* Step 1: DependencyPairs WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            less_leaves(x,leaf()) -> false()
            less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z))
            less_leaves(leaf(),cons(w,z)) -> true()
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
            shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
            shuffle(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0
            ,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app,concat,less_leaves,minus,quot,reverse
            ,shuffle} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          app#(add(n,x),y) -> c_1(app#(x,y))
          app#(nil(),y) -> c_2()
          concat#(cons(u,v),y) -> c_3(concat#(v,y))
          concat#(leaf(),y) -> c_4()
          less_leaves#(x,leaf()) -> c_5()
          less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
          less_leaves#(leaf(),cons(w,z)) -> c_7()
          minus#(x,0()) -> c_8()
          minus#(s(x),s(y)) -> c_9(minus#(x,y))
          quot#(0(),s(y)) -> c_10()
          quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
          reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
          reverse#(nil()) -> c_13()
          shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          shuffle#(nil()) -> c_15()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
            app#(nil(),y) -> c_2()
            concat#(cons(u,v),y) -> c_3(concat#(v,y))
            concat#(leaf(),y) -> c_4()
            less_leaves#(x,leaf()) -> c_5()
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
            less_leaves#(leaf(),cons(w,z)) -> c_7()
            minus#(x,0()) -> c_8()
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(0(),s(y)) -> c_10()
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
            reverse#(nil()) -> c_13()
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
            shuffle#(nil()) -> c_15()
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            less_leaves(x,leaf()) -> false()
            less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z))
            less_leaves(leaf(),cons(w,z)) -> true()
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
            shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
            shuffle(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          app(add(n,x),y) -> add(n,app(x,y))
          app(nil(),y) -> y
          concat(cons(u,v),y) -> cons(u,concat(v,y))
          concat(leaf(),y) -> y
          minus(x,0()) -> x
          minus(s(x),s(y)) -> minus(x,y)
          reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
          reverse(nil()) -> nil()
          app#(add(n,x),y) -> c_1(app#(x,y))
          app#(nil(),y) -> c_2()
          concat#(cons(u,v),y) -> c_3(concat#(v,y))
          concat#(leaf(),y) -> c_4()
          less_leaves#(x,leaf()) -> c_5()
          less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
          less_leaves#(leaf(),cons(w,z)) -> c_7()
          minus#(x,0()) -> c_8()
          minus#(s(x),s(y)) -> c_9(minus#(x,y))
          quot#(0(),s(y)) -> c_10()
          quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
          reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
          reverse#(nil()) -> c_13()
          shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          shuffle#(nil()) -> c_15()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
            app#(nil(),y) -> c_2()
            concat#(cons(u,v),y) -> c_3(concat#(v,y))
            concat#(leaf(),y) -> c_4()
            less_leaves#(x,leaf()) -> c_5()
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
            less_leaves#(leaf(),cons(w,z)) -> c_7()
            minus#(x,0()) -> c_8()
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(0(),s(y)) -> c_10()
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
            reverse#(nil()) -> c_13()
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
            shuffle#(nil()) -> c_15()
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4,5,7,8,10,13,15}
        by application of
          Pre({2,4,5,7,8,10,13,15}) = {1,3,6,9,11,12,14}.
        Here rules are labelled as follows:
          1: app#(add(n,x),y) -> c_1(app#(x,y))
          2: app#(nil(),y) -> c_2()
          3: concat#(cons(u,v),y) -> c_3(concat#(v,y))
          4: concat#(leaf(),y) -> c_4()
          5: less_leaves#(x,leaf()) -> c_5()
          6: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
          7: less_leaves#(leaf(),cons(w,z)) -> c_7()
          8: minus#(x,0()) -> c_8()
          9: minus#(s(x),s(y)) -> c_9(minus#(x,y))
          10: quot#(0(),s(y)) -> c_10()
          11: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
          12: reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
          13: reverse#(nil()) -> c_13()
          14: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          15: shuffle#(nil()) -> c_15()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
            concat#(cons(u,v),y) -> c_3(concat#(v,y))
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak DPs:
            app#(nil(),y) -> c_2()
            concat#(leaf(),y) -> c_4()
            less_leaves#(x,leaf()) -> c_5()
            less_leaves#(leaf(),cons(w,z)) -> c_7()
            minus#(x,0()) -> c_8()
            quot#(0(),s(y)) -> c_10()
            reverse#(nil()) -> c_13()
            shuffle#(nil()) -> c_15()
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:app#(add(n,x),y) -> c_1(app#(x,y))
             -->_1 app#(nil(),y) -> c_2():8
             -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
          
          2:S:concat#(cons(u,v),y) -> c_3(concat#(v,y))
             -->_1 concat#(leaf(),y) -> c_4():9
             -->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):2
          
          3:S:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                      ,concat#(u,v)
                                                      ,concat#(w,z))
             -->_1 less_leaves#(leaf(),cons(w,z)) -> c_7():11
             -->_1 less_leaves#(x,leaf()) -> c_5():10
             -->_3 concat#(leaf(),y) -> c_4():9
             -->_2 concat#(leaf(),y) -> c_4():9
             -->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                           ,concat#(u,v)
                                                           ,concat#(w,z)):3
             -->_3 concat#(cons(u,v),y) -> c_3(concat#(v,y)):2
             -->_2 concat#(cons(u,v),y) -> c_3(concat#(v,y)):2
          
          4:S:minus#(s(x),s(y)) -> c_9(minus#(x,y))
             -->_1 minus#(x,0()) -> c_8():12
             -->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):4
          
          5:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(0(),s(y)) -> c_10():13
             -->_2 minus#(x,0()) -> c_8():12
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):5
             -->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):4
          
          6:S:reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
             -->_2 reverse#(nil()) -> c_13():14
             -->_1 app#(nil(),y) -> c_2():8
             -->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):6
             -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
          
          7:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(nil()) -> c_15():15
             -->_2 reverse#(nil()) -> c_13():14
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):7
             -->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):6
          
          8:W:app#(nil(),y) -> c_2()
             
          
          9:W:concat#(leaf(),y) -> c_4()
             
          
          10:W:less_leaves#(x,leaf()) -> c_5()
             
          
          11:W:less_leaves#(leaf(),cons(w,z)) -> c_7()
             
          
          12:W:minus#(x,0()) -> c_8()
             
          
          13:W:quot#(0(),s(y)) -> c_10()
             
          
          14:W:reverse#(nil()) -> c_13()
             
