/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1),O(n^3))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^3))
    + Considered Problem:
        - Strict 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()
            shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
            shuffle(nil()) -> nil()
        - Signature:
            {app/2,reverse/1,shuffle/1} / {add/2,nil/0}
        - Obligation:
             runtime complexity wrt. defined symbols {app,reverse,shuffle} and constructors {add,nil}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict 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()
            shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
            shuffle(nil()) -> nil()
        - Signature:
            {app/2,reverse/1,shuffle/1} / {add/2,nil/0}
        - Obligation:
             runtime complexity wrt. defined symbols {app,reverse,shuffle} and constructors {add,nil}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict 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()
            shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
            shuffle(nil()) -> nil()
        - Signature:
            {app/2,reverse/1,shuffle/1} / {add/2,nil/0}
        - Obligation:
             runtime complexity wrt. defined symbols {app,reverse,shuffle} and constructors {add,nil}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          app(y,z){y -> add(x,y)} =
            app(add(x,y),z) ->^+ add(x,app(y,z))
              = C[app(y,z) = app(y,z){}]

** Step 1.b:1: ToInnermost.  WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict 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()
            shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
            shuffle(nil()) -> nil()
        - Signature:
            {app/2,reverse/1,shuffle/1} / {add/2,nil/0}
        - Obligation:
             runtime complexity wrt. defined symbols {app,reverse,shuffle} and constructors {add,nil}
    + Applied Processor:
        ToInnermost
    + Details:
        switch to innermost, as the system is overlay and right linear and does not contain weak rules
** Step 1.b:2: DependencyPairs.  WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict 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()
            shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
            shuffle(nil()) -> nil()
        - Signature:
            {app/2,reverse/1,shuffle/1} / {add/2,nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app,reverse,shuffle} and constructors {add,nil}
    + 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()
          reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
          reverse#(nil()) -> c_4()
          shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
          shuffle#(nil()) -> c_6()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:3: 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()
            reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
            reverse#(nil()) -> c_4()
            shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
            shuffle#(nil()) -> c_6()
        - 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()
            shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
            shuffle(nil()) -> nil()
        - Signature:
            {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + 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))
          app#(nil(),y) -> c_2()
          reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
          reverse#(nil()) -> c_4()
          shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
          shuffle#(nil()) -> c_6()
** Step 1.b:4: 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()
            reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
            reverse#(nil()) -> c_4()
            shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
            shuffle#(nil()) -> c_6()
        - 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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4,6}
        by application of
          Pre({2,4,6}) = {1,3,5}.
        Here rules are labelled as follows:
          1: app#(add(n,x),y) -> c_1(app#(x,y))
          2: app#(nil(),y) -> c_2()
          3: reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
          4: reverse#(nil()) -> c_4()
          5: shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
          6: shuffle#(nil()) -> c_6()
** Step 1.b:5: RemoveWeakSuffixes.  WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
            reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
        - Weak DPs:
            app#(nil(),y) -> c_2()
            reverse#(nil()) -> c_4()
            shuffle#(nil()) -> c_6()
        - 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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + 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():4
             -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
          
          2:S:reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
             -->_2 reverse#(nil()) -> c_4():5
             -->_1 app#(nil(),y) -> c_2():4
             -->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):2
             -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
          
          3:S:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(nil()) -> c_6():6
             -->_2 reverse#(nil()) -> c_4():5
             -->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):3
             -->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):2
          
          4:W:app#(nil(),y) -> c_2()
             
          
          5:W:reverse#(nil()) -> c_4()
             
          
          6:W:shuffle#(nil()) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: shuffle#(nil()) -> c_6()
          5: reverse#(nil()) -> c_4()
          4: app#(nil(),y) -> c_2()
** Step 1.b:6: Decompose.  WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            app#(add(n,x),y) -> c_1(app#(x,y))
            reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + 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:
              reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
              shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2
              ,c_6/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
        
        Problem (S)
          - Strict DPs:
              reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
              shuffle#(add(n,x)) -> c_5(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
              reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
              reverse(nil()) -> nil()
          - Signature:
              {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2
              ,c_6/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
*** Step 1.b:6.a:1: 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_3(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + 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_5(shuffle#(reverse(x)),reverse#(x))
        and a lower component
          app#(add(n,x),y) -> c_1(app#(x,y))
          reverse#(add(n,x)) -> c_3(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 1.b:6.a:1.a:1: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + 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_5(shuffle#(reverse(x)),reverse#(x))
          
        The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.a:1.a:1.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + 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_5) = {1}
        
