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


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


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

** Step 1.b:1: ToInnermost MAYBE
    + Considered Problem:
        - Strict TRS:
            int(x,x) -> cons(x,nil())
            int(0(),s(y)) -> cons(0(),int(s(0()),s(y)))
            int(s(x),0()) -> nil()
            int(s(x),s(y)) -> intlist(int(x,y))
            intlist(cons(x,y)) -> cons(s(x),intlist(y))
            intlist(cons(x,nil())) -> cons(s(x),nil())
            intlist(nil()) -> nil()
        - Signature:
            {int/2,intlist/1} / {0/0,cons/2,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {int,intlist} and constructors {0,cons,nil,s}
    + 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 MAYBE
    + Considered Problem:
        - Strict TRS:
            int(x,x) -> cons(x,nil())
            int(0(),s(y)) -> cons(0(),int(s(0()),s(y)))
            int(s(x),0()) -> nil()
            int(s(x),s(y)) -> intlist(int(x,y))
            intlist(cons(x,y)) -> cons(s(x),intlist(y))
            intlist(cons(x,nil())) -> cons(s(x),nil())
            intlist(nil()) -> nil()
        - Signature:
            {int/2,intlist/1} / {0/0,cons/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {int,intlist} and constructors {0,cons,nil,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          int#(x,x) -> c_1()
          int#(0(),s(y)) -> c_2(int#(s(0()),s(y)))
          int#(s(x),0()) -> c_3()
          int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y))
          intlist#(cons(x,y)) -> c_5(intlist#(y))
          intlist#(cons(x,nil())) -> c_6()
          intlist#(nil()) -> c_7()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            int#(x,x) -> c_1()
            int#(0(),s(y)) -> c_2(int#(s(0()),s(y)))
            int#(s(x),0()) -> c_3()
            int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y))
            intlist#(cons(x,y)) -> c_5(intlist#(y))
            intlist#(cons(x,nil())) -> c_6()
            intlist#(nil()) -> c_7()
        - Weak TRS:
            int(x,x) -> cons(x,nil())
            int(0(),s(y)) -> cons(0(),int(s(0()),s(y)))
            int(s(x),0()) -> nil()
            int(s(x),s(y)) -> intlist(int(x,y))
            intlist(cons(x,y)) -> cons(s(x),intlist(y))
            intlist(cons(x,nil())) -> cons(s(x),nil())
            intlist(nil()) -> nil()
        - Signature:
            {int/2,intlist/1,int#/2,intlist#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {int#,intlist#} and constructors {0,cons,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,6,7}
        by application of
          Pre({1,3,6,7}) = {2,4,5}.
        Here rules are labelled as follows:
          1: int#(x,x) -> c_1()
          2: int#(0(),s(y)) -> c_2(int#(s(0()),s(y)))
          3: int#(s(x),0()) -> c_3()
          4: int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y))
          5: intlist#(cons(x,y)) -> c_5(intlist#(y))
          6: intlist#(cons(x,nil())) -> c_6()
          7: intlist#(nil()) -> c_7()
** Step 1.b:4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            int#(0(),s(y)) -> c_2(int#(s(0()),s(y)))
            int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y))
            intlist#(cons(x,y)) -> c_5(intlist#(y))
        - Weak DPs:
            int#(x,x) -> c_1()
            int#(s(x),0()) -> c_3()
            intlist#(cons(x,nil())) -> c_6()
            intlist#(nil()) -> c_7()
        - Weak TRS:
            int(x,x) -> cons(x,nil())
            int(0(),s(y)) -> cons(0(),int(s(0()),s(y)))
            int(s(x),0()) -> nil()
            int(s(x),s(y)) -> intlist(int(x,y))
            intlist(cons(x,y)) -> cons(s(x),intlist(y))
            intlist(cons(x,nil())) -> cons(s(x),nil())
            intlist(nil()) -> nil()
        - Signature:
            {int/2,intlist/1,int#/2,intlist#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {int#,intlist#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:int#(0(),s(y)) -> c_2(int#(s(0()),s(y)))
             -->_1 int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y)):2
             -->_1 int#(x,x) -> c_1():4
          
