/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:
            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: 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:
             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:
             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_ = WIDP}
    + 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: WeightGap.  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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
    + Details:
        The weightgap principle applies using the following constant 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]                  
             p(admit) = [4] x1 + [4] x2 + [2]
             p(carry) = [1] x1 + [0]         
              p(cond) = [2] x1 + [1] x2 + [0]
               p(nil) = [2]                  
               p(sum) = [1] x1 + [1] x2 + [0]
              p(true) = [5]                  
                 p(w) = [2]                  
            p(admit#) = [6] x1 + [4] x2 + [0]
             p(cond#) = [1] x1 + [1] x2 + [3]
               p(c_1) = [1] x1 + [4]         
               p(c_2) = [0]                  
               p(c_3) = [1] x1 + [1]         
          
          Following rules are strictly oriented:
                      admit#(x,nil()) = [6] x + [8]                                                   
                                      > [0]                                                           
                                      = c_2()                                                         
          
                      cond#(true(),y) = [1] y + [8]                                                   
                                      > [1] y + [1]                                                   
                                      = c_3(y)                                                        
          
          admit(x,.(u,.(v,.(w(),z)))) = [4] u + [4] v + [4] x + [4] z + [10]                          
                                      > [1] u + [1] v + [4] x + [4] z + [6]                           
                                      = cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
          
                       admit(x,nil()) = [4] x + [10]                                                  
                                      > [2]                                                           
                                      = nil()                                                         
          
          
          Following rules are (at-least) weakly oriented:
          admit#(x,.(u,.(v,.(w(),z)))) =  [4] u + [4] v + [6] x + [4] z + [8]                                 
                                       >= [1] u + [1] v + [4] x + [4] z + [12]                                
                                       =  c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: RemoveWeakSuffixes.  WORST_CASE(?,O(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))))))
        - Weak DPs:
            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:
        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))))))
             
          
          2:W:admit#(x,nil()) -> c_2()
             
          
          3:W:cond#(true(),y) -> c_3(y)
             -->_1 cond#(true(),y) -> c_3(y):3
             -->_1 admit#(x,nil()) -> c_2():2
             -->_1 admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w())
                                                            ,.(u,.(v,.(w(),admit(carry(x,u,v),z)))))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: admit#(x,nil()) -> c_2()
** Step 1.b:5: SimplifyRHS.  WORST_CASE(?,O(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))))))
        - Weak DPs:
            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:
        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))))))
             
          
          3:W:cond#(true(),y) -> c_3(y)
             -->_1 cond#(true(),y) -> c_3(y):3
             -->_1 admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w())
                                                            ,.(u,.(v,.(w(),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()
** Step 1.b:6: UsableRules.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
        - Weak DPs:
            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_1/0,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)))) -> c_1()
          cond#(true(),y) -> c_3(y)
** Step 1.b:7: PredecessorEstimationCP.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
        - Weak DPs:
            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_1/0,c_2/0,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + 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: admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
          
        The strictly oriented rules are moved into the weak component.
*** Step 1.b:7.a:1: NaturalMI.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
        - Weak DPs:
            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_1/0,c_2/0,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + 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]         
               p(=) = [0]         
           p(admit) = [0]         
           p(carry) = [0]         
            p(cond) = [0]         
             p(nil) = [0]         
             p(sum) = [0]         
            p(true) = [0]         
               p(w) = [0]         
          p(admit#) = [2]         
           p(cond#) = [1] x2 + [0]
             p(c_1) = [0]         
             p(c_2) = [0]         
             p(c_3) = [1] x1 + [0]
        
        Following rules are strictly oriented:
        admit#(x,.(u,.(v,.(w(),z)))) = [2]  
                                     > [0]  
                                     = c_1()
        
        
        Following rules are (at-least) weakly oriented:
        cond#(true(),y) =  [1] y + [0]
                        >= [1] y + [0]
                        =  c_3(y)     
        
*** Step 1.b:7.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
            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_1/0,c_2/0,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

*** Step 1.b:7.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
            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_1/0,c_2/0,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
             
          
          2:W:cond#(true(),y) -> c_3(y)
             -->_1 cond#(true(),y) -> c_3(y):2
             -->_1 admit#(x,.(u,.(v,.(w(),z)))) -> c_1():1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: cond#(true(),y) -> c_3(y)
          1: admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
*** Step 1.b:7.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/0,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))