          
          15:W:shuffle#(nil()) -> c_15()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          15: shuffle#(nil()) -> c_15()
          14: reverse#(nil()) -> c_13()
          13: quot#(0(),s(y)) -> c_10()
          12: minus#(x,0()) -> c_8()
          10: less_leaves#(x,leaf()) -> c_5()
          11: less_leaves#(leaf(),cons(w,z)) -> c_7()
          9: concat#(leaf(),y) -> c_4()
          8: app#(nil(),y) -> c_2()
* Step 5: Decompose WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
            concat#(cons(u,v),y) -> c_3(concat#(v,y))
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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:
              app#(add(n,x),y) -> c_1(app#(x,y))
          - Weak DPs:
              concat#(cons(u,v),y) -> c_3(concat#(v,y))
              less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
              minus#(s(x),s(y)) -> c_9(minus#(x,y))
              quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
              reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
              shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          - Weak TRS:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              concat(cons(u,v),y) -> cons(u,concat(v,y))
              concat(leaf(),y) -> y
              minus(x,0()) -> x
              minus(s(x),s(y)) -> minus(x,y)
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
          - Signature:
              {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
              ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
              ,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
              ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              concat#(cons(u,v),y) -> c_3(concat#(v,y))
              less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
              minus#(s(x),s(y)) -> c_9(minus#(x,y))
              quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
              reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
              shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          - Weak DPs:
              app#(add(n,x),y) -> c_1(app#(x,y))
          - Weak TRS:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              concat(cons(u,v),y) -> cons(u,concat(v,y))
              concat(leaf(),y) -> y
              minus(x,0()) -> x
              minus(s(x),s(y)) -> minus(x,y)
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
          - Signature:
              {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
              ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
              ,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
              ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
** Step 5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
        - Weak DPs:
            concat#(cons(u,v),y) -> c_3(concat#(v,y))
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:app#(add(n,x),y) -> c_1(app#(x,y))
             -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
          
          2:W:concat#(cons(u,v),y) -> c_3(concat#(v,y))
             -->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):2
          
          3:W:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                      ,concat#(u,v)
                                                      ,concat#(w,z))
             -->_3 concat#(cons(u,v),y) -> c_3(concat#(v,y)):2
             -->_2 concat#(cons(u,v),y) -> c_3(concat#(v,y)):2
             -->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                           ,concat#(u,v)
                                                           ,concat#(w,z)):3
          
          4:W:minus#(s(x),s(y)) -> c_9(minus#(x,y))
             -->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):4
          
          5:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):4
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):5
          
          6:W:reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
             -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
             -->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):6
          
          7:W:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):6
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):7
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
          4: minus#(s(x),s(y)) -> c_9(minus#(x,y))
          3: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
          2: concat#(cons(u,v),y) -> c_3(concat#(v,y))
** Step 5.a:2: UsableRules WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
        - Weak DPs:
            reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          app(add(n,x),y) -> add(n,app(x,y))
          app(nil(),y) -> y
          reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
          reverse(nil()) -> nil()
          app#(add(n,x),y) -> c_1(app#(x,y))
          reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
          shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
** Step 5.a:3: DecomposeDG WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
        - Weak DPs:
            reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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
          shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        and a lower component
          app#(add(n,x),y) -> c_1(app#(x,y))
          reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
        Further, following extension rules are added to the lower component.
          shuffle#(add(n,x)) -> reverse#(x)
          shuffle#(add(n,x)) -> shuffle#(reverse(x))
*** Step 5.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          
        The strictly oriented rules are moved into the weak component.
**** Step 5.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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_14) = {1}
        
        Following symbols are considered usable:
          {app,reverse,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
        TcT has computed the following interpretation:
                     p(0) = [0]                  
                   p(add) = [1] x1 + [1] x2 + [4]
                   p(app) = [1] x1 + [1] x2 + [0]
                p(concat) = [4] x1 + [4] x2 + [0]
                  p(cons) = [1] x2 + [0]         
                 p(false) = [1]                  
                  p(leaf) = [2]                  
           p(less_leaves) = [1] x1 + [4] x2 + [1]
                 p(minus) = [1] x1 + [1] x2 + [1]
                   p(nil) = [0]                  
                  p(quot) = [1] x1 + [8]         
               p(reverse) = [1] x1 + [0]         
                     p(s) = [1]                  
               p(shuffle) = [1]                  
                  p(true) = [1]                  
                  p(app#) = [2] x1 + [2] x2 + [1]
               p(concat#) = [1]                  
          p(less_leaves#) = [2] x1 + [1] x2 + [0]
                p(minus#) = [1] x1 + [1]         
                 p(quot#) = [2]                  
              p(reverse#) = [3]                  
              p(shuffle#) = [4] x1 + [0]         
                   p(c_1) = [1] x1 + [2]         
                   p(c_2) = [1]                  
                   p(c_3) = [1] x1 + [0]         
                   p(c_4) = [0]                  
                   p(c_5) = [0]                  
                   p(c_6) = [1] x3 + [1]         
                   p(c_7) = [0]                  
                   p(c_8) = [0]                  
                   p(c_9) = [4] x1 + [1]         
                  p(c_10) = [2]                  
                  p(c_11) = [2] x2 + [1]         
                  p(c_12) = [1] x1 + [0]         
                  p(c_13) = [2]                  
                  p(c_14) = [1] x1 + [1] x2 + [0]
                  p(c_15) = [1]                  
        
        Following rules are strictly oriented:
        shuffle#(add(n,x)) = [4] n + [4] x + [16]                  
                           > [4] x + [3]                           
                           = c_14(shuffle#(reverse(x)),reverse#(x))
        
        
        Following rules are (at-least) weakly oriented:
          app(add(n,x),y) =  [1] n + [1] x + [1] y + [4] 
                          >= [1] n + [1] x + [1] y + [4] 
                          =  add(n,app(x,y))             
        
             app(nil(),y) =  [1] y + [0]                 
                          >= [1] y + [0]                 
                          =  y                           
        
        reverse(add(n,x)) =  [1] n + [1] x + [4]         
                          >= [1] n + [1] x + [4]         
                          =  app(reverse(x),add(n,nil()))
        
           reverse(nil()) =  [0]                         
                          >= [0]                         
                          =  nil()                       
        
**** Step 5.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 5.a:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
**** Step 5.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 5.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
        - Weak DPs:
            reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> reverse#(x)
            shuffle#(add(n,x)) -> shuffle#(reverse(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: app#(add(n,x),y) -> c_1(app#(x,y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 5.a:3.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
        - Weak DPs:
            reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> reverse#(x)
            shuffle#(add(n,x)) -> shuffle#(reverse(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_1) = {1},
          uargs(c_12) = {1,2}
        
