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


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WORST_CASE(Omega(n^1),?)
* Step 1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            +(X,0()) -> X
            +(X,s(Y)) -> s(+(X,Y))
            double(X) -> +(X,X)
            f(0(),s(0()),X) -> f(X,double(X),X)
            g(X,Y) -> X
            g(X,Y) -> Y
        - Signature:
            {+/2,double/1,f/3,g/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,double,f,g} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            +(X,0()) -> X
            +(X,s(Y)) -> s(+(X,Y))
            double(X) -> +(X,X)
            f(0(),s(0()),X) -> f(X,double(X),X)
            g(X,Y) -> X
            g(X,Y) -> Y
        - Signature:
            {+/2,double/1,f/3,g/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,double,f,g} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            +(X,0()) -> X
            +(X,s(Y)) -> s(+(X,Y))
            double(X) -> +(X,X)
            f(0(),s(0()),X) -> f(X,double(X),X)
            g(X,Y) -> X
            g(X,Y) -> Y
        - Signature:
            {+/2,double/1,f/3,g/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,double,f,g} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          +(x,y){y -> s(y)} =
            +(x,s(y)) ->^+ s(+(x,y))
              = C[+(x,y) = +(x,y){}]

** Step 1.b:1: DependencyPairs.  MAYBE
    + Considered Problem:
        - Strict TRS:
            +(X,0()) -> X
            +(X,s(Y)) -> s(+(X,Y))
            double(X) -> +(X,X)
            f(0(),s(0()),X) -> f(X,double(X),X)
            g(X,Y) -> X
            g(X,Y) -> Y
        - Signature:
            {+/2,double/1,f/3,g/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,double,f,g} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          +#(X,0()) -> c_1()
          +#(X,s(Y)) -> c_2(+#(X,Y))
          double#(X) -> c_3(+#(X,X))
          f#(0(),s(0()),X) -> c_4(f#(X,double(X),X),double#(X))
          g#(X,Y) -> c_5()
          g#(X,Y) -> c_6()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  MAYBE
    + Considered Problem:
        - Strict DPs:
            +#(X,0()) -> c_1()
            +#(X,s(Y)) -> c_2(+#(X,Y))
            double#(X) -> c_3(+#(X,X))
            f#(0(),s(0()),X) -> c_4(f#(X,double(X),X),double#(X))
            g#(X,Y) -> c_5()
            g#(X,Y) -> c_6()
        - Weak TRS:
            +(X,0()) -> X
            +(X,s(Y)) -> s(+(X,Y))
            double(X) -> +(X,X)
            f(0(),s(0()),X) -> f(X,double(X),X)
            g(X,Y) -> X
            g(X,Y) -> Y
        - Signature:
            {+/2,double/1,f/3,g/2,+#/2,double#/1,f#/3,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,f#,g#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,5,6}
        by application of
          Pre({1,5,6}) = {2,3}.
        Here rules are labelled as follows:
          1: +#(X,0()) -> c_1()
          2: +#(X,s(Y)) -> c_2(+#(X,Y))
          3: double#(X) -> c_3(+#(X,X))
          4: f#(0(),s(0()),X) -> c_4(f#(X,double(X),X),double#(X))
          5: g#(X,Y) -> c_5()
          6: g#(X,Y) -> c_6()
** Step 1.b:3: RemoveWeakSuffixes.  MAYBE
    + Considered Problem:
        - Strict DPs:
            +#(X,s(Y)) -> c_2(+#(X,Y))
            double#(X) -> c_3(+#(X,X))
            f#(0(),s(0()),X) -> c_4(f#(X,double(X),X),double#(X))
        - Weak DPs:
            +#(X,0()) -> c_1()
            g#(X,Y) -> c_5()
            g#(X,Y) -> c_6()
        - Weak TRS:
            +(X,0()) -> X
            +(X,s(Y)) -> s(+(X,Y))
            double(X) -> +(X,X)
            f(0(),s(0()),X) -> f(X,double(X),X)
            g(X,Y) -> X
            g(X,Y) -> Y
        - Signature:
            {+/2,double/1,f/3,g/2,+#/2,double#/1,f#/3,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,f#,g#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:+#(X,s(Y)) -> c_2(+#(X,Y))
             -->_1 +#(X,0()) -> c_1():4
             -->_1 +#(X,s(Y)) -> c_2(+#(X,Y)):1
          
