/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(?,O(1))
* Step 1: Sum.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            =(x,y) -> xor(x,xor(y,true()))
            implies(x,y) -> xor(and(x,y),xor(x,true()))
            not(x) -> xor(x,true())
            or(x,y) -> xor(and(x,y),xor(x,y))
        - Signature:
            {=/2,implies/2,not/1,or/2} / {and/2,true/0,xor/2}
        - Obligation:
             runtime complexity wrt. defined symbols {=,implies,not,or} and constructors {and,true,xor}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            =(x,y) -> xor(x,xor(y,true()))
            implies(x,y) -> xor(and(x,y),xor(x,true()))
            not(x) -> xor(x,true())
            or(x,y) -> xor(and(x,y),xor(x,y))
        - Signature:
            {=/2,implies/2,not/1,or/2} / {and/2,true/0,xor/2}
        - Obligation:
             runtime complexity wrt. defined symbols {=,implies,not,or} and constructors {and,true,xor}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          =#(x,y) -> c_1(x,y)
          implies#(x,y) -> c_2(x,y,x)
          not#(x) -> c_3(x)
          or#(x,y) -> c_4(x,y,x,y)
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            =#(x,y) -> c_1(x,y)
            implies#(x,y) -> c_2(x,y,x)
            not#(x) -> c_3(x)
            or#(x,y) -> c_4(x,y,x,y)
        - Strict TRS:
            =(x,y) -> xor(x,xor(y,true()))
            implies(x,y) -> xor(and(x,y),xor(x,true()))
            not(x) -> xor(x,true())
            or(x,y) -> xor(and(x,y),xor(x,y))
        - Signature:
            {=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
        - Obligation:
             runtime complexity wrt. defined symbols {=#,implies#,not#,or#} and constructors {and,true,xor}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          =#(x,y) -> c_1(x,y)
          implies#(x,y) -> c_2(x,y,x)
          not#(x) -> c_3(x)
          or#(x,y) -> c_4(x,y,x,y)
* Step 4: PredecessorEstimationCP.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            =#(x,y) -> c_1(x,y)
            implies#(x,y) -> c_2(x,y,x)
            not#(x) -> c_3(x)
            or#(x,y) -> c_4(x,y,x,y)
        - Signature:
            {=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
        - Obligation:
             runtime complexity wrt. defined symbols {=#,implies#,not#,or#} and constructors {and,true,xor}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: =#(x,y) -> c_1(x,y)
          2: implies#(x,y) -> c_2(x,y,x)
          3: not#(x) -> c_3(x)
          4: or#(x,y) -> c_4(x,y,x,y)
          
        The strictly oriented rules are moved into the weak component.
** Step 4.a:1: NaturalMI.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            =#(x,y) -> c_1(x,y)
            implies#(x,y) -> c_2(x,y,x)
            not#(x) -> c_3(x)
            or#(x,y) -> c_4(x,y,x,y)
        - Signature:
            {=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
        - Obligation:
             runtime complexity wrt. defined symbols {=#,implies#,not#,or#} and constructors {and,true,xor}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 0, 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 (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          none
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                 p(=) = [1] x1 + [0]
               p(and) = [0]         
           p(implies) = [0]         
               p(not) = [0]         
                p(or) = [0]         
              p(true) = [0]         
               p(xor) = [0]         
                p(=#) = [1]         
          p(implies#) = [5] x1 + [1]
              p(not#) = [1]         
               p(or#) = [1]         
               p(c_1) = [0]         
               p(c_2) = [5] x1 + [0]
               p(c_3) = [0]         
               p(c_4) = [0]         
        
