true()
              eq(0(),s(m)) -> false()
              eq(s(n),0()) -> false()
              eq(s(n),s(m)) -> eq(n,m)
              if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
              if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
              if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
              if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
              le(0(),m) -> true()
              le(s(n),0()) -> false()
              le(s(n),s(m)) -> le(n,m)
              min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
              min(cons(0(),nil())) -> 0()
              min(cons(s(n),nil())) -> s(n)
              replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
              replace(n,m,nil()) -> nil()
          - Signature:
              {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
              ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1
              ,c_8/0,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
              ,sort#} and constructors {0,cons,false,nil,s,true}
***** Step 1.b:5.b:3.b:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
        - Weak DPs:
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
          
        The strictly oriented rules are moved into the weak component.
****** Step 1.b:5.b:3.b:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
        - Weak DPs:
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 2, 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},
          uargs(c_15) = {1},
          uargs(c_17) = {1,2}
        
        Following symbols are considered usable:
          {if_replace,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#}
        TcT has computed the following interpretation:
                    p(0) = [2]                      
                           [0]                      
                 p(cons) = [1 1] x2 + [1]           
                           [0 1]      [2]           
                   p(eq) = [0]                      
                           [0]                      
                p(false) = [0]                      
                           [0]                      
               p(if_min) = [0 0] x1 + [0]           
                           [1 1]      [1]           
           p(if_replace) = [1 0] x4 + [0]           
                           [0 1]      [0]           
                   p(le) = [0 1] x1 + [2]           
                           [1 1]      [3]           
                  p(min) = [0]                      
                           [0]                      
                  p(nil) = [0]                      
                           [0]                      
              p(replace) = [1 0] x3 + [0]           
                           [0 1]      [0]           
                    p(s) = [1]                      
                           [0]                      
                 p(sort) = [1]                      
                           [0]                      
                 p(true) = [0]                      
                           [3]                      
                  p(eq#) = [1 0] x2 + [0]           
                           [0 0]      [0]           
              p(if_min#) = [1 1] x1 + [0 1] x2 + [0]
                           [2 1]      [2 0]      [2]
          p(if_replace#) = [0 0] x2 + [0 1] x4 + [0]
                           [1 1]      [1 2]      [1]
                  p(le#) = [0 0] x1 + [2 0] x2 + [0]
                           [0 1]      [0 2]      [0]
                 p(min#) = [0]                      
                           [0]                      
             p(replace#) = [0 1] x3 + [0]           
                           [0 0]      [1]           
                p(sort#) = [2 2] x1 + [1]           
                           [0 2]      [1]           
                  p(c_1) = [0]                      
                           [0]                      
                  p(c_2) = [0]                      
                           [0]                      
                  p(c_3) = [0]                      
                           [0]                      
                  p(c_4) = [0]                      
                           [0]                      
                  p(c_5) = [1]                      
                           [2]                      
                  p(c_6) = [1]                      
                           [0]                      
                  p(c_7) = [1 0] x1 + [0]           
                           [0 0]      [0]           
                  p(c_8) = [0]                      
                           [0]                      
                  p(c_9) = [0]                      
                           [0]                      
                 p(c_10) = [2]                      
                           [1]                      
                 p(c_11) = [2]                      
                           [1]                      
                 p(c_12) = [2 2] x1 + [1 1] x2 + [0]
                           [0 1]      [0 0]      [1]
                 p(c_13) = [0]                      
                           [1]                      
                 p(c_14) = [1]                      
                           [1]                      
                 p(c_15) = [1 0] x1 + [0]           
                           [0 0]      [1]           
                 p(c_16) = [0]                      
                           [1]                      
                 p(c_17) = [1 0] x1 + [2 3] x2 + [0]
                           [0 1]      [0 0]      [2]
                 p(c_18) = [0]                      
                           [2]                      
        
