/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:
            quot(x,0(),s(z)) -> s(quot(x,s(z),s(z)))
            quot(0(),s(y),s(z)) -> 0()
            quot(s(x),s(y),z) -> quot(x,y,z)
        - Signature:
            {quot/3} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {quot} 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:
            quot(x,0(),s(z)) -> s(quot(x,s(z),s(z)))
            quot(0(),s(y),s(z)) -> 0()
            quot(s(x),s(y),z) -> quot(x,y,z)
        - Signature:
            {quot/3} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {quot} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            quot(x,0(),s(z)) -> s(quot(x,s(z),s(z)))
            quot(0(),s(y),s(z)) -> 0()
            quot(s(x),s(y),z) -> quot(x,y,z)
        - Signature:
            {quot/3} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {quot} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          quot(x,y,z){x -> s(x),y -> s(y)} =
            quot(s(x),s(y),z) ->^+ quot(x,y,z)
              = C[quot(x,y,z) = quot(x,y,z){}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            quot(x,0(),s(z)) -> s(quot(x,s(z),s(z)))
            quot(0(),s(y),s(z)) -> 0()
            quot(s(x),s(y),z) -> quot(x,y,z)
        - Signature:
            {quot/3} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {quot} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z)))
          quot#(0(),s(y),s(z)) -> c_2()
          quot#(s(x),s(y),z) -> c_3(quot#(x,y,z))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z)))
            quot#(0(),s(y),s(z)) -> c_2()
            quot#(s(x),s(y),z) -> c_3(quot#(x,y,z))
        - Strict TRS:
            quot(x,0(),s(z)) -> s(quot(x,s(z),s(z)))
            quot(0(),s(y),s(z)) -> 0()
            quot(s(x),s(y),z) -> quot(x,y,z)
        - Signature:
            {quot/3,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {quot#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z)))
          quot#(0(),s(y),s(z)) -> c_2()
          quot#(s(x),s(y),z) -> c_3(quot#(x,y,z))
** Step 1.b:3: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z)))
            quot#(0(),s(y),s(z)) -> c_2()
            quot#(s(x),s(y),z) -> c_3(quot#(x,y,z))
        - Signature:
            {quot/3,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {quot#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {1,3}.
        Here rules are labelled as follows:
          1: quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z)))
          2: quot#(0(),s(y),s(z)) -> c_2()
          3: quot#(s(x),s(y),z) -> c_3(quot#(x,y,z))
** Step 1.b:4: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z)))
            quot#(s(x),s(y),z) -> c_3(quot#(x,y,z))
        - Weak DPs:
            quot#(0(),s(y),s(z)) -> c_2()
        - Signature:
            {quot/3,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {quot#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z)))
             -->_1 quot#(s(x),s(y),z) -> c_3(quot#(x,y,z)):2
             -->_1 quot#(0(),s(y),s(z)) -> c_2():3
          
          2:S:quot#(s(x),s(y),z) -> c_3(quot#(x,y,z))
             -->_1 quot#(0(),s(y),s(z)) -> c_2():3
             -->_1 quot#(s(x),s(y),z) -> c_3(quot#(x,y,z)):2
             -->_1 quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z))):1
          
          3:W:quot#(0(),s(y),s(z)) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: quot#(0(),s(y),s(z)) -> c_2()
** Step 1.b:5: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z)))
            quot#(s(x),s(y),z) -> c_3(quot#(x,y,z))
        - Signature:
            {quot/3,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {quot#} and constructors {0,s}
    + 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: quot#(s(x),s(y),z) -> c_3(quot#(x,y,z))
          
        Consider the set of all dependency pairs
          1: quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z)))
          2: quot#(s(x),s(y),z) -> c_3(quot#(x,y,z))
        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:5.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z)))
            quot#(s(x),s(y),z) -> c_3(quot#(x,y,z))
        - Signature:
            {quot/3,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {quot#} and constructors {0,s}
    + 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_3) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
              p(0) = [2]         
           p(quot) = [1] x1 + [1]
              p(s) = [1] x1 + [4]
          p(quot#) = [1] x1 + [2]
            p(c_1) = [1] x1 + [0]
            p(c_2) = [1]         
            p(c_3) = [1] x1 + [2]
        
        Following rules are strictly oriented:
        quot#(s(x),s(y),z) = [1] x + [6]      
                           > [1] x + [4]      
                           = c_3(quot#(x,y,z))
        
        
        Following rules are (at-least) weakly oriented:
        quot#(x,0(),s(z)) =  [1] x + [2]            
                          >= [1] x + [2]            
                          =  c_1(quot#(x,s(z),s(z)))
        
*** Step 1.b:5.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z)))
        - Weak DPs:
            quot#(s(x),s(y),z) -> c_3(quot#(x,y,z))
        - Signature:
            {quot/3,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {quot#} and constructors {0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

*** Step 1.b:5.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z)))
            quot#(s(x),s(y),z) -> c_3(quot#(x,y,z))
        - Signature:
            {quot/3,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {quot#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z)))
             -->_1 quot#(s(x),s(y),z) -> c_3(quot#(x,y,z)):2
          
          2:W:quot#(s(x),s(y),z) -> c_3(quot#(x,y,z))
             -->_1 quot#(s(x),s(y),z) -> c_3(quot#(x,y,z)):2
             -->_1 quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: quot#(x,0(),s(z)) -> c_1(quot#(x,s(z),s(z)))
          2: quot#(s(x),s(y),z) -> c_3(quot#(x,y,z))
*** Step 1.b:5.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {quot/3,quot#/3} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {quot#} 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^1))