/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /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:
            admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
            admit(x,nil()) -> nil()
            cond(true(),y) -> y
        - Signature:
            {admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit,cond} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
            admit(x,nil()) -> nil()
            cond(true(),y) -> y
        - Signature:
            {admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit,cond} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
            admit(x,nil()) -> nil()
            cond(true(),y) -> y
        - Signature:
            {admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit,cond} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          admit(x,u){u -> .(y,.(z,.(w(),u)))} =
            admit(x,.(y,.(z,.(w(),u)))) ->^+ cond(=(sum(x,y,z),w()),.(y,.(z,.(w(),admit(carry(x,y,z),u)))))
              = C[admit(carry(x,y,z),u) = admit(x,u){x -> carry(x,y,z)}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
            admit(x,nil()) -> nil()
            cond(true(),y) -> y
        - Signature:
            {admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit,cond} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                             ,admit#(carry(x,u,v),z))
          admit#(x,nil()) -> c_2()
          cond#(true(),y) -> c_3()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                               ,admit#(carry(x,u,v),z))
            admit#(x,nil()) -> c_2()
            cond#(true(),y) -> c_3()
        - Weak TRS:
            admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
            admit(x,nil()) -> nil()
            cond(true(),y) -> y
        - Signature:
            {admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/2,c_2/0,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3}
        by application of
          Pre({2,3}) = {1}.
        Here rules are labelled as follows:
          1: admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                                ,admit#(carry(x,u,v),z))
          2: admit#(x,nil()) -> c_2()
          3: cond#(true(),y) -> c_3()
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                               ,admit#(carry(x,u,v),z))
        - Weak DPs:
            admit#(x,nil()) -> c_2()
            cond#(true(),y) -> c_3()
        - Weak TRS:
            admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
            admit(x,nil()) -> nil()
            cond(true(),y) -> y
        - Signature:
            {admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/2,c_2/0,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                                 ,admit#(carry(x,u,v),z))
             -->_2 admit#(x,nil()) -> c_2():2
             -->_2 admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                                      ,admit#(carry(x,u,v),z)):1
          
          2:W:admit#(x,nil()) -> c_2()
             
          
          3:W:cond#(true(),y) -> c_3()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: cond#(true(),y) -> c_3()
          2: admit#(x,nil()) -> c_2()
** Step 1.b:4: SimplifyRHS.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                               ,admit#(carry(x,u,v),z))
        - Weak TRS:
            admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
            admit(x,nil()) -> nil()
            cond(true(),y) -> y
        - Signature:
            {admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/2,c_2/0,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                                 ,admit#(carry(x,u,v),z))
             -->_2 admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
                                                      ,admit#(carry(x,u,v),z)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          admit#(x,.(u,.(v,.(w(),z)))) -> c_1(admit#(carry(x,u,v),z))
** Step 1.b:5: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1(admit#(carry(x,u,v),z))
        - Weak TRS:
            admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
            admit(x,nil()) -> nil()
            cond(true(),y) -> y
        - Signature:
            {admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          admit#(x,.(u,.(v,.(w(),z)))) -> c_1(admit#(carry(x,u,v),z))
** Step 1.b:6: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1(admit#(carry(x,u,v),z))
        - Signature:
            {admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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}
        
        Following symbols are considered usable:
          {admit#,cond#}
        TcT has computed the following interpretation:
               p(.) = [1] x1 + [1] x2 + [3]
               p(=) = [1] x2 + [1]         
           p(admit) = [1] x1 + [2]         
           p(carry) = [0]                  
            p(cond) = [2]                  
             p(nil) = [2]                  
             p(sum) = [1] x2 + [1] x3 + [4]
            p(true) = [0]                  
               p(w) = [2]                  
          p(admit#) = [8] x1 + [1] x2 + [0]
           p(cond#) = [2] x1 + [1] x2 + [0]
             p(c_1) = [1] x1 + [0]         
             p(c_2) = [8]                  
             p(c_3) = [1]                  
        
        Following rules are strictly oriented:
        admit#(x,.(u,.(v,.(w(),z)))) = [1] u + [1] v + [8] x + [1] z + [11]
                                     > [1] z + [0]                         
                                     = c_1(admit#(carry(x,u,v),z))         
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:7: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1(admit#(carry(x,u,v),z))
        - Signature:
            {admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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