          2:S:int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y))
             -->_1 intlist#(cons(x,y)) -> c_5(intlist#(y)):3
             -->_1 intlist#(nil()) -> c_7():7
             -->_1 intlist#(cons(x,nil())) -> c_6():6
             -->_2 int#(s(x),0()) -> c_3():5
             -->_2 int#(x,x) -> c_1():4
             -->_2 int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y)):2
             -->_2 int#(0(),s(y)) -> c_2(int#(s(0()),s(y))):1
          
          3:S:intlist#(cons(x,y)) -> c_5(intlist#(y))
             -->_1 intlist#(nil()) -> c_7():7
             -->_1 intlist#(cons(x,nil())) -> c_6():6
             -->_1 intlist#(cons(x,y)) -> c_5(intlist#(y)):3
          
          4:W:int#(x,x) -> c_1()
             
          
          5:W:int#(s(x),0()) -> c_3()
             
          
          6:W:intlist#(cons(x,nil())) -> c_6()
             
          
          7:W:intlist#(nil()) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: int#(x,x) -> c_1()
          5: int#(s(x),0()) -> c_3()
          6: intlist#(cons(x,nil())) -> c_6()
          7: intlist#(nil()) -> c_7()
** Step 1.b:5: DecomposeDG MAYBE
    + Considered Problem:
        - Strict DPs:
            int#(0(),s(y)) -> c_2(int#(s(0()),s(y)))
            int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y))
            intlist#(cons(x,y)) -> c_5(intlist#(y))
        - Weak TRS:
            int(x,x) -> cons(x,nil())
            int(0(),s(y)) -> cons(0(),int(s(0()),s(y)))
            int(s(x),0()) -> nil()
            int(s(x),s(y)) -> intlist(int(x,y))
            intlist(cons(x,y)) -> cons(s(x),intlist(y))
            intlist(cons(x,nil())) -> cons(s(x),nil())
            intlist(nil()) -> nil()
        - Signature:
            {int/2,intlist/1,int#/2,intlist#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {int#,intlist#} and constructors {0,cons,nil,s}
    + 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
          int#(0(),s(y)) -> c_2(int#(s(0()),s(y)))
          int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y))
        and a lower component
          intlist#(cons(x,y)) -> c_5(intlist#(y))
        Further, following extension rules are added to the lower component.
          int#(0(),s(y)) -> int#(s(0()),s(y))
          int#(s(x),s(y)) -> int#(x,y)
          int#(s(x),s(y)) -> intlist#(int(x,y))
*** Step 1.b:5.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            int#(0(),s(y)) -> c_2(int#(s(0()),s(y)))
            int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y))
        - Weak TRS:
            int(x,x) -> cons(x,nil())
            int(0(),s(y)) -> cons(0(),int(s(0()),s(y)))
            int(s(x),0()) -> nil()
            int(s(x),s(y)) -> intlist(int(x,y))
            intlist(cons(x,y)) -> cons(s(x),intlist(y))
            intlist(cons(x,nil())) -> cons(s(x),nil())
            intlist(nil()) -> nil()
        - Signature:
            {int/2,intlist/1,int#/2,intlist#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {int#,intlist#} and constructors {0,cons,nil,s}
    + 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:
          2: int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y))
          
        Consider the set of all dependency pairs
          1: int#(0(),s(y)) -> c_2(int#(s(0()),s(y)))
          2: int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {2}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
**** Step 1.b:5.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            int#(0(),s(y)) -> c_2(int#(s(0()),s(y)))
            int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y))
        - Weak TRS:
            int(x,x) -> cons(x,nil())
            int(0(),s(y)) -> cons(0(),int(s(0()),s(y)))
            int(s(x),0()) -> nil()
            int(s(x),s(y)) -> intlist(int(x,y))
            intlist(cons(x,y)) -> cons(s(x),intlist(y))
            intlist(cons(x,nil())) -> cons(s(x),nil())
            intlist(nil()) -> nil()
        - Signature:
            {int/2,intlist/1,int#/2,intlist#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {int#,intlist#} and constructors {0,cons,nil,s}
    + 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_2) = {1},
          uargs(c_4) = {1,2}
        