        Following symbols are considered usable:
          {app,reverse,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
        TcT has computed the following interpretation:
                     p(0) = 1                            
                   p(add) = 1 + x2                       
                   p(app) = x1 + x2                      
                p(concat) = x1 + 4*x1^2                  
                  p(cons) = x2                           
                 p(false) = 1                            
                  p(leaf) = 0                            
           p(less_leaves) = 2 + 2*x1 + x2^2              
                 p(minus) = 1                            
                   p(nil) = 0                            
                  p(quot) = 1 + x1 + 4*x1^2 + x2 + 2*x2^2
               p(reverse) = x1                           
                     p(s) = 0                            
               p(shuffle) = 2*x1 + x1^2                  
                  p(true) = 0                            
                  p(app#) = 1 + x1 + x2 + x2^2           
               p(concat#) = x1 + 4*x1*x2 + x2^2          
          p(less_leaves#) = 2*x1*x2 + x1^2 + x2          
                p(minus#) = x1*x2 + 4*x1^2 + x2 + x2^2   
                 p(quot#) = 1                            
              p(reverse#) = 5 + 6*x1 + x1^2              
              p(shuffle#) = 2 + 2*x1 + 2*x1^2            
                   p(c_1) = x1                           
                   p(c_2) = 0                            
                   p(c_3) = x1                           
                   p(c_4) = 0                            
                   p(c_5) = 0                            
                   p(c_6) = x2                           
                   p(c_7) = 0                            
                   p(c_8) = 1                            
                   p(c_9) = 0                            
                  p(c_10) = 0                            
                  p(c_11) = x2                           
                  p(c_12) = x1 + x2                      
                  p(c_13) = 1                            
                  p(c_14) = x1 + x2                      
                  p(c_15) = 0                            
        
        Following rules are strictly oriented:
        app#(add(n,x),y) = 2 + x + y + y^2
                         > 1 + x + y + y^2
                         = c_1(app#(x,y)) 
        
        
        Following rules are (at-least) weakly oriented:
        reverse#(add(n,x)) =  12 + 8*x + x^2                                 
                           >= 8 + 7*x + x^2                                  
                           =  c_12(app#(reverse(x),add(n,nil())),reverse#(x))
        
        shuffle#(add(n,x)) =  6 + 6*x + 2*x^2                                
                           >= 5 + 6*x + x^2                                  
                           =  reverse#(x)                                    
        
        shuffle#(add(n,x)) =  6 + 6*x + 2*x^2                                
                           >= 2 + 2*x + 2*x^2                                
                           =  shuffle#(reverse(x))                           
        
           app(add(n,x),y) =  1 + x + y                                      
                           >= 1 + x + y                                      
                           =  add(n,app(x,y))                                
        
              app(nil(),y) =  y                                              
                           >= y                                              
                           =  y                                              
        
         reverse(add(n,x)) =  1 + x                                          
                           >= 1 + x                                          
                           =  app(reverse(x),add(n,nil()))                   
        
            reverse(nil()) =  0                                              
                           >= 0                                              
                           =  nil()                                          
        
**** Step 5.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
            reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> reverse#(x)
            shuffle#(add(n,x)) -> shuffle#(reverse(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

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

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            concat#(cons(u,v),y) -> c_3(concat#(v,y))
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:concat#(cons(u,v),y) -> c_3(concat#(v,y))
             -->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
          
          2:S:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                      ,concat#(u,v)
                                                      ,concat#(w,z))
             -->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                           ,concat#(u,v)
                                                           ,concat#(w,z)):2
             -->_3 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
             -->_2 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
          
          3:S:minus#(s(x),s(y)) -> c_9(minus#(x,y))
             -->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):3
          
          4:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):4
             -->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):3
          
          5:S:reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
             -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):7
             -->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):5
          
          6:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):6
             -->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):5
          
          7:W:app#(add(n,x),y) -> c_1(app#(x,y))
             -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):7
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: app#(add(n,x),y) -> c_1(app#(x,y))
** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            concat#(cons(u,v),y) -> c_3(concat#(v,y))
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:concat#(cons(u,v),y) -> c_3(concat#(v,y))
             -->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
          
          2:S:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                      ,concat#(u,v)
                                                      ,concat#(w,z))
             -->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                           ,concat#(u,v)
                                                           ,concat#(w,z)):2
             -->_3 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
             -->_2 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
          
          3:S:minus#(s(x),s(y)) -> c_9(minus#(x,y))
             -->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):3
          
          4:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):4
             -->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):3
          
          5:S:reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
             -->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):5
          
          6:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):6
             -->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):5
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          reverse#(add(n,x)) -> c_12(reverse#(x))
** Step 5.b:3: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            concat#(cons(u,v),y) -> c_3(concat#(v,y))
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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:
              concat#(cons(u,v),y) -> c_3(concat#(v,y))
          - Weak DPs:
              less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
              minus#(s(x),s(y)) -> c_9(minus#(x,y))
              quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
              reverse#(add(n,x)) -> c_12(reverse#(x))
              shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          - Weak TRS:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              concat(cons(u,v),y) -> cons(u,concat(v,y))
              concat(leaf(),y) -> y
              minus(x,0()) -> x
              minus(s(x),s(y)) -> minus(x,y)
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
          - Signature:
              {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
              ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
              ,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
              ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
              minus#(s(x),s(y)) -> c_9(minus#(x,y))
              quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
              reverse#(add(n,x)) -> c_12(reverse#(x))
              shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          - Weak DPs:
              concat#(cons(u,v),y) -> c_3(concat#(v,y))
          - Weak TRS:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              concat(cons(u,v),y) -> cons(u,concat(v,y))
              concat(leaf(),y) -> y
              minus(x,0()) -> x
              minus(s(x),s(y)) -> minus(x,y)
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
          - Signature:
              {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
              ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
              ,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
              ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
*** Step 5.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            concat#(cons(u,v),y) -> c_3(concat#(v,y))
        - Weak DPs:
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:concat#(cons(u,v),y) -> c_3(concat#(v,y))
             -->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
          
          2:W:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                      ,concat#(u,v)
                                                      ,concat#(w,z))
             -->_3 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
             -->_2 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
             -->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                           ,concat#(u,v)
                                                           ,concat#(w,z)):2
          
          3:W:minus#(s(x),s(y)) -> c_9(minus#(x,y))
             -->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):3
          
          4:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):3
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):4
          
          5:W:reverse#(add(n,x)) -> c_12(reverse#(x))
             -->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):5
          
          6:W:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):5
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):6
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          5: reverse#(add(n,x)) -> c_12(reverse#(x))
          4: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
          3: minus#(s(x),s(y)) -> c_9(minus#(x,y))
*** Step 5.b:3.a:2: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            concat#(cons(u,v),y) -> c_3(concat#(v,y))
        - Weak DPs:
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          concat(cons(u,v),y) -> cons(u,concat(v,y))
          concat(leaf(),y) -> y
          concat#(cons(u,v),y) -> c_3(concat#(v,y))
          less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
*** Step 5.b:3.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            concat#(cons(u,v),y) -> c_3(concat#(v,y))
        - Weak DPs:
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
        - Weak TRS:
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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: concat#(cons(u,v),y) -> c_3(concat#(v,y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 5.b:3.a:3.a:1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            concat#(cons(u,v),y) -> c_3(concat#(v,y))
        - Weak DPs:
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
        - Weak TRS:
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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_3) = {1},
          uargs(c_6) = {1,2,3}
        