          2:S:double#(X) -> c_3(+#(X,X))
             -->_1 +#(X,0()) -> c_1():4
             -->_1 +#(X,s(Y)) -> c_2(+#(X,Y)):1
          
          3:S:f#(0(),s(0()),X) -> c_4(f#(X,double(X),X),double#(X))
             -->_1 f#(0(),s(0()),X) -> c_4(f#(X,double(X),X),double#(X)):3
             -->_2 double#(X) -> c_3(+#(X,X)):2
          
          4:W:+#(X,0()) -> c_1()
             
          
          5:W:g#(X,Y) -> c_5()
             
          
          6:W:g#(X,Y) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: g#(X,Y) -> c_6()
          5: g#(X,Y) -> c_5()
          4: +#(X,0()) -> c_1()
** Step 1.b:4: UsableRules.  MAYBE
    + Considered Problem:
        - Strict DPs:
            +#(X,s(Y)) -> c_2(+#(X,Y))
            double#(X) -> c_3(+#(X,X))
            f#(0(),s(0()),X) -> c_4(f#(X,double(X),X),double#(X))
        - Weak TRS:
            +(X,0()) -> X
            +(X,s(Y)) -> s(+(X,Y))
            double(X) -> +(X,X)
            f(0(),s(0()),X) -> f(X,double(X),X)
            g(X,Y) -> X
            g(X,Y) -> Y
        - Signature:
            {+/2,double/1,f/3,g/2,+#/2,double#/1,f#/3,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,f#,g#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          +(X,0()) -> X
          +(X,s(Y)) -> s(+(X,Y))
          double(X) -> +(X,X)
          +#(X,s(Y)) -> c_2(+#(X,Y))
          double#(X) -> c_3(+#(X,X))
          f#(0(),s(0()),X) -> c_4(f#(X,double(X),X),double#(X))
** Step 1.b:5: DecomposeDG.  MAYBE
    + Considered Problem:
        - Strict DPs:
            +#(X,s(Y)) -> c_2(+#(X,Y))
            double#(X) -> c_3(+#(X,X))
            f#(0(),s(0()),X) -> c_4(f#(X,double(X),X),double#(X))
        - Weak TRS:
            +(X,0()) -> X
            +(X,s(Y)) -> s(+(X,Y))
            double(X) -> +(X,X)
        - Signature:
            {+/2,double/1,f/3,g/2,+#/2,double#/1,f#/3,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,f#,g#} and constructors {0,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          f#(0(),s(0()),X) -> c_4(f#(X,double(X),X),double#(X))
        and a lower component
          +#(X,s(Y)) -> c_2(+#(X,Y))
          double#(X) -> c_3(+#(X,X))
        Further, following extension rules are added to the lower component.
          f#(0(),s(0()),X) -> double#(X)
          f#(0(),s(0()),X) -> f#(X,double(X),X)
*** Step 1.b:5.a:1: SimplifyRHS.  MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(0(),s(0()),X) -> c_4(f#(X,double(X),X),double#(X))
        - Weak TRS:
            +(X,0()) -> X
            +(X,s(Y)) -> s(+(X,Y))
            double(X) -> +(X,X)
        - Signature:
            {+/2,double/1,f/3,g/2,+#/2,double#/1,f#/3,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,f#,g#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(0(),s(0()),X) -> c_4(f#(X,double(X),X),double#(X))
             -->_1 f#(0(),s(0()),X) -> c_4(f#(X,double(X),X),double#(X)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(0(),s(0()),X) -> c_4(f#(X,double(X),X))
*** Step 1.b:5.a:2: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          f#(0(),s(0()),X) -> c_4(f#(X,double(X),X))
      - Weak TRS:
          +(X,0()) -> X
          +(X,s(Y)) -> s(+(X,Y))
          double(X) -> +(X,X)
      - Signature:
          {+/2,double/1,f/3,g/2,+#/2,double#/1,f#/3,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {+#,double#,f#,g#} and constructors {0,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
*** Step 1.b:5.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            +#(X,s(Y)) -> c_2(+#(X,Y))
            double#(X) -> c_3(+#(X,X))
        - Weak DPs:
            f#(0(),s(0()),X) -> double#(X)
            f#(0(),s(0()),X) -> f#(X,double(X),X)
        - Weak TRS:
            +(X,0()) -> X
            +(X,s(Y)) -> s(+(X,Y))
            double(X) -> +(X,X)
        - Signature:
            {+/2,double/1,f/3,g/2,+#/2,double#/1,f#/3,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,f#,g#} and constructors {0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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_3) = {1}
        