        Following symbols are considered usable:
          {int#,intlist#}
        TcT has computed the following interpretation:
                 p(0) = [4]                  
              p(cons) = [1] x2 + [10]        
               p(int) = [4]                  
           p(intlist) = [2] x1 + [0]         
               p(nil) = [2]                  
                 p(s) = [1] x1 + [8]         
              p(int#) = [2] x2 + [0]         
          p(intlist#) = [1]                  
               p(c_1) = [8]                  
               p(c_2) = [1] x1 + [0]         
               p(c_3) = [0]                  
               p(c_4) = [4] x1 + [1] x2 + [4]
               p(c_5) = [1]                  
               p(c_6) = [0]                  
               p(c_7) = [0]                  
        
        Following rules are strictly oriented:
        int#(s(x),s(y)) = [2] y + [16]                     
                        > [2] y + [8]                      
                        = c_4(intlist#(int(x,y)),int#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        int#(0(),s(y)) =  [2] y + [16]          
                       >= [2] y + [16]          
                       =  c_2(int#(s(0()),s(y)))
        
**** Step 1.b:5.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            int#(0(),s(y)) -> c_2(int#(s(0()),s(y)))
        - Weak DPs:
            int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y))
        - Weak TRS:
            int(x,x) -> cons(x,nil())
            int(0(),s(y)) -> cons(0(),int(s(0()),s(y)))
            int(s(x),0()) -> nil()
            int(s(x),s(y)) -> intlist(int(x,y))
            intlist(cons(x,y)) -> cons(s(x),intlist(y))
            intlist(cons(x,nil())) -> cons(s(x),nil())
            intlist(nil()) -> nil()
        - Signature:
            {int/2,intlist/1,int#/2,intlist#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {int#,intlist#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 1.b:5.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            int#(0(),s(y)) -> c_2(int#(s(0()),s(y)))
            int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y))
        - Weak TRS:
            int(x,x) -> cons(x,nil())
            int(0(),s(y)) -> cons(0(),int(s(0()),s(y)))
            int(s(x),0()) -> nil()
            int(s(x),s(y)) -> intlist(int(x,y))
            intlist(cons(x,y)) -> cons(s(x),intlist(y))
            intlist(cons(x,nil())) -> cons(s(x),nil())
            intlist(nil()) -> nil()
        - Signature:
            {int/2,intlist/1,int#/2,intlist#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {int#,intlist#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:int#(0(),s(y)) -> c_2(int#(s(0()),s(y)))
             -->_1 int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y)):2
          
          2:W:int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y))
             -->_2 int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y)):2
             -->_2 int#(0(),s(y)) -> c_2(int#(s(0()),s(y))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: int#(0(),s(y)) -> c_2(int#(s(0()),s(y)))
          2: int#(s(x),s(y)) -> c_4(intlist#(int(x,y)),int#(x,y))
**** Step 1.b:5.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            int(x,x) -> cons(x,nil())
            int(0(),s(y)) -> cons(0(),int(s(0()),s(y)))
            int(s(x),0()) -> nil()
            int(s(x),s(y)) -> intlist(int(x,y))
            intlist(cons(x,y)) -> cons(s(x),intlist(y))
            intlist(cons(x,nil())) -> cons(s(x),nil())
            intlist(nil()) -> nil()
        - Signature:
            {int/2,intlist/1,int#/2,intlist#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {int#,intlist#} and constructors {0,cons,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:5.b:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          intlist#(cons(x,y)) -> c_5(intlist#(y))
      - Weak DPs:
          int#(0(),s(y)) -> int#(s(0()),s(y))
          int#(s(x),s(y)) -> int#(x,y)
          int#(s(x),s(y)) -> intlist#(int(x,y))
      - Weak TRS:
          int(x,x) -> cons(x,nil())
          int(0(),s(y)) -> cons(0(),int(s(0()),s(y)))
          int(s(x),0()) -> nil()
          int(s(x),s(y)) -> intlist(int(x,y))
          intlist(cons(x,y)) -> cons(s(x),intlist(y))
          intlist(cons(x,nil())) -> cons(s(x),nil())
          intlist(nil()) -> nil()
      - Signature:
          {int/2,intlist/1,int#/2,intlist#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {int#,intlist#} and constructors {0,cons,nil,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
WORST_CASE(Omega(n^1),?)