        Following symbols are considered usable:
          {concat,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
        TcT has computed the following interpretation:
                     p(0) = [0]                                 
                            [2]                                 
                   p(add) = [0 1] x1 + [1]                      
                            [0 0]      [1]                      
                   p(app) = [1 0] x1 + [0]                      
                            [4 4]      [0]                      
                p(concat) = [1 0] x1 + [1 0] x2 + [1]           
                            [0 4]      [0 1]      [0]           
                  p(cons) = [1 4] x1 + [1 0] x2 + [0]           
                            [0 1]      [0 1]      [2]           
                 p(false) = [1]                                 
                            [1]                                 
                  p(leaf) = [2]                                 
                            [3]                                 
           p(less_leaves) = [4 0] x1 + [0 0] x2 + [1]           
                            [1 0]      [4 1]      [1]           
                 p(minus) = [2 1] x1 + [0 0] x2 + [0]           
                            [1 0]      [0 1]      [4]           
                   p(nil) = [0]                                 
                            [1]                                 
                  p(quot) = [1 4] x2 + [0]                      
                            [1 4]      [0]                      
               p(reverse) = [0]                                 
                            [0]                                 
                     p(s) = [0 1] x1 + [1]                      
                            [0 0]      [4]                      
               p(shuffle) = [0]                                 
                            [1]                                 
                  p(true) = [0]                                 
                            [0]                                 
                  p(app#) = [0 1] x2 + [1]                      
                            [0 2]      [0]                      
               p(concat#) = [0 1] x1 + [0]                      
                            [0 0]      [1]                      
          p(less_leaves#) = [1 1] x1 + [2 2] x2 + [6]           
                            [0 0]      [1 0]      [1]           
                p(minus#) = [0 0] x1 + [1 1] x2 + [0]           
                            [0 1]      [0 1]      [2]           
                 p(quot#) = [1 4] x2 + [0]                      
                            [0 1]      [1]                      
              p(reverse#) = [2 0] x1 + [0]                      
                            [1 1]      [0]                      
              p(shuffle#) = [1]                                 
                            [0]                                 
                   p(c_1) = [1]                                 
                            [1]                                 
                   p(c_2) = [1]                                 
                            [0]                                 
                   p(c_3) = [1 1] x1 + [0]                      
                            [0 1]      [0]                      
                   p(c_4) = [1]                                 
                            [1]                                 
                   p(c_5) = [0]                                 
                            [1]                                 
                   p(c_6) = [1 0] x1 + [1 0] x2 + [2 2] x3 + [1]
                            [0 0]      [0 1]      [0 0]      [0]
                   p(c_7) = [1]                                 
                            [1]                                 
                   p(c_8) = [1]                                 
                            [1]                                 
                   p(c_9) = [1]                                 
                            [1]                                 
                  p(c_10) = [0]                                 
                            [0]                                 
                  p(c_11) = [4 0] x2 + [0]                      
                            [2 1]      [1]                      
                  p(c_12) = [1 2] x1 + [0]                      
                            [1 0]      [1]                      
                  p(c_13) = [4]                                 
                            [1]                                 
                  p(c_14) = [0 0] x1 + [1 2] x2 + [0]           
                            [0 1]      [4 1]      [0]           
                  p(c_15) = [1]                                 
                            [1]                                 
        
        Following rules are strictly oriented:
        concat#(cons(u,v),y) = [0 1] u + [0 1] v + [2]
                               [0 0]     [0 0]     [1]
                             > [0 1] v + [1]          
                               [0 0]     [1]          
                             = c_3(concat#(v,y))      
        
        
        Following rules are (at-least) weakly oriented:
        less_leaves#(cons(u,v),cons(w,z)) =  [1 5] u + [1 1] v + [2 10] w + [2 2] z + [12]                       
                                             [0 0]     [0 0]     [1  4]     [1 0]     [1]                        
                                          >= [1 5] u + [1 1] v + [2 10] w + [2 2] z + [12]                       
                                             [0 0]     [0 0]     [0  0]     [0 0]     [1]                        
                                          =  c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
        
                      concat(cons(u,v),y) =  [1 4] u + [1 0] v + [1 0] y + [1]                                   
                                             [0 4]     [0 4]     [0 1]     [8]                                   
                                          >= [1 4] u + [1 0] v + [1 0] y + [1]                                   
                                             [0 1]     [0 4]     [0 1]     [2]                                   
                                          =  cons(u,concat(v,y))                                                 
        
                         concat(leaf(),y) =  [1 0] y + [3]                                                       
                                             [0 1]     [12]                                                      
                                          >= [1 0] y + [0]                                                       
                                             [0 1]     [0]                                                       
                                          =  y                                                                   
        
**** Step 5.b:3.a:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            concat#(cons(u,v),y) -> c_3(concat#(v,y))
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
        - Weak TRS:
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 5.b:3.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            concat#(cons(u,v),y) -> c_3(concat#(v,y))
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
        - Weak TRS:
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:concat#(cons(u,v),y) -> c_3(concat#(v,y))
             -->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
          
          2:W:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                      ,concat#(u,v)
                                                      ,concat#(w,z))
             -->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                           ,concat#(u,v)
                                                           ,concat#(w,z)):2
             -->_3 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
             -->_2 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                     ,concat#(u,v)
                                                     ,concat#(w,z))
          1: concat#(cons(u,v),y) -> c_3(concat#(v,y))
**** Step 5.b:3.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak DPs:
            concat#(cons(u,v),y) -> c_3(concat#(v,y))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                      ,concat#(u,v)
                                                      ,concat#(w,z))
             -->_3 concat#(cons(u,v),y) -> c_3(concat#(v,y)):6
             -->_2 concat#(cons(u,v),y) -> c_3(concat#(v,y)):6
             -->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                           ,concat#(u,v)
                                                           ,concat#(w,z)):1
          
          2:S:minus#(s(x),s(y)) -> c_9(minus#(x,y))
             -->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):2
          
          3:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):3
             -->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):2
          
          4:S:reverse#(add(n,x)) -> c_12(reverse#(x))
             -->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):4
          
          5:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):5
             -->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):4
          
          6:W:concat#(cons(u,v),y) -> c_3(concat#(v,y))
             -->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):6
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: concat#(cons(u,v),y) -> c_3(concat#(v,y))
*** Step 5.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                      ,concat#(u,v)
                                                      ,concat#(w,z))
             -->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
                                                           ,concat#(u,v)
                                                           ,concat#(w,z)):1
          
