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

** Step 1.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            p(f(f(x))) -> q(f(g(x)))
            p(g(g(x))) -> q(g(f(x)))
            q(f(f(x))) -> p(f(g(x)))
            q(g(g(x))) -> p(g(f(x)))
        - Signature:
            {p/1,q/1} / {f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {p,q} and constructors {f,g}
    + 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:
          none
        
        Following symbols are considered usable:
          {p,q}
        TcT has computed the following interpretation:
          p(f) = [1] x1 + [2]
          p(g) = [0]         
          p(p) = [1] x1 + [4]
          p(q) = [1] x1 + [4]
        
        Following rules are strictly oriented:
        p(f(f(x))) = [1] x + [8]
                   > [6]        
                   = q(f(g(x))) 
        
        q(f(f(x))) = [1] x + [8]
                   > [6]        
                   = p(f(g(x))) 
        
        
        Following rules are (at-least) weakly oriented:
        p(g(g(x))) =  [4]       
                   >= [4]       
                   =  q(g(f(x)))
        
        q(g(g(x))) =  [4]       
                   >= [4]       
                   =  p(g(f(x)))
        
** Step 1.b:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            p(g(g(x))) -> q(g(f(x)))
            q(g(g(x))) -> p(g(f(x)))
        - Weak TRS:
            p(f(f(x))) -> q(f(g(x)))
            q(f(f(x))) -> p(f(g(x)))
        - Signature:
            {p/1,q/1} / {f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {p,q} and constructors {f,g}
    + 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:
          none
        
        Following symbols are considered usable:
          {p,q}
        TcT has computed the following interpretation:
          p(f) = [0]         
          p(g) = [1] x1 + [4]
          p(p) = [1] x1 + [9]
          p(q) = [2] x1 + [9]
        
        Following rules are strictly oriented:
        q(g(g(x))) = [2] x + [25]
                   > [13]        
                   = p(g(f(x)))  
        
        
        Following rules are (at-least) weakly oriented:
        p(f(f(x))) =  [9]         
                   >= [9]         
                   =  q(f(g(x)))  
        
        p(g(g(x))) =  [1] x + [17]
                   >= [17]        
                   =  q(g(f(x)))  
        
        q(f(f(x))) =  [9]         
                   >= [9]         
                   =  p(f(g(x)))  
        
** Step 1.b:3: NaturalPI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            p(g(g(x))) -> q(g(f(x)))
        - Weak TRS:
            p(f(f(x))) -> q(f(g(x)))
            q(f(f(x))) -> p(f(g(x)))
            q(g(g(x))) -> p(g(f(x)))
        - Signature:
            {p/1,q/1} / {f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {p,q} and constructors {f,g}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          none
        
        Following symbols are considered usable:
          {p,q}
        TcT has computed the following interpretation:
          p(f) = 0       
          p(g) = 1 + x1  
          p(p) = 2 + 8*x1
          p(q) = 2 + 8*x1
        
        Following rules are strictly oriented:
        p(g(g(x))) = 18 + 8*x  
                   > 10        
                   = q(g(f(x)))
        
        
        Following rules are (at-least) weakly oriented:
        p(f(f(x))) =  2         
                   >= 2         
                   =  q(f(g(x)))
        
        q(f(f(x))) =  2         
                   >= 2         
                   =  p(f(g(x)))
        
        q(g(g(x))) =  18 + 8*x  
                   >= 10        
                   =  p(g(f(x)))
        
** Step 1.b:4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            p(f(f(x))) -> q(f(g(x)))
            p(g(g(x))) -> q(g(f(x)))
            q(f(f(x))) -> p(f(g(x)))
            q(g(g(x))) -> p(g(f(x)))
        - Signature:
            {p/1,q/1} / {f/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {p,q} and constructors {f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))