/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:
            compS_f#1(compS_f(x2),x1) -> compS_f#1(x2,S(x1))
            compS_f#1(id(),x3) -> S(x3)
            iter#3(0()) -> id()
            iter#3(S(x6)) -> compS_f(iter#3(x6))
            main(0()) -> 0()
            main(S(x9)) -> compS_f#1(iter#3(x9),0())
        - Signature:
            {compS_f#1/2,iter#3/1,main/1} / {0/0,S/1,compS_f/1,id/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {compS_f#1,iter#3,main} and constructors {0,S,compS_f,id}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            compS_f#1(compS_f(x2),x1) -> compS_f#1(x2,S(x1))
            compS_f#1(id(),x3) -> S(x3)
            iter#3(0()) -> id()
            iter#3(S(x6)) -> compS_f(iter#3(x6))
            main(0()) -> 0()
            main(S(x9)) -> compS_f#1(iter#3(x9),0())
        - Signature:
            {compS_f#1/2,iter#3/1,main/1} / {0/0,S/1,compS_f/1,id/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {compS_f#1,iter#3,main} and constructors {0,S,compS_f,id}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
*** Step 1.a:1.a:1: Decompose.  MAYBE
    + Considered Problem:
        - Strict TRS:
            compS_f#1(compS_f(x2),x1) -> compS_f#1(x2,S(x1))
            compS_f#1(id(),x3) -> S(x3)
            iter#3(0()) -> id()
            iter#3(S(x6)) -> compS_f(iter#3(x6))
            main(0()) -> 0()
            main(S(x9)) -> compS_f#1(iter#3(x9),0())
        - Signature:
            {compS_f#1/2,iter#3/1,main/1} / {0/0,S/1,compS_f/1,id/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {compS_f#1,iter#3,main} and constructors {0,S,compS_f,id}
    + Applied Processor:
        Decompose {onSelection = any filtered on not main function of strict-rules, withBound = BestCaseLBMax}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        For the best-case lower bound analysis the maximum derivation length of either sub-problem provides a lower
        bound in terms of derivation length of the original TRS.
        
        Problem (R)
          - Strict TRS:
              compS_f#1(compS_f(x2),x1) -> compS_f#1(x2,S(x1))
              compS_f#1(id(),x3) -> S(x3)
              iter#3(0()) -> id()
              iter#3(S(x6)) -> compS_f(iter#3(x6))
          - Weak TRS:
              main(0()) -> 0()
              main(S(x9)) -> compS_f#1(iter#3(x9),0())
          - Signature:
              {compS_f#1/2,iter#3/1,main/1} / {0/0,S/1,compS_f/1,id/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {compS_f#1,iter#3,main} and constructors {0,S,compS_f
              ,id}
        
        Problem (S)
          - Strict TRS:
              main(0()) -> 0()
              main(S(x9)) -> compS_f#1(iter#3(x9),0())
          - Signature:
              {compS_f#1/2,iter#3/1,main/1} / {0/0,S/1,compS_f/1,id/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {compS_f#1,iter#3,main} and constructors {0,S,compS_f
              ,id}
**** Step 1.a:1.a:1.a:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            compS_f#1(compS_f(x2),x1) -> compS_f#1(x2,S(x1))
            compS_f#1(id(),x3) -> S(x3)
            iter#3(0()) -> id()
            iter#3(S(x6)) -> compS_f(iter#3(x6))
        - Weak TRS:
            main(0()) -> 0()
            main(S(x9)) -> compS_f#1(iter#3(x9),0())
        - Signature:
            {compS_f#1/2,iter#3/1,main/1} / {0/0,S/1,compS_f/1,id/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {compS_f#1,iter#3,main} and constructors {0,S,compS_f,id}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "0") :: [] -(0)-> "A"(1)
          F (TrsFun "0") :: [] -(0)-> "A"(0)
          F (TrsFun "S") :: ["A"(1)] -(1)-> "A"(1)
          F (TrsFun "S") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "compS_f") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "compS_f#1") :: ["A"(0) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "id") :: [] -(0)-> "A"(0)
          F (TrsFun "iter#3") :: ["A"(1)] -(1)-> "A"(0)
          F (TrsFun "main") :: ["A"(1)] -(0)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          compS_f#1(compS_f(x2),x1) -> compS_f#1(x2,S(x1))
          compS_f#1(id(),x3) -> S(x3)
          iter#3(0()) -> id()
          iter#3(S(x6)) -> compS_f(iter#3(x6))
        2. Weak:
          