          2:S:minus#(s(x),s(y)) -> c_9(minus#(x,y))
             -->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):2
          
          3:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):3
             -->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):2
          
          4:S:reverse#(add(n,x)) -> c_12(reverse#(x))
             -->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):4
          
          5:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):5
             -->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):4
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
*** Step 5.b:3.b:3: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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:
              less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
          - Weak DPs:
              minus#(s(x),s(y)) -> c_9(minus#(x,y))
              quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
              reverse#(add(n,x)) -> c_12(reverse#(x))
              shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          - Weak TRS:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              concat(cons(u,v),y) -> cons(u,concat(v,y))
              concat(leaf(),y) -> y
              minus(x,0()) -> x
              minus(s(x),s(y)) -> minus(x,y)
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
          - Signature:
              {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
              ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
              ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
              ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              minus#(s(x),s(y)) -> c_9(minus#(x,y))
              quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
              reverse#(add(n,x)) -> c_12(reverse#(x))
              shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          - Weak DPs:
              less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
          - Weak TRS:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              concat(cons(u,v),y) -> cons(u,concat(v,y))
              concat(leaf(),y) -> y
              minus(x,0()) -> x
              minus(s(x),s(y)) -> minus(x,y)
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
          - Signature:
              {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
              ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
              ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
              ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
**** Step 5.b:3.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
        - Weak DPs:
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
             -->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))):1
          
          2:W:minus#(s(x),s(y)) -> c_9(minus#(x,y))
             -->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):2
          
          3:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):2
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):3
          
          4:W:reverse#(add(n,x)) -> c_12(reverse#(x))
             -->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):4
          
          5:W:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):4
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          4: reverse#(add(n,x)) -> c_12(reverse#(x))
          3: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
          2: minus#(s(x),s(y)) -> c_9(minus#(x,y))
**** Step 5.b:3.b:3.a:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          concat(cons(u,v),y) -> cons(u,concat(v,y))
          concat(leaf(),y) -> y
          less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
**** Step 5.b:3.b:3.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
        - Weak TRS:
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
          
        The strictly oriented rules are moved into the weak component.
***** Step 5.b:3.b:3.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
        - Weak TRS:
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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_6) = {1}
        
        Following symbols are considered usable:
          {concat,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
        TcT has computed the following interpretation:
                     p(0) = [0]                   
                   p(add) = [1]                   
                   p(app) = [8] x1 + [1] x2 + [0] 
                p(concat) = [1] x1 + [1] x2 + [0] 
                  p(cons) = [1] x1 + [1] x2 + [1] 
                 p(false) = [2]                   
                  p(leaf) = [1]                   
           p(less_leaves) = [1] x1 + [1] x2 + [0] 
                 p(minus) = [1] x2 + [0]          
                   p(nil) = [2]                   
                  p(quot) = [1] x1 + [1]          
               p(reverse) = [1] x1 + [4]          
                     p(s) = [1]                   
               p(shuffle) = [1]                   
                  p(true) = [1]                   
                  p(app#) = [1] x1 + [1] x2 + [0] 
               p(concat#) = [4] x2 + [4]          
          p(less_leaves#) = [12] x1 + [8] x2 + [9]
                p(minus#) = [1] x2 + [2]          
                 p(quot#) = [2] x1 + [1] x2 + [1] 
              p(reverse#) = [1] x1 + [0]          
              p(shuffle#) = [2] x1 + [8]          
                   p(c_1) = [1] x1 + [0]          
                   p(c_2) = [0]                   
                   p(c_3) = [1] x1 + [2]          
                   p(c_4) = [1]                   
                   p(c_5) = [0]                   
                   p(c_6) = [1] x1 + [1]          
                   p(c_7) = [4]                   
                   p(c_8) = [0]                   
                   p(c_9) = [1] x1 + [4]          
                  p(c_10) = [2]                   
                  p(c_11) = [2] x1 + [8]          
                  p(c_12) = [1]                   
                  p(c_13) = [0]                   
                  p(c_14) = [1] x1 + [1] x2 + [1] 
                  p(c_15) = [1]                   
        
        Following rules are strictly oriented:
        less_leaves#(cons(u,v),cons(w,z)) = [12] u + [12] v + [8] w + [8] z + [29]    
                                          > [12] u + [12] v + [8] w + [8] z + [10]    
                                          = c_6(less_leaves#(concat(u,v),concat(w,z)))
        
        
        Following rules are (at-least) weakly oriented:
        concat(cons(u,v),y) =  [1] u + [1] v + [1] y + [1]
                            >= [1] u + [1] v + [1] y + [1]
                            =  cons(u,concat(v,y))        
        
           concat(leaf(),y) =  [1] y + [1]                
                            >= [1] y + [0]                
                            =  y                          
        
***** Step 5.b:3.b:3.a:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
        - Weak TRS:
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 5.b:3.b:3.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
        - Weak TRS:
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
             -->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
***** Step 5.b:3.b:3.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 5.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak DPs:
            less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:minus#(s(x),s(y)) -> c_9(minus#(x,y))
             -->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):1
          
          2:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):2
             -->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):1
          
          3:S:reverse#(add(n,x)) -> c_12(reverse#(x))
             -->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):3
          
          4:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):4
             -->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):3
          
          5:W:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
             -->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
**** Step 5.b:3.b:3.b:2: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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:
              minus#(s(x),s(y)) -> c_9(minus#(x,y))
          - Weak DPs:
              quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
              reverse#(add(n,x)) -> c_12(reverse#(x))
              shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          - Weak TRS:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              concat(cons(u,v),y) -> cons(u,concat(v,y))
              concat(leaf(),y) -> y
              minus(x,0()) -> x
              minus(s(x),s(y)) -> minus(x,y)
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
          - Signature:
              {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
              ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
              ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
              ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
              reverse#(add(n,x)) -> c_12(reverse#(x))
              shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          - Weak DPs:
              minus#(s(x),s(y)) -> c_9(minus#(x,y))
          - Weak TRS:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              concat(cons(u,v),y) -> cons(u,concat(v,y))
              concat(leaf(),y) -> y
              minus(x,0()) -> x
              minus(s(x),s(y)) -> minus(x,y)
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
          - Signature:
              {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
              ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
              ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
              ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
***** Step 5.b:3.b:3.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
        - Weak DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:minus#(s(x),s(y)) -> c_9(minus#(x,y))
             -->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):1
          
          2:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):1
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):2
          
          3:W:reverse#(add(n,x)) -> c_12(reverse#(x))
             -->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):3
          
