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


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WORST_CASE(?,O(1))
* Step 1: Sum.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            f(k(a()),k(b()),X) -> f(X,X,X)
            g(X) -> u(h(X),h(X),X)
            h(d()) -> c(a())
            h(d()) -> c(b())
            u(d(),c(Y),X) -> k(Y)
        - Signature:
            {f/3,g/1,h/1,u/3} / {a/0,b/0,c/1,d/0,k/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h,u} and constructors {a,b,c,d,k}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  MAYBE
    + Considered Problem:
        - Strict TRS:
            f(k(a()),k(b()),X) -> f(X,X,X)
            g(X) -> u(h(X),h(X),X)
            h(d()) -> c(a())
            h(d()) -> c(b())
            u(d(),c(Y),X) -> k(Y)
        - Signature:
            {f/3,g/1,h/1,u/3} / {a/0,b/0,c/1,d/0,k/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h,u} and constructors {a,b,c,d,k}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            f(k(a()),k(b()),X) -> f(X,X,X)
            g(X) -> u(h(X),h(X),X)
            h(d()) -> c(a())
            h(d()) -> c(b())
            u(d(),c(Y),X) -> k(Y)
        - Signature:
            {f/3,g/1,h/1,u/3} / {a/0,b/0,c/1,d/0,k/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h,u} and constructors {a,b,c,d,k}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "a") :: [] -(0)-> "A"(0, 0, 0)
          F (TrsFun "b") :: [] -(0)-> "A"(0, 0, 0)
          F (TrsFun "c") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "d") :: [] -(0)-> "A"(0, 0, 0)
          F (TrsFun "d") :: [] -(0)-> "A"(1, 1, 1)
          F (TrsFun "f") :: ["A"(0, 0, 0) x "A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "g") :: ["A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "h") :: ["A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "k") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "main") :: ["A"(1, 1, 1) x "A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "u") :: ["A"(1, 1, 1) x "A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          f(k(a()),k(b()),X) -> f(X,X,X)
          g(X) -> u(h(X),h(X),X)
          h(d()) -> c(a())
          h(d()) -> c(b())
          u(d(),c(Y),X) -> k(Y)
          main(x1,x2,x3) -> u(x1,x2,x3)
        2. Weak:
          
        

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            f(k(a()),k(b()),X) -> f(X,X,X)
            g(X) -> u(h(X),h(X),X)
            h(d()) -> c(a())
            h(d()) -> c(b())
            u(d(),c(Y),X) -> k(Y)
        - Signature:
            {f/3,g/1,h/1,u/3} / {a/0,b/0,c/1,d/0,k/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h,u} and constructors {a,b,c,d,k}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
          g#(X) -> c_2(u#(h(X),h(X),X),h#(X),h#(X))
          h#(d()) -> c_3()
          h#(d()) -> c_4()
          u#(d(),c(Y),X) -> c_5()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
            g#(X) -> c_2(u#(h(X),h(X),X),h#(X),h#(X))
            h#(d()) -> c_3()
            h#(d()) -> c_4()
            u#(d(),c(Y),X) -> c_5()
        - Weak TRS:
            f(k(a()),k(b()),X) -> f(X,X,X)
            g(X) -> u(h(X),h(X),X)
            h(d()) -> c(a())
            h(d()) -> c(b())
            u(d(),c(Y),X) -> k(Y)
        - Signature:
            {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,4,5}
        by application of
          Pre({1,3,4,5}) = {2}.
        Here rules are labelled as follows:
          1: f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
          2: g#(X) -> c_2(u#(h(X),h(X),X),h#(X),h#(X))
          3: h#(d()) -> c_3()
          4: h#(d()) -> c_4()
          5: u#(d(),c(Y),X) -> c_5()
** Step 1.b:3: PredecessorEstimation.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(X) -> c_2(u#(h(X),h(X),X),h#(X),h#(X))
        - Weak DPs:
            f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
            h#(d()) -> c_3()
            h#(d()) -> c_4()
            u#(d(),c(Y),X) -> c_5()
        - Weak TRS:
            f(k(a()),k(b()),X) -> f(X,X,X)
            g(X) -> u(h(X),h(X),X)
            h(d()) -> c(a())
            h(d()) -> c(b())
            u(d(),c(Y),X) -> k(Y)
        - Signature:
            {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {}.
        Here rules are labelled as follows:
          1: g#(X) -> c_2(u#(h(X),h(X),X),h#(X),h#(X))
          2: f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
          3: h#(d()) -> c_3()
          4: h#(d()) -> c_4()
          5: u#(d(),c(Y),X) -> c_5()
** Step 1.b:4: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
            g#(X) -> c_2(u#(h(X),h(X),X),h#(X),h#(X))
            h#(d()) -> c_3()
            h#(d()) -> c_4()
            u#(d(),c(Y),X) -> c_5()
        - Weak TRS:
            f(k(a()),k(b()),X) -> f(X,X,X)
            g(X) -> u(h(X),h(X),X)
            h(d()) -> c(a())
            h(d()) -> c(b())
            u(d(),c(Y),X) -> k(Y)
        - Signature:
            {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
             
          
          2:W:g#(X) -> c_2(u#(h(X),h(X),X),h#(X),h#(X))
             -->_1 u#(d(),c(Y),X) -> c_5():5
             -->_3 h#(d()) -> c_4():4
             -->_2 h#(d()) -> c_4():4
             -->_3 h#(d()) -> c_3():3
             -->_2 h#(d()) -> c_3():3
          
          3:W:h#(d()) -> c_3()
             
          
          4:W:h#(d()) -> c_4()
             
          
          5:W:u#(d(),c(Y),X) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: g#(X) -> c_2(u#(h(X),h(X),X),h#(X),h#(X))
          3: h#(d()) -> c_3()
          4: h#(d()) -> c_4()
          5: u#(d(),c(Y),X) -> c_5()
          1: f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
** Step 1.b:5: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(k(a()),k(b()),X) -> f(X,X,X)
            g(X) -> u(h(X),h(X),X)
            h(d()) -> c(a())
            h(d()) -> c(b())
            u(d(),c(Y),X) -> k(Y)
        - Signature:
            {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))