/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^2))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^2))
    + Considered Problem:
        - Strict TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,-,exp} 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:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,-,exp} and constructors {+,0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
*** Step 1.a:1.a:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,-,exp} 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)] -(1)-> "A"(0)
          F (TrsFun "+") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "-") :: ["A"(0) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "0") :: [] -(0)-> "A"(0)
          F (TrsFun "0") :: [] -(0)-> "A"(1)
          F (TrsFun "exp") :: ["A"(0) x "A"(1)] -(1)-> "A"(0)
          F (TrsFun "main") :: ["A"(0) x "A"(1)] -(1)-> "A"(0)
          F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "s") :: ["A"(1)] -(1)-> "A"(1)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          *(0(),y) -> 0()
          *(s(x),y) -> +(y,*(x,y))
          -(x,0()) -> x
          -(0(),y) -> 0()
          -(s(x),s(y)) -> -(x,y)
          exp(x,0()) -> s(0())
          exp(x,s(y)) -> *(x,exp(x,y))
          main(x1,x2) -> exp(x1,x2)
        2. Weak:
          
        

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

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,-,exp} and constructors {+,0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          *#(0(),y) -> c_1()
          *#(s(x),y) -> c_2(*#(x,y))
          -#(x,0()) -> c_3()
          -#(0(),y) -> c_4()
          -#(s(x),s(y)) -> c_5(-#(x,y))
          exp#(x,0()) -> c_6()
          exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            *#(0(),y) -> c_1()
            *#(s(x),y) -> c_2(*#(x,y))
            -#(x,0()) -> c_3()
            -#(0(),y) -> c_4()
            -#(s(x),s(y)) -> c_5(-#(x,y))
            exp#(x,0()) -> c_6()
            exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,4,6}
        by application of
          Pre({1,3,4,6}) = {2,5,7}.
        Here rules are labelled as follows:
          1: *#(0(),y) -> c_1()
          2: *#(s(x),y) -> c_2(*#(x,y))
          3: -#(x,0()) -> c_3()
          4: -#(0(),y) -> c_4()
          5: -#(s(x),s(y)) -> c_5(-#(x,y))
          6: exp#(x,0()) -> c_6()
          7: exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),y) -> c_2(*#(x,y))
            -#(s(x),s(y)) -> c_5(-#(x,y))
            exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
        - Weak DPs:
            *#(0(),y) -> c_1()
            -#(x,0()) -> c_3()
            -#(0(),y) -> c_4()
            exp#(x,0()) -> c_6()
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:*#(s(x),y) -> c_2(*#(x,y))
             -->_1 *#(0(),y) -> c_1():4
             -->_1 *#(s(x),y) -> c_2(*#(x,y)):1
          
          2:S:-#(s(x),s(y)) -> c_5(-#(x,y))
             -->_1 -#(0(),y) -> c_4():6
             -->_1 -#(x,0()) -> c_3():5
             -->_1 -#(s(x),s(y)) -> c_5(-#(x,y)):2
          
          3:S:exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
             -->_2 exp#(x,0()) -> c_6():7
             -->_1 *#(0(),y) -> c_1():4
             -->_2 exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)):3
             -->_1 *#(s(x),y) -> c_2(*#(x,y)):1
          
          4:W:*#(0(),y) -> c_1()
             
          
          5:W:-#(x,0()) -> c_3()
             
          
          6:W:-#(0(),y) -> c_4()
             
          
          7:W:exp#(x,0()) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: exp#(x,0()) -> c_6()
          5: -#(x,0()) -> c_3()
          6: -#(0(),y) -> c_4()
          4: *#(0(),y) -> c_1()
** Step 1.b:4: UsableRules.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),y) -> c_2(*#(x,y))
            -#(s(x),s(y)) -> c_5(-#(x,y))
            exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            -(x,0()) -> x
            -(0(),y) -> 0()
            -(s(x),s(y)) -> -(x,y)
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          *(0(),y) -> 0()
          *(s(x),y) -> +(y,*(x,y))
          exp(x,0()) -> s(0())
          exp(x,s(y)) -> *(x,exp(x,y))
          *#(s(x),y) -> c_2(*#(x,y))
          -#(s(x),s(y)) -> c_5(-#(x,y))
          exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
** Step 1.b:5: DecomposeDG.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),y) -> c_2(*#(x,y))
            -#(s(x),s(y)) -> c_5(-#(x,y))
            exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          -#(s(x),s(y)) -> c_5(-#(x,y))
          exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
        and a lower component
          *#(s(x),y) -> c_2(*#(x,y))
        Further, following extension rules are added to the lower component.
          -#(s(x),s(y)) -> -#(x,y)
          exp#(x,s(y)) -> *#(x,exp(x,y))
          exp#(x,s(y)) -> exp#(x,y)
*** Step 1.b:5.a:1: SimplifyRHS.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            -#(s(x),s(y)) -> c_5(-#(x,y))
            exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:-#(s(x),s(y)) -> c_5(-#(x,y))
             -->_1 -#(s(x),s(y)) -> c_5(-#(x,y)):1
          