          4:W:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):3
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          3: reverse#(add(n,x)) -> c_12(reverse#(x))
***** Step 5.b:3.b:3.b:2.a:2: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
        - Weak DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          minus(x,0()) -> x
          minus(s(x),s(y)) -> minus(x,y)
          minus#(s(x),s(y)) -> c_9(minus#(x,y))
          quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
***** Step 5.b:3.b:3.b:2.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
        - Weak DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: minus#(s(x),s(y)) -> c_9(minus#(x,y))
          
        The strictly oriented rules are moved into the weak component.
****** Step 5.b:3.b:3.b:2.a:3.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
        - Weak DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_9) = {1},
          uargs(c_11) = {1,2}
        
        Following symbols are considered usable:
          {minus,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
        TcT has computed the following interpretation:
                     p(0) = 1                         
                   p(add) = 1 + x1 + x2               
                   p(app) = 0                         
                p(concat) = 0                         
                  p(cons) = 1 + x1 + x2               
                 p(false) = 1                         
                  p(leaf) = 1                         
           p(less_leaves) = x1 + x1*x2 + x1^2 + 2*x2  
                 p(minus) = x1                        
                   p(nil) = 0                         
                  p(quot) = 2 + x1                    
               p(reverse) = 1                         
                     p(s) = 1 + x1                    
               p(shuffle) = 4                         
                  p(true) = 0                         
                  p(app#) = 2 + x1 + 2*x1^2 + 4*x2^2  
               p(concat#) = 2 + x1 + 4*x1^2           
          p(less_leaves#) = x1^2 + x2 + x2^2          
                p(minus#) = 5 + 3*x2                  
                 p(quot#) = 4*x1 + 4*x1*x2 + x2 + x2^2
              p(reverse#) = 1 + x1 + 2*x1^2           
              p(shuffle#) = x1 + x1^2                 
                   p(c_1) = 1                         
                   p(c_2) = 1                         
                   p(c_3) = x1                        
                   p(c_4) = 1                         
                   p(c_5) = 1                         
                   p(c_6) = 0                         
                   p(c_7) = 0                         
                   p(c_8) = 0                         
                   p(c_9) = x1                        
                  p(c_10) = 1                         
                  p(c_11) = x1 + x2                   
                  p(c_12) = 1                         
                  p(c_13) = 0                         
                  p(c_14) = x1                        
                  p(c_15) = 1                         
        
        Following rules are strictly oriented:
        minus#(s(x),s(y)) = 8 + 3*y         
                          > 5 + 3*y         
                          = c_9(minus#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        quot#(s(x),s(y)) =  10 + 8*x + 4*x*y + 7*y + y^2            
                         >= 7 + 8*x + 4*x*y + 6*y + y^2             
                         =  c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        
            minus(x,0()) =  x                                       
                         >= x                                       
                         =  x                                       
        
        minus(s(x),s(y)) =  1 + x                                   
                         >= x                                       
                         =  minus(x,y)                              
        
****** Step 5.b:3.b:3.b:2.a:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 5.b:3.b:3.b:2.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:minus#(s(x),s(y)) -> c_9(minus#(x,y))
             -->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):1
          
          2:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):2
             -->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
          1: minus#(s(x),s(y)) -> c_9(minus#(x,y))
****** Step 5.b:3.b:3.b:2.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

***** Step 5.b:3.b:3.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak DPs:
            minus#(s(x),s(y)) -> c_9(minus#(x,y))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):4
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):1
          
          2:S:reverse#(add(n,x)) -> c_12(reverse#(x))
             -->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):2
          
          3:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):3
             -->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):2
          
          4:W:minus#(s(x),s(y)) -> c_9(minus#(x,y))
             -->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: minus#(s(x),s(y)) -> c_9(minus#(x,y))
***** Step 5.b:3.b:3.b:2.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):1
          
          2:S:reverse#(add(n,x)) -> c_12(reverse#(x))
             -->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):2
          
          3:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):3
             -->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
***** Step 5.b:3.b:3.b:2.b:3: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            concat(cons(u,v),y) -> cons(u,concat(v,y))
            concat(leaf(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          app(add(n,x),y) -> add(n,app(x,y))
          app(nil(),y) -> y
          minus(x,0()) -> x
          minus(s(x),s(y)) -> minus(x,y)
          reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
          reverse(nil()) -> nil()
          quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
          reverse#(add(n,x)) -> c_12(reverse#(x))
          shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
***** Step 5.b:3.b:3.b:2.b:4: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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:
              quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
          - Weak DPs:
              reverse#(add(n,x)) -> c_12(reverse#(x))
              shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          - Weak TRS:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              minus(x,0()) -> x
              minus(s(x),s(y)) -> minus(x,y)
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
          - Signature:
              {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
              ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
              ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
              ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              reverse#(add(n,x)) -> c_12(reverse#(x))
              shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          - Weak DPs:
              quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
          - Weak TRS:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              minus(x,0()) -> x
              minus(s(x),s(y)) -> minus(x,y)
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
          - Signature:
              {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
              ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
              ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
              ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
****** Step 5.b:3.b:3.b:2.b:4.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
        - Weak DPs:
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y))):1
          
          2:W:reverse#(add(n,x)) -> c_12(reverse#(x))
             -->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):2
          
          3:W:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):2
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          2: reverse#(add(n,x)) -> c_12(reverse#(x))
****** Step 5.b:3.b:3.b:2.b:4.a:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          minus(x,0()) -> x
          minus(s(x),s(y)) -> minus(x,y)
          quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
****** Step 5.b:3.b:3.b:2.b:4.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
        - Weak TRS:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
          
        The strictly oriented rules are moved into the weak component.
******* Step 5.b:3.b:3.b:2.b:4.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
        - Weak TRS:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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_11) = {1}
        
        Following symbols are considered usable:
          {minus,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
        TcT has computed the following interpretation:
                     p(0) = [2]                  
                   p(add) = [1]                  
                   p(app) = [8] x1 + [4]         
                p(concat) = [2] x1 + [1] x2 + [1]
                  p(cons) = [2]                  
                 p(false) = [1]                  
                  p(leaf) = [8]                  
           p(less_leaves) = [0]                  
                 p(minus) = [1] x1 + [0]         
                   p(nil) = [0]                  
                  p(quot) = [1] x1 + [0]         
               p(reverse) = [0]                  
                     p(s) = [1] x1 + [1]         
               p(shuffle) = [0]                  
                  p(true) = [1]                  
                  p(app#) = [1] x1 + [1] x2 + [4]
               p(concat#) = [1] x2 + [1]         
          p(less_leaves#) = [1]                  
                p(minus#) = [1] x1 + [1]         
                 p(quot#) = [4] x1 + [0]         
              p(reverse#) = [1] x1 + [1]         
              p(shuffle#) = [2]                  
                   p(c_1) = [8] x1 + [2]         
                   p(c_2) = [4]                  
                   p(c_3) = [0]                  
                   p(c_4) = [1]                  
                   p(c_5) = [1]                  
                   p(c_6) = [1]                  
                   p(c_7) = [1]                  
                   p(c_8) = [1]                  
                   p(c_9) = [1] x1 + [1]         
                  p(c_10) = [0]                  
                  p(c_11) = [1] x1 + [2]         
                  p(c_12) = [8] x1 + [1]         
                  p(c_13) = [1]                  
                  p(c_14) = [1] x1 + [8]         
                  p(c_15) = [0]                  
        
