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

** Step 1.b:1: ToInnermost.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(f(x),y) -> f(h(x,y))
            h(x,y) -> g(x,f(y))
        - Signature:
            {g/2,h/2} / {f/1}
        - Obligation:
             runtime complexity wrt. defined symbols {g,h} and constructors {f}
    + Applied Processor:
        ToInnermost
    + Details:
        switch to innermost, as the system is overlay and right linear and does not contain weak rules
** Step 1.b:2: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(f(x),y) -> f(h(x,y))
            h(x,y) -> g(x,f(y))
        - Signature:
            {g/2,h/2} / {f/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g,h} and constructors {f}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          g#(f(x),y) -> c_1(h#(x,y))
          h#(x,y) -> c_2(g#(x,f(y)))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:3: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(f(x),y) -> c_1(h#(x,y))
            h#(x,y) -> c_2(g#(x,f(y)))
        - Strict TRS:
            g(f(x),y) -> f(h(x,y))
            h(x,y) -> g(x,f(y))
        - Signature:
            {g/2,h/2,g#/2,h#/2} / {f/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {f}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          g#(f(x),y) -> c_1(h#(x,y))
          h#(x,y) -> c_2(g#(x,f(y)))
** Step 1.b:4: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(f(x),y) -> c_1(h#(x,y))
            h#(x,y) -> c_2(g#(x,f(y)))
        - Signature:
            {g/2,h/2,g#/2,h#/2} / {f/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {f}
    + 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:
          2: h#(x,y) -> c_2(g#(x,f(y)))
          
        Consider the set of all dependency pairs
          1: g#(f(x),y) -> c_1(h#(x,y))
          2: h#(x,y) -> c_2(g#(x,f(y)))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        BEST_CASE TIME (?,?)
        SPACE(?,?)on application of the dependency pairs
          {2}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
*** Step 1.b:4.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(f(x),y) -> c_1(h#(x,y))
            h#(x,y) -> c_2(g#(x,f(y)))
        - Signature:
            {g/2,h/2,g#/2,h#/2} / {f/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {f}
    + 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},
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {g#,h#}
        TcT has computed the following interpretation:
            p(f) = [1] x1 + [4]         
            p(g) = [1]                  
            p(h) = [1] x1 + [1] x2 + [2]
           p(g#) = [4] x1 + [0]         
           p(h#) = [4] x1 + [4]         
          p(c_1) = [1] x1 + [12]        
          p(c_2) = [1] x1 + [2]         
        
        Following rules are strictly oriented:
        h#(x,y) = [4] x + [4]    
                > [4] x + [2]    
                = c_2(g#(x,f(y)))
        
        
        Following rules are (at-least) weakly oriented:
        g#(f(x),y) =  [4] x + [16]
                   >= [4] x + [16]
                   =  c_1(h#(x,y))
        
*** Step 1.b:4.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(f(x),y) -> c_1(h#(x,y))
        - Weak DPs:
            h#(x,y) -> c_2(g#(x,f(y)))
        - Signature:
            {g/2,h/2,g#/2,h#/2} / {f/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {f}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

*** Step 1.b:4.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(f(x),y) -> c_1(h#(x,y))
            h#(x,y) -> c_2(g#(x,f(y)))
        - Signature:
            {g/2,h/2,g#/2,h#/2} / {f/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {f}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:g#(f(x),y) -> c_1(h#(x,y))
             -->_1 h#(x,y) -> c_2(g#(x,f(y))):2
          
          2:W:h#(x,y) -> c_2(g#(x,f(y)))
             -->_1 g#(f(x),y) -> c_1(h#(x,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: g#(f(x),y) -> c_1(h#(x,y))
          2: h#(x,y) -> c_2(g#(x,f(y)))
*** Step 1.b:4.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {g/2,h/2,g#/2,h#/2} / {f/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g#,h#} and constructors {f}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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