/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(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr,sum} 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:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
*** Step 1.a:1.a:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
    + 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) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "+") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "0") :: [] -(0)-> "A"(1)
          F (TrsFun "0") :: [] -(0)-> "A"(0)
          F (TrsFun "main") :: ["A"(1)] -(1)-> "A"(0)
          F (TrsFun "s") :: ["A"(1)] -(1)-> "A"(1)
          F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "sqr") :: ["A"(0)] -(1)-> "A"(0)
          F (TrsFun "sum") :: ["A"(1)] -(1)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          sqr(x) -> *(x,x)
          sum(0()) -> 0()
          sum(s(x)) -> +(*(s(x),s(x)),sum(x))
          sum(s(x)) -> +(sqr(s(x)),sum(x))
          main(x1) -> sum(x1)
        2. Weak:
          
        

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

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          sqr#(x) -> c_1()
          sum#(0()) -> c_2()
          sum#(s(x)) -> c_3(sum#(x))
          sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sqr#(x) -> c_1()
            sum#(0()) -> c_2()
            sum#(s(x)) -> c_3(sum#(x))
            sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x))
        - Weak TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1,sqr#/1,sum#/1} / {*/2,+/2,0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr#,sum#} and constructors {*,+,0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2}
        by application of
          Pre({1,2}) = {3,4}.
        Here rules are labelled as follows:
          1: sqr#(x) -> c_1()
          2: sum#(0()) -> c_2()
          3: sum#(s(x)) -> c_3(sum#(x))
          4: sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x))
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_3(sum#(x))
            sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x))
        - Weak DPs:
            sqr#(x) -> c_1()
            sum#(0()) -> c_2()
        - Weak TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1,sqr#/1,sum#/1} / {*/2,+/2,0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr#,sum#} and constructors {*,+,0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sum#(s(x)) -> c_3(sum#(x))
             -->_1 sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x)):2
             -->_1 sum#(0()) -> c_2():4
             -->_1 sum#(s(x)) -> c_3(sum#(x)):1
          
          2:S:sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x))
             -->_2 sum#(0()) -> c_2():4
             -->_1 sqr#(x) -> c_1():3
             -->_2 sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x)):2
             -->_2 sum#(s(x)) -> c_3(sum#(x)):1
          
          3:W:sqr#(x) -> c_1()
             
          
          4:W:sum#(0()) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: sqr#(x) -> c_1()
          4: sum#(0()) -> c_2()
** Step 1.b:4: SimplifyRHS.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_3(sum#(x))
            sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x))
        - Weak TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1,sqr#/1,sum#/1} / {*/2,+/2,0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr#,sum#} and constructors {*,+,0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:sum#(s(x)) -> c_3(sum#(x))
             -->_1 sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x)):2
             -->_1 sum#(s(x)) -> c_3(sum#(x)):1
          
          2:S:sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x))
             -->_2 sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x)):2
             -->_2 sum#(s(x)) -> c_3(sum#(x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sum#(s(x)) -> c_4(sum#(x))
** Step 1.b:5: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_3(sum#(x))
            sum#(s(x)) -> c_4(sum#(x))
        - Weak TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1,sqr#/1,sum#/1} / {*/2,+/2,0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr#,sum#} and constructors {*,+,0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          sum#(s(x)) -> c_3(sum#(x))
          sum#(s(x)) -> c_4(sum#(x))
** Step 1.b:6: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_3(sum#(x))
            sum#(s(x)) -> c_4(sum#(x))
        - Signature:
            {sqr/1,sum/1,sqr#/1,sum#/1} / {*/2,+/2,0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr#,sum#} 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_3) = {1},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {sqr#,sum#}
        TcT has computed the following interpretation:
             p(*) = [1] x1 + [1] x2 + [1]
             p(+) = [1] x1 + [1] x2 + [1]
             p(0) = [1]                  
             p(s) = [1] x1 + [1]         
           p(sqr) = [1] x1 + [2]         
           p(sum) = [2] x1 + [1]         
          p(sqr#) = [1] x1 + [1]         
          p(sum#) = [8] x1 + [0]         
           p(c_1) = [0]                  
           p(c_2) = [0]                  
           p(c_3) = [1] x1 + [0]         
           p(c_4) = [1] x1 + [8]         
        
        Following rules are strictly oriented:
        sum#(s(x)) = [8] x + [8] 
                   > [8] x + [0] 
                   = c_3(sum#(x))
        
        
        Following rules are (at-least) weakly oriented:
        sum#(s(x)) =  [8] x + [8] 
                   >= [8] x + [8] 
                   =  c_4(sum#(x))
        
** Step 1.b:7: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_4(sum#(x))
        - Weak DPs:
            sum#(s(x)) -> c_3(sum#(x))
        - Signature:
            {sqr/1,sum/1,sqr#/1,sum#/1} / {*/2,+/2,0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr#,sum#} 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_3) = {1},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {sqr#,sum#}
        TcT has computed the following interpretation:
             p(*) = [1] x2 + [1]
             p(+) = [2]         
             p(0) = [1]         
             p(s) = [1] x1 + [8]
           p(sqr) = [0]         
           p(sum) = [2] x1 + [2]
          p(sqr#) = [2] x1 + [0]
          p(sum#) = [1] x1 + [0]
           p(c_1) = [1]         
           p(c_2) = [0]         
           p(c_3) = [1] x1 + [3]
           p(c_4) = [1] x1 + [6]
        
        Following rules are strictly oriented:
        sum#(s(x)) = [1] x + [8] 
                   > [1] x + [6] 
                   = c_4(sum#(x))
        
        
        Following rules are (at-least) weakly oriented:
        sum#(s(x)) =  [1] x + [8] 
                   >= [1] x + [3] 
                   =  c_3(sum#(x))
        
** Step 1.b:8: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum#(s(x)) -> c_3(sum#(x))
            sum#(s(x)) -> c_4(sum#(x))
        - Signature:
            {sqr/1,sum/1,sqr#/1,sum#/1} / {*/2,+/2,0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr#,sum#} and constructors {*,+,0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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