        Following symbols are considered usable:
          {+#,double#,f#,g#}
        TcT has computed the following interpretation:
                p(+) = [0]         
                p(0) = [0]         
           p(double) = [0]         
                p(f) = [0]         
                p(g) = [0]         
                p(s) = [0]         
               p(+#) = [0]         
          p(double#) = [10]        
               p(f#) = [10]        
               p(g#) = [1]         
              p(c_1) = [0]         
              p(c_2) = [4] x1 + [0]
              p(c_3) = [2] x1 + [2]
              p(c_4) = [2]         
              p(c_5) = [0]         
              p(c_6) = [0]         
        
        Following rules are strictly oriented:
        double#(X) = [10]        
                   > [2]         
                   = c_3(+#(X,X))
        
        
        Following rules are (at-least) weakly oriented:
              +#(X,s(Y)) =  [0]              
                         >= [0]              
                         =  c_2(+#(X,Y))     
        
        f#(0(),s(0()),X) =  [10]             
                         >= [10]             
                         =  double#(X)       
        
        f#(0(),s(0()),X) =  [10]             
                         >= [10]             
                         =  f#(X,double(X),X)
        
*** Step 1.b:5.b:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            +#(X,s(Y)) -> c_2(+#(X,Y))
        - Weak DPs:
            double#(X) -> c_3(+#(X,X))
            f#(0(),s(0()),X) -> double#(X)
            f#(0(),s(0()),X) -> f#(X,double(X),X)
        - Weak TRS:
            +(X,0()) -> X
            +(X,s(Y)) -> s(+(X,Y))
            double(X) -> +(X,X)
        - Signature:
            {+/2,double/1,f/3,g/2,+#/2,double#/1,f#/3,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,f#,g#} and constructors {0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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_3) = {1}
        
        Following symbols are considered usable:
          {+#,double#,f#,g#}
        TcT has computed the following interpretation:
                p(+) = [6] x1 + [4] x2 + [0]
                p(0) = [0]                  
           p(double) = [1]                  
                p(f) = [1] x2 + [2] x3 + [8]
                p(g) = [1] x2 + [0]         
                p(s) = [1] x1 + [4]         
               p(+#) = [1] x2 + [0]         
          p(double#) = [4] x1 + [10]        
               p(f#) = [8] x3 + [10]        
               p(g#) = [2] x2 + [2]         
              p(c_1) = [1]                  
              p(c_2) = [1] x1 + [0]         
              p(c_3) = [4] x1 + [0]         
              p(c_4) = [1] x1 + [1]         
              p(c_5) = [2]                  
              p(c_6) = [1]                  
        
        Following rules are strictly oriented:
        +#(X,s(Y)) = [1] Y + [4] 
                   > [1] Y + [0] 
                   = c_2(+#(X,Y))
        
        
        Following rules are (at-least) weakly oriented:
              double#(X) =  [4] X + [10]     
                         >= [4] X + [0]      
                         =  c_3(+#(X,X))     
        
        f#(0(),s(0()),X) =  [8] X + [10]     
                         >= [4] X + [10]     
                         =  double#(X)       
        
        f#(0(),s(0()),X) =  [8] X + [10]     
                         >= [8] X + [10]     
                         =  f#(X,double(X),X)
        
*** Step 1.b:5.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            +#(X,s(Y)) -> c_2(+#(X,Y))
            double#(X) -> c_3(+#(X,X))
            f#(0(),s(0()),X) -> double#(X)
            f#(0(),s(0()),X) -> f#(X,double(X),X)
        - Weak TRS:
            +(X,0()) -> X
            +(X,s(Y)) -> s(+(X,Y))
            double(X) -> +(X,X)
        - Signature:
            {+/2,double/1,f/3,g/2,+#/2,double#/1,f#/3,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,double#,f#,g#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),?)