        Following symbols are considered usable:
          {app,reverse,app#,reverse#,shuffle#}
        TcT has computed the following interpretation:
               p(add) = [1] x2 + [1]         
               p(app) = [1] x1 + [1] x2 + [0]
               p(nil) = [0]                  
           p(reverse) = [1] x1 + [0]         
           p(shuffle) = [1] x1 + [1]         
              p(app#) = [1] x1 + [1] x2 + [1]
          p(reverse#) = [0]                  
          p(shuffle#) = [8] x1 + [0]         
               p(c_1) = [1]                  
               p(c_2) = [0]                  
               p(c_3) = [4] x1 + [1] x2 + [0]
               p(c_4) = [1]                  
               p(c_5) = [1] x1 + [4] x2 + [4]
               p(c_6) = [8]                  
        
        Following rules are strictly oriented:
        shuffle#(add(n,x)) = [8] x + [8]                          
                           > [8] x + [4]                          
                           = c_5(shuffle#(reverse(x)),reverse#(x))
        
        
        Following rules are (at-least) weakly oriented:
          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 1.b:6.a:1.a:1.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

***** Step 1.b:6.a:1.a:1.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_5(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_5(shuffle#(reverse(x)),reverse#(x))
***** Step 1.b:6.a:1.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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 1.b:6.a:1.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_3(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + 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 1.b:6.a:1.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_3(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + 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_3) = {1,2}
        
        Following symbols are considered usable:
          {app,reverse,app#,reverse#,shuffle#}
        TcT has computed the following interpretation:
               p(add) = 1 + x2           
               p(app) = x1 + x2          
               p(nil) = 0                
           p(reverse) = x1               
           p(shuffle) = 1 + x1 + x1^2    
              p(app#) = 2*x1 + 5*x2^2    
          p(reverse#) = 1 + 7*x1 + 2*x1^2
          p(shuffle#) = 3 + 6*x1 + 2*x1^2
               p(c_1) = x1               
               p(c_2) = 0                
               p(c_3) = 4 + x1 + x2      
               p(c_4) = 0                
               p(c_5) = 0                
               p(c_6) = 1                
        
        Following rules are strictly oriented:
        app#(add(n,x),y) = 2 + 2*x + 5*y^2
                         > 2*x + 5*y^2    
                         = c_1(app#(x,y)) 
        
        
        Following rules are (at-least) weakly oriented:
        reverse#(add(n,x)) =  10 + 11*x + 2*x^2                             
                           >= 10 + 9*x + 2*x^2                              
                           =  c_3(app#(reverse(x),add(n,nil())),reverse#(x))
        
        shuffle#(add(n,x)) =  11 + 10*x + 2*x^2                             
                           >= 1 + 7*x + 2*x^2                               
                           =  reverse#(x)                                   
        
        shuffle#(add(n,x)) =  11 + 10*x + 2*x^2                             
                           >= 3 + 6*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 1.b:6.a:1.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_3(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

***** Step 1.b:6.a:1.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_3(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + 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_3(app#(reverse(x),add(n,nil())),reverse#(x))
             -->_2 reverse#(add(n,x)) -> c_3(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_3(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_3(app#(reverse(x),add(n,nil())),reverse#(x))
          1: app#(add(n,x),y) -> c_1(app#(x,y))
***** Step 1.b:6.a:1.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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:6.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> c_5(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
            reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
            reverse(nil()) -> nil()
        - Signature:
            {app/2,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
             -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):3
             -->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):1
          
          2:S:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):2
             -->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):1
          
          3:W:app#(add(n,x),y) -> c_1(app#(x,y))
             -->_1 app#(add(n,x),y) -> c_1(app#(x,y)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: app#(add(n,x),y) -> c_1(app#(x,y))
*** Step 1.b:6.b:2: SimplifyRHS.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x))
             -->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):1
          
          2:S:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):2
             -->_2 reverse#(add(n,x)) -> c_3(app#(reverse(x),add(n,nil())),reverse#(x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          reverse#(add(n,x)) -> c_3(reverse#(x))
*** Step 1.b:6.b:3: Decompose.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            reverse#(add(n,x)) -> c_3(reverse#(x))
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + 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_3(reverse#(x))
          - Weak DPs:
              shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2
              ,c_6/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
        
        Problem (S)
          - Strict DPs:
              shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
          - Weak DPs:
              reverse#(add(n,x)) -> c_3(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2
              ,c_6/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
**** Step 1.b:6.b:3.a:1: PredecessorEstimationCP.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            reverse#(add(n,x)) -> c_3(reverse#(x))
        - Weak DPs:
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + 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: reverse#(add(n,x)) -> c_3(reverse#(x))
          
        The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.b:3.a:1.a:1: NaturalPI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            reverse#(add(n,x)) -> c_3(reverse#(x))
        - Weak DPs:
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + 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_3) = {1},
          uargs(c_5) = {1,2}
        