        Following rules are strictly oriented:
              =#(x,y) = [1]         
                      > [0]         
                      = c_1(x,y)    
        
        implies#(x,y) = [5] x + [1] 
                      > [5] x + [0] 
                      = c_2(x,y,x)  
        
              not#(x) = [1]         
                      > [0]         
                      = c_3(x)      
        
             or#(x,y) = [1]         
                      > [0]         
                      = c_4(x,y,x,y)
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 4.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            =#(x,y) -> c_1(x,y)
            implies#(x,y) -> c_2(x,y,x)
            not#(x) -> c_3(x)
            or#(x,y) -> c_4(x,y,x,y)
        - Signature:
            {=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
        - Obligation:
             runtime complexity wrt. defined symbols {=#,implies#,not#,or#} and constructors {and,true,xor}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

** Step 4.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            =#(x,y) -> c_1(x,y)
            implies#(x,y) -> c_2(x,y,x)
            not#(x) -> c_3(x)
            or#(x,y) -> c_4(x,y,x,y)
        - Signature:
            {=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
        - Obligation:
             runtime complexity wrt. defined symbols {=#,implies#,not#,or#} and constructors {and,true,xor}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:=#(x,y) -> c_1(x,y)
             -->_2 or#(x,y) -> c_4(x,y,x,y):4
             -->_1 or#(x,y) -> c_4(x,y,x,y):4
             -->_2 not#(x) -> c_3(x):3
             -->_1 not#(x) -> c_3(x):3
             -->_2 implies#(x,y) -> c_2(x,y,x):2
             -->_1 implies#(x,y) -> c_2(x,y,x):2
             -->_2 =#(x,y) -> c_1(x,y):1
             -->_1 =#(x,y) -> c_1(x,y):1
          
          2:W:implies#(x,y) -> c_2(x,y,x)
             -->_3 or#(x,y) -> c_4(x,y,x,y):4
             -->_2 or#(x,y) -> c_4(x,y,x,y):4
             -->_1 or#(x,y) -> c_4(x,y,x,y):4
             -->_3 not#(x) -> c_3(x):3
             -->_2 not#(x) -> c_3(x):3
             -->_1 not#(x) -> c_3(x):3
             -->_3 implies#(x,y) -> c_2(x,y,x):2
             -->_2 implies#(x,y) -> c_2(x,y,x):2
             -->_1 implies#(x,y) -> c_2(x,y,x):2
             -->_3 =#(x,y) -> c_1(x,y):1
             -->_2 =#(x,y) -> c_1(x,y):1
             -->_1 =#(x,y) -> c_1(x,y):1
          
          3:W:not#(x) -> c_3(x)
             -->_1 or#(x,y) -> c_4(x,y,x,y):4
             -->_1 not#(x) -> c_3(x):3
             -->_1 implies#(x,y) -> c_2(x,y,x):2
             -->_1 =#(x,y) -> c_1(x,y):1
          
          4:W:or#(x,y) -> c_4(x,y,x,y)
             -->_4 or#(x,y) -> c_4(x,y,x,y):4
             -->_3 or#(x,y) -> c_4(x,y,x,y):4
             -->_2 or#(x,y) -> c_4(x,y,x,y):4
             -->_1 or#(x,y) -> c_4(x,y,x,y):4
             -->_4 not#(x) -> c_3(x):3
             -->_3 not#(x) -> c_3(x):3
             -->_2 not#(x) -> c_3(x):3
             -->_1 not#(x) -> c_3(x):3
             -->_4 implies#(x,y) -> c_2(x,y,x):2
             -->_3 implies#(x,y) -> c_2(x,y,x):2
             -->_2 implies#(x,y) -> c_2(x,y,x):2
             -->_1 implies#(x,y) -> c_2(x,y,x):2
             -->_4 =#(x,y) -> c_1(x,y):1
             -->_3 =#(x,y) -> c_1(x,y):1
             -->_2 =#(x,y) -> c_1(x,y):1
             -->_1 =#(x,y) -> c_1(x,y):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: =#(x,y) -> c_1(x,y)
          4: or#(x,y) -> c_4(x,y,x,y)
          3: not#(x) -> c_3(x)
          2: implies#(x,y) -> c_2(x,y,x)
** Step 4.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
        - Obligation:
             runtime complexity wrt. defined symbols {=#,implies#,not#,or#} and constructors {and,true,xor}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))