/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(?,O(1))
* Step 1: Sum.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            h(x,c(y,z),t(w)) -> h(c(s(y),x),z,t(c(t(w),w)))
            h(c(x,y),c(s(z),z),t(w)) -> h(z,c(y,x),t(t(c(x,c(y,t(w))))))
            h(c(s(x),c(s(0()),y)),z,t(x)) -> h(y,c(s(0()),c(x,z)),t(t(c(x,s(x)))))
            t(x) -> x
            t(x) -> c(0(),c(0(),c(0(),c(0(),c(0(),x)))))
            t(t(x)) -> t(c(t(x),x))
        - Signature:
            {h/3,t/1} / {0/0,c/2,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {h,t} and constructors {0,c,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            h(x,c(y,z),t(w)) -> h(c(s(y),x),z,t(c(t(w),w)))
            h(c(x,y),c(s(z),z),t(w)) -> h(z,c(y,x),t(t(c(x,c(y,t(w))))))
            h(c(s(x),c(s(0()),y)),z,t(x)) -> h(y,c(s(0()),c(x,z)),t(t(c(x,s(x)))))
            t(x) -> x
            t(x) -> c(0(),c(0(),c(0(),c(0(),c(0(),x)))))
            t(t(x)) -> t(c(t(x),x))
        - Signature:
            {h/3,t/1} / {0/0,c/2,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {h,t} and constructors {0,c,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          h#(x,c(y,z),t(w)) -> c_1(h#(c(s(y),x),z,t(c(t(w),w))))
          h#(c(x,y),c(s(z),z),t(w)) -> c_2(h#(z,c(y,x),t(t(c(x,c(y,t(w)))))))
          h#(c(s(x),c(s(0()),y)),z,t(x)) -> c_3(h#(y,c(s(0()),c(x,z)),t(t(c(x,s(x))))))
          t#(x) -> c_4(x)
          t#(x) -> c_5(x)
          t#(t(x)) -> c_6(t#(c(t(x),x)))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            h#(x,c(y,z),t(w)) -> c_1(h#(c(s(y),x),z,t(c(t(w),w))))
            h#(c(x,y),c(s(z),z),t(w)) -> c_2(h#(z,c(y,x),t(t(c(x,c(y,t(w)))))))
            h#(c(s(x),c(s(0()),y)),z,t(x)) -> c_3(h#(y,c(s(0()),c(x,z)),t(t(c(x,s(x))))))
            t#(x) -> c_4(x)
            t#(x) -> c_5(x)
            t#(t(x)) -> c_6(t#(c(t(x),x)))
        - Strict TRS:
            h(x,c(y,z),t(w)) -> h(c(s(y),x),z,t(c(t(w),w)))
            h(c(x,y),c(s(z),z),t(w)) -> h(z,c(y,x),t(t(c(x,c(y,t(w))))))
            h(c(s(x),c(s(0()),y)),z,t(x)) -> h(y,c(s(0()),c(x,z)),t(t(c(x,s(x)))))
            t(x) -> x
            t(x) -> c(0(),c(0(),c(0(),c(0(),c(0(),x)))))
            t(t(x)) -> t(c(t(x),x))
        - Signature:
            {h/3,t/1,h#/3,t#/1} / {0/0,c/2,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {h#,t#} and constructors {0,c,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          t#(x) -> c_4(x)
          t#(x) -> c_5(x)
* Step 4: PredecessorEstimationCP.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            t#(x) -> c_4(x)
            t#(x) -> c_5(x)
        - Signature:
            {h/3,t/1,h#/3,t#/1} / {0/0,c/2,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {h#,t#} and constructors {0,c,s}
    + 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: t#(x) -> c_4(x)
          2: t#(x) -> c_5(x)
          
        The strictly oriented rules are moved into the weak component.
** Step 4.a:1: NaturalMI.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            t#(x) -> c_4(x)
            t#(x) -> c_5(x)
        - Signature:
            {h/3,t/1,h#/3,t#/1} / {0/0,c/2,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {h#,t#} and constructors {0,c,s}
    + 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(0) = [0]
            p(c) = [0]
            p(h) = [0]
            p(s) = [0]
            p(t) = [0]
           p(h#) = [0]
           p(t#) = [1]
          p(c_1) = [0]
          p(c_2) = [0]
          p(c_3) = [0]
          p(c_4) = [0]
          p(c_5) = [0]
          p(c_6) = [0]
        
        Following rules are strictly oriented:
        t#(x) = [1]   
              > [0]   
              = c_4(x)
        
        t#(x) = [1]   
              > [0]   
              = c_5(x)
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 4.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            t#(x) -> c_4(x)
            t#(x) -> c_5(x)
        - Signature:
            {h/3,t/1,h#/3,t#/1} / {0/0,c/2,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {h#,t#} and constructors {0,c,s}
    + 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:
            t#(x) -> c_4(x)
            t#(x) -> c_5(x)
        - Signature:
            {h/3,t/1,h#/3,t#/1} / {0/0,c/2,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {h#,t#} and constructors {0,c,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:t#(x) -> c_4(x)
             -->_1 t#(x) -> c_5(x):2
             -->_1 t#(x) -> c_4(x):1
          
          2:W:t#(x) -> c_5(x)
             -->_1 t#(x) -> c_5(x):2
             -->_1 t#(x) -> c_4(x):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: t#(x) -> c_4(x)
          2: t#(x) -> c_5(x)
** Step 4.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {h/3,t/1,h#/3,t#/1} / {0/0,c/2,s/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1}
        - Obligation:
             runtime complexity wrt. defined symbols {h#,t#} and constructors {0,c,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))