/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/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:
             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: 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:
             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:
             runtime complexity wrt. defined symbols {admit,cond} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following weak dependency pairs:
        
        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#(x,nil()) -> c_2()
          cond#(true(),y) -> c_3(y)
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules 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#(x,nil()) -> c_2()
            cond#(true(),y) -> c_3(y)
        - 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,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
             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)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
          admit(x,nil()) -> nil()
          admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
          admit#(x,nil()) -> c_2()
          cond#(true(),y) -> c_3(y)
** Step 1.b:3: 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#(x,nil()) -> c_2()
            cond#(true(),y) -> c_3(y)
        - 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()
        - 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_3/1}
        - Obligation:
             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
          {1,2}
        by application of
          Pre({1,2}) = {3}.
        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))))))
          2: admit#(x,nil()) -> c_2()
          3: cond#(true(),y) -> c_3(y)
** Step 1.b:4: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            cond#(true(),y) -> c_3(y)
        - 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()
        - Weak 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#(x,nil()) -> c_2()
        - 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_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(.) = {2},
            uargs(cond) = {2},
            uargs(cond#) = {2},
            uargs(c_1) = {1},
            uargs(c_3) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(.) = [1] x2 + [0]         
                 p(=) = [2]                  
             p(admit) = [0]                  
             p(carry) = [1] x3 + [0]         
              p(cond) = [4] x1 + [1] x2 + [4]
               p(nil) = [1]                  
               p(sum) = [1] x2 + [1]         
              p(true) = [0]                  
                 p(w) = [0]                  
            p(admit#) = [4]                  
             p(cond#) = [1] x2 + [1]         
               p(c_1) = [1] x1 + [3]         
               p(c_2) = [0]                  
               p(c_3) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          cond#(true(),y) = [1] y + [1]
                          > [1] y + [0]
                          = c_3(y)     
          
          
          Following rules are (at-least) weakly oriented:
          admit#(x,.(u,.(v,.(w(),z)))) =  [4]                                                                 
                                       >= [4]                                                                 
                                       =  c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
          
                       admit#(x,nil()) =  [4]                                                                 
                                       >= [0]                                                                 
                                       =  c_2()                                                               
          
           admit(x,.(u,.(v,.(w(),z)))) =  [0]                                                                 
                                       >= [12]                                                                
                                       =  cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))      
          
                        admit(x,nil()) =  [0]                                                                 
                                       >= [1]                                                                 
                                       =  nil()                                                               
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: WeightGap 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()
        - Weak 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#(x,nil()) -> c_2()
            cond#(true(),y) -> c_3(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_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(.) = {2},
            uargs(cond) = {2},
            uargs(cond#) = {2},
            uargs(c_1) = {1},
            uargs(c_3) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(.) = [1] x1 + [1] x2 + [0]
                 p(=) = [1] x1 + [0]         
             p(admit) = [2] x2 + [0]         
             p(carry) = [1] x1 + [1] x3 + [1]
              p(cond) = [1] x2 + [1]         
               p(nil) = [4]                  
               p(sum) = [1] x1 + [1] x3 + [1]
              p(true) = [3]                  
                 p(w) = [4]                  
            p(admit#) = [1] x1 + [2] x2 + [4]
             p(cond#) = [1] x1 + [1] x2 + [0]
               p(c_1) = [1] x1 + [0]         
               p(c_2) = [1]                  
               p(c_3) = [1] x1 + [3]         
          
          Following rules are strictly oriented:
          admit(x,.(u,.(v,.(w(),z)))) = [2] u + [2] v + [2] z + [8]                                   
                                      > [1] u + [1] v + [2] z + [5]                                   
                                      = cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
          
                       admit(x,nil()) = [8]                                                           
                                      > [4]                                                           
                                      = nil()                                                         
          
          
          Following rules are (at-least) weakly oriented:
          admit#(x,.(u,.(v,.(w(),z)))) =  [2] u + [2] v + [1] x + [2] z + [12]                                
                                       >= [1] u + [2] v + [1] x + [2] z + [5]                                 
                                       =  c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
          
                       admit#(x,nil()) =  [1] x + [12]                                                        
                                       >= [1]                                                                 
                                       =  c_2()                                                               
          
                       cond#(true(),y) =  [1] y + [3]                                                         
                                       >= [1] y + [3]                                                         
                                       =  c_3(y)                                                              
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:6: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak 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#(x,nil()) -> c_2()
            cond#(true(),y) -> c_3(y)
        - 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()
        - 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_3/1}
        - Obligation:
             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))