        Following symbols are considered usable:
          {app,reverse,app#,reverse#,shuffle#}
        TcT has computed the following interpretation:
               p(add) = 2 + x2                  
               p(app) = x1 + x2                 
               p(nil) = 0                       
           p(reverse) = x1                      
           p(shuffle) = 1                       
              p(app#) = x1 + 4*x1*x2 + x2 + x2^2
          p(reverse#) = 4*x1                    
          p(shuffle#) = x1^2                    
               p(c_1) = 1 + x1                  
               p(c_2) = 0                       
               p(c_3) = x1                      
               p(c_4) = 0                       
               p(c_5) = x1 + x2                 
               p(c_6) = 0                       
        
        Following rules are strictly oriented:
        reverse#(add(n,x)) = 8 + 4*x         
                           > 4*x             
                           = c_3(reverse#(x))
        
        
        Following rules are (at-least) weakly oriented:
        shuffle#(add(n,x)) =  4 + 4*x + x^2                        
                           >= 4*x + x^2                            
                           =  c_5(shuffle#(reverse(x)),reverse#(x))
        
           app(add(n,x),y) =  2 + x + y                            
                           >= 2 + x + y                            
                           =  add(n,app(x,y))                      
        
              app(nil(),y) =  y                                    
                           >= y                                    
                           =  y                                    
        
         reverse(add(n,x)) =  2 + x                                
                           >= 2 + x                                
                           =  app(reverse(x),add(n,nil()))         
        
            reverse(nil()) =  0                                    
                           >= 0                                    
                           =  nil()                                
        
***** Step 1.b:6.b:3.a:1.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            reverse#(add(n,x)) -> c_3(reverse#(x))
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

***** Step 1.b:6.b:3.a:1.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            reverse#(add(n,x)) -> c_3(reverse#(x))
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:reverse#(add(n,x)) -> c_3(reverse#(x))
             -->_1 reverse#(add(n,x)) -> c_3(reverse#(x)):1
          
          2:W:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):2
             -->_2 reverse#(add(n,x)) -> c_3(reverse#(x)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
          1: reverse#(add(n,x)) -> c_3(reverse#(x))
***** Step 1.b:6.b: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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 1.b:6.b:3.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
        - Weak DPs:
            reverse#(add(n,x)) -> c_3(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
             -->_2 reverse#(add(n,x)) -> c_3(reverse#(x)):2
             -->_1 shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x)):1
          
          2:W:reverse#(add(n,x)) -> c_3(reverse#(x))
             -->_1 reverse#(add(n,x)) -> c_3(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_3(reverse#(x))
**** Step 1.b:6.b:3.b:2: SimplifyRHS.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)),reverse#(x))
             -->_1 shuffle#(add(n,x)) -> c_5(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_5(shuffle#(reverse(x)))
**** Step 1.b:6.b:3.b:3: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + 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_5(shuffle#(reverse(x)))
          
        The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.b:3.b:3.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + 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_5) = {1}
        
        Following symbols are considered usable:
          {app,reverse,app#,reverse#,shuffle#}
        TcT has computed the following interpretation:
               p(add) = [1] x2 + [4]         
               p(app) = [1] x1 + [1] x2 + [0]
               p(nil) = [0]                  
           p(reverse) = [1] x1 + [0]         
           p(shuffle) = [1]                  
              p(app#) = [8] x1 + [1] x2 + [1]
          p(reverse#) = [8]                  
          p(shuffle#) = [4] x1 + [4]         
               p(c_1) = [0]                  
               p(c_2) = [1]                  
               p(c_3) = [0]                  
               p(c_4) = [2]                  
               p(c_5) = [1] x1 + [14]        
               p(c_6) = [8]                  
        
        Following rules are strictly oriented:
        shuffle#(add(n,x)) = [4] x + [20]             
                           > [4] x + [18]             
                           = c_5(shuffle#(reverse(x)))
        
        
        Following rules are (at-least) weakly oriented:
          app(add(n,x),y) =  [1] x + [1] y + [4]         
                          >= [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] x + [4]                 
                          >= [1] x + [4]                 
                          =  app(reverse(x),add(n,nil()))
        
           reverse(nil()) =  [0]                         
                          >= [0]                         
                          =  nil()                       
        
***** Step 1.b:6.b:3.b:3.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

***** Step 1.b:6.b:3.b:3.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            shuffle#(add(n,x)) -> c_5(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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:shuffle#(add(n,x)) -> c_5(shuffle#(reverse(x)))
             -->_1 shuffle#(add(n,x)) -> c_5(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_5(shuffle#(reverse(x)))
***** Step 1.b:6.b:3.b:3.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,reverse/1,shuffle/1,app#/2,reverse#/1,shuffle#/1} / {add/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,reverse#,shuffle#} and constructors {add,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^3))