        Following rules are strictly oriented:
        quot#(s(x),s(y)) = [4] x + [4]                 
                         > [4] x + [2]                 
                         = c_11(quot#(minus(x,y),s(y)))
        
        
        Following rules are (at-least) weakly oriented:
            minus(x,0()) =  [1] x + [0]
                         >= [1] x + [0]
                         =  x          
        
        minus(s(x),s(y)) =  [1] x + [1]
                         >= [1] x + [0]
                         =  minus(x,y) 
        
******* Step 5.b:3.b:3.b:2.b:4.a:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
        - Weak TRS:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

******* Step 5.b:3.b:3.b:2.b:4.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
        - Weak TRS:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
******* Step 5.b:3.b:3.b:2.b:4.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

****** Step 5.b:3.b:3.b:2.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak DPs:
            quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:reverse#(add(n,x)) -> c_12(reverse#(x))
             -->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):1
          
          2:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):2
             -->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):1
          
          3:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
             -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y))):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
****** Step 5.b:3.b:3.b:2.b:4.b:2: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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:
              reverse#(add(n,x)) -> c_12(reverse#(x))
          - Weak DPs:
              shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          - Weak TRS:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              minus(x,0()) -> x
              minus(s(x),s(y)) -> minus(x,y)
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
          - Signature:
              {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
              ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
              ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
              ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
        
        Problem (S)
          - Strict DPs:
              shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          - Weak DPs:
              reverse#(add(n,x)) -> c_12(reverse#(x))
          - Weak TRS:
              app(add(n,x),y) -> add(n,app(x,y))
              app(nil(),y) -> y
              minus(x,0()) -> x
              minus(s(x),s(y)) -> minus(x,y)
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
          - Signature:
              {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
              ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
              ,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
              ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
******* Step 5.b:3.b:3.b:2.b:4.b:2.a:1: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            reverse#(add(n,x)) -> c_12(reverse#(x))
        - Weak DPs:
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          app(add(n,x),y) -> add(n,app(x,y))
          app(nil(),y) -> y
          reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
          reverse(nil()) -> nil()
          reverse#(add(n,x)) -> c_12(reverse#(x))
          shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
******* Step 5.b:3.b:3.b:2.b:4.b:2.a:2: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            reverse#(add(n,x)) -> c_12(reverse#(x))
        - Weak DPs:
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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
          shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        and a lower component
          reverse#(add(n,x)) -> c_12(reverse#(x))
        Further, following extension rules are added to the lower component.
          shuffle#(add(n,x)) -> reverse#(x)
          shuffle#(add(n,x)) -> shuffle#(reverse(x))
******** Step 5.b:3.b:3.b:2.b:4.b:2.a:2.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
          
        The strictly oriented rules are moved into the weak component.
********* Step 5.b:3.b:3.b:2.b:4.b:2.a:2.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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_14) = {1}
        
        Following symbols are considered usable:
          {app,reverse,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
        TcT has computed the following interpretation:
                     p(0) = [1]                  
                   p(add) = [1] x1 + [1] x2 + [2]
                   p(app) = [1] x1 + [1] x2 + [0]
                p(concat) = [2] x1 + [2]         
                  p(cons) = [0]                  
                 p(false) = [1]                  
                  p(leaf) = [0]                  
           p(less_leaves) = [1] x1 + [0]         
                 p(minus) = [1] x1 + [1] x2 + [0]
                   p(nil) = [0]                  
                  p(quot) = [4]                  
               p(reverse) = [1] x1 + [1]         
                     p(s) = [1]                  
               p(shuffle) = [2] x1 + [1]         
                  p(true) = [0]                  
                  p(app#) = [2] x2 + [0]         
               p(concat#) = [1] x1 + [1]         
          p(less_leaves#) = [1] x1 + [2]         
                p(minus#) = [1] x2 + [2]         
                 p(quot#) = [0]                  
              p(reverse#) = [1]                  
              p(shuffle#) = [12] x1 + [0]        
                   p(c_1) = [0]                  
                   p(c_2) = [2]                  
                   p(c_3) = [0]                  
                   p(c_4) = [0]                  
                   p(c_5) = [0]                  
                   p(c_6) = [1] x1 + [0]         
                   p(c_7) = [1]                  
                   p(c_8) = [2]                  
                   p(c_9) = [1] x1 + [1]         
                  p(c_10) = [0]                  
                  p(c_11) = [1]                  
                  p(c_12) = [1] x1 + [4]         
                  p(c_13) = [2]                  
                  p(c_14) = [1] x1 + [9] x2 + [0]
                  p(c_15) = [4]                  
        
        Following rules are strictly oriented:
        shuffle#(add(n,x)) = [12] n + [12] x + [24]                
                           > [12] x + [21]                         
                           = c_14(shuffle#(reverse(x)),reverse#(x))
        
        
        Following rules are (at-least) weakly oriented:
          app(add(n,x),y) =  [1] n + [1] x + [1] y + [2] 
                          >= [1] n + [1] x + [1] y + [2] 
                          =  add(n,app(x,y))             
        
             app(nil(),y) =  [1] y + [0]                 
                          >= [1] y + [0]                 
                          =  y                           
        
        reverse(add(n,x)) =  [1] n + [1] x + [3]         
                          >= [1] n + [1] x + [3]         
                          =  app(reverse(x),add(n,nil()))
        
           reverse(nil()) =  [1]                         
                          >= [0]                         
                          =  nil()                       
        
********* Step 5.b:3.b:3.b:2.b:4.b:2.a:2.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

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

******** Step 5.b:3.b:3.b:2.b:4.b:2.a:2.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            reverse#(add(n,x)) -> c_12(reverse#(x))
        - Weak DPs:
            shuffle#(add(n,x)) -> reverse#(x)
            shuffle#(add(n,x)) -> shuffle#(reverse(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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: reverse#(add(n,x)) -> c_12(reverse#(x))
          
        The strictly oriented rules are moved into the weak component.
********* Step 5.b:3.b:3.b:2.b:4.b:2.a:2.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            reverse#(add(n,x)) -> c_12(reverse#(x))
        - Weak DPs:
            shuffle#(add(n,x)) -> reverse#(x)
            shuffle#(add(n,x)) -> shuffle#(reverse(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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_12) = {1}
        
