/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),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
            append(Nil(),ys) -> ys
            goal(x,y) -> append(x,y)
        - Signature:
            {append/2,goal/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,goal} and constructors {Cons,Nil}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
            append(Nil(),ys) -> ys
            goal(x,y) -> append(x,y)
        - Signature:
            {append/2,goal/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,goal} and constructors {Cons,Nil}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
*** Step 1.a:1.a:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
            append(Nil(),ys) -> ys
            goal(x,y) -> append(x,y)
        - Signature:
            {append/2,goal/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,goal} and constructors {Cons,Nil}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "Cons") :: ["A"(0) x "A"(1)] -(1)-> "A"(1)
          F (TrsFun "Cons") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "Nil") :: [] -(0)-> "A"(1)
          F (TrsFun "append") :: ["A"(1) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "goal") :: ["A"(1) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "main") :: ["A"(1) x "A"(0)] -(1)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
          append(Nil(),ys) -> ys
          goal(x,y) -> append(x,y)
          main(x1,x2) -> goal(x1,x2)
        2. Weak:
          
        

*** Step 1.a:1.b:1: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
            append(Nil(),ys) -> ys
            goal(x,y) -> append(x,y)
        - Signature:
            {append/2,goal/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,goal} and constructors {Cons,Nil}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          append(y,z){y -> Cons(x,y)} =
            append(Cons(x,y),z) ->^+ Cons(x,append(y,z))
              = C[append(y,z) = append(y,z){}]

** Step 1.b:1: Bounds.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
            append(Nil(),ys) -> ys
            goal(x,y) -> append(x,y)
        - Signature:
            {append/2,goal/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,goal} and constructors {Cons,Nil}
    + Applied Processor:
        Bounds {initialAutomaton = minimal, enrichment = match}
    + Details:
        The problem is match-bounded by 1.
        The enriched problem is compatible with follwoing automaton.
          Cons_0(2,2) -> 1
          Cons_0(2,2) -> 2
          Cons_0(2,2) -> 3
          Cons_1(2,3) -> 1
          Cons_1(2,3) -> 3
          Nil_0() -> 1
          Nil_0() -> 2
          Nil_0() -> 3
          append_0(2,2) -> 1
          append_1(2,2) -> 1
          append_1(2,2) -> 3
          goal_0(2,2) -> 1
          2 -> 1
          2 -> 3
** Step 1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            append(Cons(x,xs),ys) -> Cons(x,append(xs,ys))
            append(Nil(),ys) -> ys
            goal(x,y) -> append(x,y)
        - Signature:
            {append/2,goal/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,goal} and constructors {Cons,Nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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