          2:S:exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
             -->_2 exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          exp#(x,s(y)) -> c_7(exp#(x,y))
*** Step 1.b:5.a:2: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            -#(s(x),s(y)) -> c_5(-#(x,y))
            exp#(x,s(y)) -> c_7(exp#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          -#(s(x),s(y)) -> c_5(-#(x,y))
          exp#(x,s(y)) -> c_7(exp#(x,y))
*** Step 1.b:5.a:3: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            -#(s(x),s(y)) -> c_5(-#(x,y))
            exp#(x,s(y)) -> c_7(exp#(x,y))
        - Signature:
            {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_5) = {1},
            uargs(c_7) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(*) = [0]                  
               p(+) = [1] x1 + [1] x2 + [0]
               p(-) = [0]                  
               p(0) = [0]                  
             p(exp) = [0]                  
               p(s) = [1] x1 + [2]         
              p(*#) = [0]                  
              p(-#) = [2] x2 + [0]         
            p(exp#) = [8] x2 + [0]         
             p(c_1) = [0]                  
             p(c_2) = [0]                  
             p(c_3) = [0]                  
             p(c_4) = [0]                  
             p(c_5) = [1] x1 + [0]         
             p(c_6) = [0]                  
             p(c_7) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          -#(s(x),s(y)) = [2] y + [4]   
                        > [2] y + [0]   
                        = c_5(-#(x,y))  
          
           exp#(x,s(y)) = [8] y + [16]  
                        > [8] y + [0]   
                        = c_7(exp#(x,y))
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 1.b:5.a:4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            -#(s(x),s(y)) -> c_5(-#(x,y))
            exp#(x,s(y)) -> c_7(exp#(x,y))
        - Signature:
            {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:5.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),y) -> c_2(*#(x,y))
        - Weak DPs:
            -#(s(x),s(y)) -> -#(x,y)
            exp#(x,s(y)) -> *#(x,exp(x,y))
            exp#(x,s(y)) -> exp#(x,y)
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:*#(s(x),y) -> c_2(*#(x,y))
             -->_1 *#(s(x),y) -> c_2(*#(x,y)):1
          
          2:W:-#(s(x),s(y)) -> -#(x,y)
             -->_1 -#(s(x),s(y)) -> -#(x,y):2
          
          3:W:exp#(x,s(y)) -> *#(x,exp(x,y))
             -->_1 *#(s(x),y) -> c_2(*#(x,y)):1
          
          4:W:exp#(x,s(y)) -> exp#(x,y)
             -->_1 exp#(x,s(y)) -> exp#(x,y):4
             -->_1 exp#(x,s(y)) -> *#(x,exp(x,y)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: -#(s(x),s(y)) -> -#(x,y)
*** Step 1.b:5.b:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),y) -> c_2(*#(x,y))
        - Weak DPs:
            exp#(x,s(y)) -> *#(x,exp(x,y))
            exp#(x,s(y)) -> exp#(x,y)
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,-#,exp#} 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_2) = {1}
        
        Following symbols are considered usable:
          {*#,-#,exp#}
        TcT has computed the following interpretation:
             p(*) = [5]                  
             p(+) = [1] x1 + [1] x2 + [0]
             p(-) = [0]                  
             p(0) = [0]                  
           p(exp) = [0]                  
             p(s) = [1] x1 + [9]         
            p(*#) = [2] x1 + [0]         
            p(-#) = [0]                  
          p(exp#) = [8] x1 + [1]         
           p(c_1) = [0]                  
           p(c_2) = [1] x1 + [10]        
           p(c_3) = [0]                  
           p(c_4) = [0]                  
           p(c_5) = [1] x1 + [0]         
           p(c_6) = [0]                  
           p(c_7) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        *#(s(x),y) = [2] x + [18]
                   > [2] x + [10]
                   = c_2(*#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        exp#(x,s(y)) =  [8] x + [1]   
                     >= [2] x + [0]   
                     =  *#(x,exp(x,y))
        
        exp#(x,s(y)) =  [8] x + [1]   
                     >= [8] x + [1]   
                     =  exp#(x,y)     
        
*** Step 1.b:5.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            *#(s(x),y) -> c_2(*#(x,y))
            exp#(x,s(y)) -> *#(x,exp(x,y))
            exp#(x,s(y)) -> exp#(x,y)
        - Weak TRS:
            *(0(),y) -> 0()
            *(s(x),y) -> +(y,*(x,y))
            exp(x,0()) -> s(0())
            exp(x,s(y)) -> *(x,exp(x,y))
        - Signature:
            {*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,-#,exp#} 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^2))