        

**** Step 1.a:1.a:1.b:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            main(0()) -> 0()
            main(S(x9)) -> compS_f#1(iter#3(x9),0())
        - Signature:
            {compS_f#1/2,iter#3/1,main/1} / {0/0,S/1,compS_f/1,id/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {compS_f#1,iter#3,main} and constructors {0,S,compS_f,id}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "0") :: [] -(0)-> "A"(0, 0, 1)
          F (TrsFun "0") :: [] -(0)-> "A"(0, 0, 0)
          F (TrsFun "S") :: ["A"(0, 1, 1)] -(0)-> "A"(0, 0, 1)
          F (TrsFun "compS_f#1") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "iter#3") :: ["A"(0, 1, 1)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "main") :: ["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:
          main(0()) -> 0()
          main(S(x9)) -> compS_f#1(iter#3(x9),0())
        2. Weak:
          
        

*** Step 1.a:1.b:1: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            compS_f#1(compS_f(x2),x1) -> compS_f#1(x2,S(x1))
            compS_f#1(id(),x3) -> S(x3)
            iter#3(0()) -> id()
            iter#3(S(x6)) -> compS_f(iter#3(x6))
            main(0()) -> 0()
            main(S(x9)) -> compS_f#1(iter#3(x9),0())
        - Signature:
            {compS_f#1/2,iter#3/1,main/1} / {0/0,S/1,compS_f/1,id/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {compS_f#1,iter#3,main} and constructors {0,S,compS_f,id}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          compS_f#1(x,y){x -> compS_f(x)} =
            compS_f#1(compS_f(x),y) ->^+ compS_f#1(x,S(y))
              = C[compS_f#1(x,S(y)) = compS_f#1(x,y){y -> S(y)}]

** Step 1.b:1: Bounds.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            compS_f#1(compS_f(x2),x1) -> compS_f#1(x2,S(x1))
            compS_f#1(id(),x3) -> S(x3)
            iter#3(0()) -> id()
            iter#3(S(x6)) -> compS_f(iter#3(x6))
            main(0()) -> 0()
            main(S(x9)) -> compS_f#1(iter#3(x9),0())
        - Signature:
            {compS_f#1/2,iter#3/1,main/1} / {0/0,S/1,compS_f/1,id/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {compS_f#1,iter#3,main} and constructors {0,S,compS_f,id}
    + Applied Processor:
        Bounds {initialAutomaton = minimal, enrichment = match}
    + Details:
        The problem is match-bounded by 2.
        The enriched problem is compatible with follwoing automaton.
          0_0() -> 2
          0_1() -> 1
          S_0(2) -> 2
          S_1(1) -> 1
          S_1(2) -> 1
          S_1(2) -> 3
          S_1(3) -> 1
          S_2(1) -> 1
          S_2(1) -> 5
          S_2(5) -> 1
          compS_f_0(2) -> 2
          compS_f_1(4) -> 1
          compS_f_1(4) -> 4
          compS_f#1_0(2,2) -> 1
          compS_f#1_1(2,1) -> 1
          compS_f#1_1(2,3) -> 1
          compS_f#1_1(4,1) -> 1
          compS_f#1_2(4,1) -> 1
          compS_f#1_2(4,5) -> 1
          id_0() -> 2
          id_1() -> 1
          id_1() -> 4
          iter#3_0(2) -> 1
          iter#3_1(2) -> 4
          main_0(2) -> 1
** Step 1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            compS_f#1(compS_f(x2),x1) -> compS_f#1(x2,S(x1))
            compS_f#1(id(),x3) -> S(x3)
            iter#3(0()) -> id()
            iter#3(S(x6)) -> compS_f(iter#3(x6))
            main(0()) -> 0()
            main(S(x9)) -> compS_f#1(iter#3(x9),0())
        - Signature:
            {compS_f#1/2,iter#3/1,main/1} / {0/0,S/1,compS_f/1,id/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {compS_f#1,iter#3,main} and constructors {0,S,compS_f,id}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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