        Following rules are strictly oriented:
        if_replace#(false(),n,m,cons(k,x)) = [0 0] n + [0 1] x + [2]
                                             [1 1]     [1 3]     [6]
                                           > [0 1] x + [0]          
                                             [0 0]     [0]          
                                           = c_7(replace#(n,m,x))   
        
        
        Following rules are (at-least) weakly oriented:
                  replace#(n,m,cons(k,x)) =  [0 1] x + [2]                                                        
                                             [0 0]     [1]                                                        
                                          >= [0 1] x + [2]                                                        
                                             [0 0]     [1]                                                        
                                          =  c_15(if_replace#(eq(n,k),n,m,cons(k,x)))                             
        
                         sort#(cons(n,x)) =  [2 4] x + [7]                                                        
                                             [0 2]     [5]                                                        
                                          >= [2 4] x + [4]                                                        
                                             [0 2]     [3]                                                        
                                          =  c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        
        if_replace(false(),n,m,cons(k,x)) =  [1 1] x + [1]                                                        
                                             [0 1]     [2]                                                        
                                          >= [1 1] x + [1]                                                        
                                             [0 1]     [2]                                                        
                                          =  cons(k,replace(n,m,x))                                               
        
         if_replace(true(),n,m,cons(k,x)) =  [1 1] x + [1]                                                        
                                             [0 1]     [2]                                                        
                                          >= [1 1] x + [1]                                                        
                                             [0 1]     [2]                                                        
                                          =  cons(m,x)                                                            
        
                   replace(n,m,cons(k,x)) =  [1 1] x + [1]                                                        
                                             [0 1]     [2]                                                        
                                          >= [1 1] x + [1]                                                        
                                             [0 1]     [2]                                                        
                                          =  if_replace(eq(n,k),n,m,cons(k,x))                                    
        
                       replace(n,m,nil()) =  [0]                                                                  
                                             [0]                                                                  
                                          >= [0]                                                                  
                                             [0]                                                                  
                                          =  nil()                                                                
        
****** Step 1.b:5.b:3.b:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
        - Weak DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 1.b:5.b:3.b:3.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
        - Weak DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
          
        Consider the set of all dependency pairs
          1: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
          2: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
          3: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        Processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2))
        SPACE(?,?)on application of the dependency pairs
          {1}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
******* Step 1.b:5.b:3.b:3.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
        - Weak DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 2, 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},
          uargs(c_15) = {1},
          uargs(c_17) = {1,2}
        
        Following symbols are considered usable:
          {if_replace,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#}
        TcT has computed the following interpretation:
                    p(0) = [0]                      
                           [0]                      
                 p(cons) = [1 1] x2 + [2]           
                           [0 1]      [2]           
                   p(eq) = [0]                      
                           [0]                      
                p(false) = [0]                      
                           [0]                      
               p(if_min) = [0 2] x1 + [0]           
                           [0 2]      [2]           
           p(if_replace) = [1 0] x4 + [1]           
                           [0 1]      [0]           
                   p(le) = [2 2] x1 + [0 0] x2 + [0]
                           [0 3]      [0 1]      [2]
                  p(min) = [0]                      
                           [0]                      
                  p(nil) = [1]                      
                           [0]                      
              p(replace) = [1 0] x3 + [1]           
                           [0 1]      [0]           
                    p(s) = [1]                      
                           [0]                      
                 p(sort) = [0]                      
                           [0]                      
                 p(true) = [0]                      
                           [0]                      
                  p(eq#) = [0 0] x2 + [0]           
                           [0 2]      [0]           
              p(if_min#) = [0 0] x2 + [0]           
                           [1 2]      [2]           
          p(if_replace#) = [0 0] x2 + [0 2] x4 + [0]
                           [2 0]      [0 3]      [1]
                  p(le#) = [0 0] x2 + [0]           
                           [2 0]      [0]           
                 p(min#) = [0]                      
                           [2]                      
             p(replace#) = [0 2] x3 + [1]           
                           [0 0]      [2]           
                p(sort#) = [2 0] x1 + [3]           
                           [3 0]      [0]           
                  p(c_1) = [2]                      
                           [2]                      
                  p(c_2) = [0]                      
                           [2]                      
                  p(c_3) = [0]                      
                           [0]                      
                  p(c_4) = [0]                      
                           [0]                      
                  p(c_5) = [1]                      
                           [0]                      
                  p(c_6) = [0 0] x1 + [0]           
                           [2 0]      [0]           
                  p(c_7) = [1 1] x1 + [1]           
                           [0 0]      [0]           
                  p(c_8) = [0]                      
                           [0]                      
                  p(c_9) = [2]                      
                           [2]                      
                 p(c_10) = [0]                      
                           [0]                      
                 p(c_11) = [0 0] x1 + [1]           
                           [2 0]      [0]           
                 p(c_12) = [2 2] x1 + [0 0] x2 + [1]
                           [0 1]      [0 2]      [0]
                 p(c_13) = [0]                      
                           [0]                      
                 p(c_14) = [1]                      
                           [0]                      
                 p(c_15) = [1 0] x1 + [0]           
                           [0 0]      [1]           
                 p(c_16) = [0]                      
                           [0]                      
                 p(c_17) = [1 0] x1 + [1 0] x2 + [0]
                           [0 1]      [0 0]      [3]
                 p(c_18) = [0]                      
                           [0]                      
        
