/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /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:
            *(x,+(y,z)) -> +(*(x,y),*(x,z))
        - Signature:
            {*/2} / {+/2}
        - Obligation:
             runtime complexity wrt. defined symbols {*} and constructors {+}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            *(x,+(y,z)) -> +(*(x,y),*(x,z))
        - Signature:
            {*/2} / {+/2}
        - Obligation:
             runtime complexity wrt. defined symbols {*} and constructors {+}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            *(x,+(y,z)) -> +(*(x,y),*(x,z))
        - Signature:
            {*/2} / {+/2}
        - Obligation:
             runtime complexity wrt. defined symbols {*} and constructors {+}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          *(x,y){y -> +(y,z)} =
            *(x,+(y,z)) ->^+ +(*(x,y),*(x,z))
              = C[*(x,y) = *(x,y){}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            *(x,+(y,z)) -> +(*(x,y),*(x,z))
        - Signature:
            {*/2} / {+/2}
        - Obligation:
             runtime complexity wrt. defined symbols {*} and constructors {+}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          *#(x,+(y,z)) -> c_1(*#(x,y),*#(x,z))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(x,+(y,z)) -> c_1(*#(x,y),*#(x,z))
        - Strict TRS:
            *(x,+(y,z)) -> +(*(x,y),*(x,z))
        - Signature:
            {*/2,*#/2} / {+/2,c_1/2}
        - Obligation:
             runtime complexity wrt. defined symbols {*#} and constructors {+}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          *#(x,+(y,z)) -> c_1(*#(x,y),*#(x,z))
** Step 1.b:3: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(x,+(y,z)) -> c_1(*#(x,y),*#(x,z))
        - Signature:
            {*/2,*#/2} / {+/2,c_1/2}
        - Obligation:
             runtime complexity wrt. defined symbols {*#} and constructors {+}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: *#(x,+(y,z)) -> c_1(*#(x,y),*#(x,z))
          
        The strictly oriented rules are moved into the weak component.
*** Step 1.b:3.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(x,+(y,z)) -> c_1(*#(x,y),*#(x,z))
        - Signature:
            {*/2,*#/2} / {+/2,c_1/2}
        - Obligation:
             runtime complexity wrt. defined symbols {*#} and constructors {+}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_1) = {1,2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
            p(*) = [1] x1 + [2] x2 + [8] 
            p(+) = [1] x1 + [1] x2 + [12]
           p(*#) = [2] x2 + [1]          
          p(c_1) = [1] x1 + [1] x2 + [8] 
        
        Following rules are strictly oriented:
        *#(x,+(y,z)) = [2] y + [2] z + [25]
                     > [2] y + [2] z + [10]
                     = c_1(*#(x,y),*#(x,z))
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 1.b:3.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            *#(x,+(y,z)) -> c_1(*#(x,y),*#(x,z))
        - Signature:
            {*/2,*#/2} / {+/2,c_1/2}
        - Obligation:
             runtime complexity wrt. defined symbols {*#} and constructors {+}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

*** Step 1.b:3.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            *#(x,+(y,z)) -> c_1(*#(x,y),*#(x,z))
        - Signature:
            {*/2,*#/2} / {+/2,c_1/2}
        - Obligation:
             runtime complexity wrt. defined symbols {*#} and constructors {+}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:*#(x,+(y,z)) -> c_1(*#(x,y),*#(x,z))
             -->_2 *#(x,+(y,z)) -> c_1(*#(x,y),*#(x,z)):1
             -->_1 *#(x,+(y,z)) -> c_1(*#(x,y),*#(x,z)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: *#(x,+(y,z)) -> c_1(*#(x,y),*#(x,z))
*** Step 1.b:3.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {*/2,*#/2} / {+/2,c_1/2}
        - Obligation:
             runtime complexity wrt. defined symbols {*#} and constructors {+}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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