        Following symbols are considered usable:
          {app,reverse,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
        TcT has computed the following interpretation:
                     p(0) = [0]                  
                   p(add) = [1] x2 + [1]         
                   p(app) = [1] x1 + [1] x2 + [0]
                p(concat) = [1] x1 + [0]         
                  p(cons) = [1] x1 + [1] x2 + [1]
                 p(false) = [4]                  
                  p(leaf) = [1]                  
           p(less_leaves) = [1] x1 + [1] x2 + [1]
                 p(minus) = [8] x1 + [8]         
                   p(nil) = [0]                  
                  p(quot) = [1] x1 + [2]         
               p(reverse) = [1] x1 + [0]         
                     p(s) = [2]                  
               p(shuffle) = [1] x1 + [0]         
                  p(true) = [1]                  
                  p(app#) = [0]                  
               p(concat#) = [1] x1 + [1] x2 + [1]
          p(less_leaves#) = [0]                  
                p(minus#) = [2] x1 + [2] x2 + [1]
                 p(quot#) = [1] x1 + [1] x2 + [0]
              p(reverse#) = [1] x1 + [0]         
              p(shuffle#) = [1] x1 + [0]         
                   p(c_1) = [1]                  
                   p(c_2) = [0]                  
                   p(c_3) = [1] x1 + [1]         
                   p(c_4) = [1]                  
                   p(c_5) = [0]                  
                   p(c_6) = [1] x1 + [0]         
                   p(c_7) = [1]                  
                   p(c_8) = [0]                  
                   p(c_9) = [1] x1 + [0]         
                  p(c_10) = [1]                  
                  p(c_11) = [1] x1 + [8]         
                  p(c_12) = [1] x1 + [0]         
                  p(c_13) = [0]                  
                  p(c_14) = [1] x1 + [1]         
                  p(c_15) = [1]                  
        
        Following rules are strictly oriented:
        reverse#(add(n,x)) = [1] x + [1]      
                           > [1] x + [0]      
                           = c_12(reverse#(x))
        
        
        Following rules are (at-least) weakly oriented:
        shuffle#(add(n,x)) =  [1] x + [1]                 
                           >= [1] x + [0]                 
                           =  reverse#(x)                 
        
        shuffle#(add(n,x)) =  [1] x + [1]                 
                           >= [1] x + [0]                 
                           =  shuffle#(reverse(x))        
        
           app(add(n,x),y) =  [1] x + [1] y + [1]         
                           >= [1] x + [1] y + [1]         
                           =  add(n,app(x,y))             
        
              app(nil(),y) =  [1] y + [0]                 
                           >= [1] y + [0]                 
                           =  y                           
        
         reverse(add(n,x)) =  [1] x + [1]                 
                           >= [1] x + [1]                 
                           =  app(reverse(x),add(n,nil()))
        
            reverse(nil()) =  [0]                         
                           >= [0]                         
                           =  nil()                       
        
********* Step 5.b:3.b:3.b:2.b:4.b:2.a:2.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            reverse#(add(n,x)) -> c_12(reverse#(x))
            shuffle#(add(n,x)) -> reverse#(x)
            shuffle#(add(n,x)) -> shuffle#(reverse(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

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

******* Step 5.b:3.b:3.b:2.b:4.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak DPs:
            reverse#(add(n,x)) -> c_12(reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):2
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):1
          
          2:W:reverse#(add(n,x)) -> c_12(reverse#(x))
             -->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: reverse#(add(n,x)) -> c_12(reverse#(x))
******* Step 5.b:3.b:3.b:2.b:4.b:2.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
******* Step 5.b:3.b:3.b:2.b:4.b:2.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          app(add(n,x),y) -> add(n,app(x,y))
          app(nil(),y) -> y
          reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
          reverse(nil()) -> nil()
          shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
******* Step 5.b:3.b:3.b:2.b:4.b:2.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
          
        The strictly oriented rules are moved into the weak component.
******** Step 5.b:3.b:3.b:2.b:4.b:2.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,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_14) = {1}
        
        Following symbols are considered usable:
          {app,reverse,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
        TcT has computed the following interpretation:
                     p(0) = [1]                  
                   p(add) = [1] x2 + [2]         
                   p(app) = [1] x1 + [1] x2 + [0]
                p(concat) = [1]                  
                  p(cons) = [1] x2 + [0]         
                 p(false) = [2]                  
                  p(leaf) = [0]                  
           p(less_leaves) = [1] x1 + [0]         
                 p(minus) = [1] x1 + [2] x2 + [1]
                   p(nil) = [0]                  
                  p(quot) = [2] x1 + [1] x2 + [1]
               p(reverse) = [1] x1 + [0]         
                     p(s) = [0]                  
               p(shuffle) = [2]                  
                  p(true) = [1]                  
                  p(app#) = [1]                  
               p(concat#) = [4] x2 + [1]         
          p(less_leaves#) = [1] x1 + [4] x2 + [1]
                p(minus#) = [0]                  
                 p(quot#) = [1] x1 + [1]         
              p(reverse#) = [0]                  
              p(shuffle#) = [1] x1 + [0]         
                   p(c_1) = [1]                  
                   p(c_2) = [2]                  
                   p(c_3) = [8] x1 + [0]         
                   p(c_4) = [2]                  
                   p(c_5) = [2]                  
                   p(c_6) = [1] x1 + [1]         
                   p(c_7) = [2]                  
                   p(c_8) = [0]                  
                   p(c_9) = [1]                  
                  p(c_10) = [1]                  
                  p(c_11) = [8] x1 + [0]         
                  p(c_12) = [1] x1 + [4]         
                  p(c_13) = [4]                  
                  p(c_14) = [1] x1 + [0]         
                  p(c_15) = [0]                  
        
        Following rules are strictly oriented:
        shuffle#(add(n,x)) = [1] x + [2]               
                           > [1] x + [0]               
                           = c_14(shuffle#(reverse(x)))
        
        
        Following rules are (at-least) weakly oriented:
          app(add(n,x),y) =  [1] x + [1] y + [2]         
                          >= [1] x + [1] y + [2]         
                          =  add(n,app(x,y))             
        
             app(nil(),y) =  [1] y + [0]                 
                          >= [1] y + [0]                 
                          =  y                           
        
        reverse(add(n,x)) =  [1] x + [2]                 
                          >= [1] x + [2]                 
                          =  app(reverse(x),add(n,nil()))
        
           reverse(nil()) =  [0]                         
                          >= [0]                         
                          =  nil()                       
        
******** Step 5.b:3.b:3.b:2.b:4.b:2.b:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
        - Weak TRS:
            app(add(n,x),y) -> add(n,app(x,y))
            app(nil(),y) -> y
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
            ,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
            ,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

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

WORST_CASE(?,O(n^3))