        Following rules are strictly oriented:
        replace#(n,m,cons(k,x)) = [0 2] x + [5]                           
                                  [0 0]     [2]                           
                                > [0 2] x + [4]                           
                                  [0 0]     [1]                           
                                = c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
        
        
        Following rules are (at-least) weakly oriented:
        if_replace#(false(),n,m,cons(k,x)) =  [0 0] n + [0 2] x + [4]                                              
                                              [2 0]     [0 3]     [7]                                              
                                           >= [0 2] x + [4]                                                        
                                              [0 0]     [0]                                                        
                                           =  c_7(replace#(n,m,x))                                                 
        
                          sort#(cons(n,x)) =  [2 2] x + [7]                                                        
                                              [3 3]     [6]                                                        
                                           >= [2 2] x + [6]                                                        
                                              [3 0]     [6]                                                        
                                           =  c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        
         if_replace(false(),n,m,cons(k,x)) =  [1 1] x + [3]                                                        
                                              [0 1]     [2]                                                        
                                           >= [1 1] x + [3]                                                        
                                              [0 1]     [2]                                                        
                                           =  cons(k,replace(n,m,x))                                               
        
          if_replace(true(),n,m,cons(k,x)) =  [1 1] x + [3]                                                        
                                              [0 1]     [2]                                                        
                                           >= [1 1] x + [2]                                                        
                                              [0 1]     [2]                                                        
                                           =  cons(m,x)                                                            
        
                    replace(n,m,cons(k,x)) =  [1 1] x + [3]                                                        
                                              [0 1]     [2]                                                        
                                           >= [1 1] x + [3]                                                        
                                              [0 1]     [2]                                                        
                                           =  if_replace(eq(n,k),n,m,cons(k,x))                                    
        
                        replace(n,m,nil()) =  [2]                                                                  
                                              [0]                                                                  
                                           >= [1]                                                                  
                                              [0]                                                                  
                                           =  nil()                                                                
        
******* Step 1.b:5.b:3.b:3.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

******* Step 1.b:5.b:3.b:3.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
             -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):2
          
          2:W:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
             -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):1
          
          3:W:sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
             -->_1 sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)):3
             -->_2 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
          1: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
          2: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
******* Step 1.b:5.b:3.b:3.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

***** Step 1.b:5.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        - Weak DPs:
            if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
            replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
             -->_2 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):3
             -->_1 sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)):1
          
          2:W:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
             -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):3
          
          3:W:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
             -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)))
          2: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x))
***** Step 1.b:5.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x))
             -->_1 sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)))
***** Step 1.b:5.b:3.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + 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: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)))
          
        The strictly oriented rules are moved into the weak component.
****** Step 1.b:5.b:3.b:3.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + 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_17) = {1}
        
        Following symbols are considered usable:
          {if_replace,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#}
        TcT has computed the following interpretation:
                    p(0) = [0]                  
                 p(cons) = [1] x1 + [1] x2 + [2]
                   p(eq) = [0]                  
                p(false) = [0]                  
               p(if_min) = [0]                  
           p(if_replace) = [1] x3 + [1] x4 + [1]
                   p(le) = [0]                  
                  p(min) = [0]                  
                  p(nil) = [0]                  
              p(replace) = [1] x2 + [1] x3 + [1]
                    p(s) = [0]                  
                 p(sort) = [0]                  
                 p(true) = [0]                  
                  p(eq#) = [0]                  
              p(if_min#) = [0]                  
          p(if_replace#) = [0]                  
                  p(le#) = [0]                  
                 p(min#) = [0]                  
             p(replace#) = [0]                  
                p(sort#) = [4] x1 + [4]         
                  p(c_1) = [0]                  
                  p(c_2) = [0]                  
                  p(c_3) = [0]                  
                  p(c_4) = [1] x1 + [0]         
                  p(c_5) = [0]                  
                  p(c_6) = [4] x1 + [0]         
                  p(c_7) = [0]                  
                  p(c_8) = [0]                  
                  p(c_9) = [0]                  
                 p(c_10) = [0]                  
                 p(c_11) = [0]                  
                 p(c_12) = [1] x1 + [0]         
                 p(c_13) = [0]                  
                 p(c_14) = [0]                  
                 p(c_15) = [0]                  
                 p(c_16) = [2]                  
                 p(c_17) = [1] x1 + [0]         
                 p(c_18) = [0]                  
        
        Following rules are strictly oriented:
        sort#(cons(n,x)) = [4] n + [4] x + [12]                    
                         > [4] n + [4] x + [8]                     
                         = c_17(sort#(replace(min(cons(n,x)),n,x)))
        
        
        Following rules are (at-least) weakly oriented:
        if_replace(false(),n,m,cons(k,x)) =  [1] k + [1] m + [1] x + [3]      
                                          >= [1] k + [1] m + [1] x + [3]      
                                          =  cons(k,replace(n,m,x))           
        
         if_replace(true(),n,m,cons(k,x)) =  [1] k + [1] m + [1] x + [3]      
                                          >= [1] m + [1] x + [2]              
                                          =  cons(m,x)                        
        
                   replace(n,m,cons(k,x)) =  [1] k + [1] m + [1] x + [3]      
                                          >= [1] k + [1] m + [1] x + [3]      
                                          =  if_replace(eq(n,k),n,m,cons(k,x))
        
                       replace(n,m,nil()) =  [1] m + [1]                      
                                          >= [0]                              
                                          =  nil()                            
        
****** Step 1.b:5.b:3.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 1.b:5.b:3.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)))
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)))
             -->_1 sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)))
****** Step 1.b:5.b:3.b:3.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            eq(0(),0()) -> true()
            eq(0(),s(m)) -> false()
            eq(s(n),0()) -> false()
            eq(s(n),s(m)) -> eq(n,m)
            if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x))
            if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x))
            if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x))
            if_replace(true(),n,m,cons(k,x)) -> cons(m,x)
            le(0(),m) -> true()
            le(s(n),0()) -> false()
            le(s(n),s(m)) -> le(n,m)
            min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x)))
            min(cons(0(),nil())) -> 0()
            min(cons(s(n),nil())) -> s(n)
            replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x))
            replace(n,m,nil()) -> nil()
        - Signature:
            {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1
            ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0
            ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace#
            ,sort#} and constructors {0,cons,false,nil,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^2))