/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:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
             runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
             runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
             runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          ackin(s(x),y){y -> s(y)} =
            ackin(s(x),s(y)) ->^+ u21(ackin(s(x),y),x)
              = C[ackin(s(x),y) = ackin(s(x),y){}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
             runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          ackin#(s(X),s(Y)) -> c_1(u21#(ackin(s(X),Y),X))
          u21#(ackout(X),Y) -> c_2(ackin#(Y,X))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            ackin#(s(X),s(Y)) -> c_1(u21#(ackin(s(X),Y),X))
            u21#(ackout(X),Y) -> c_2(ackin#(Y,X))
        - Strict TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2,ackin#/2,u21#/2} / {ackout/1,s/1,u22/1,c_1/1,c_2/1}
        - Obligation:
             runtime complexity wrt. defined symbols {ackin#,u21#} and constructors {ackout,s,u22}
    + 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(u21) = {1},
            uargs(u22) = {1},
            uargs(u21#) = {1},
            uargs(c_1) = {1},
            uargs(c_2) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
             p(ackin) = [0]                  
            p(ackout) = [3]                  
                 p(s) = [1] x1 + [0]         
               p(u21) = [1] x1 + [0]         
               p(u22) = [1] x1 + [0]         
            p(ackin#) = [8] x1 + [0]         
              p(u21#) = [1] x1 + [8] x2 + [0]
               p(c_1) = [1] x1 + [0]         
               p(c_2) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          u21#(ackout(X),Y) = [8] Y + [3]     
                            > [8] Y + [0]     
                            = c_2(ackin#(Y,X))
          
           u21(ackout(X),Y) = [3]             
                            > [0]             
                            = u22(ackin(Y,X)) 
          
          
          Following rules are (at-least) weakly oriented:
          ackin#(s(X),s(Y)) =  [8] X + [0]               
                            >= [8] X + [0]               
                            =  c_1(u21#(ackin(s(X),Y),X))
          
           ackin(s(X),s(Y)) =  [0]                       
                            >= [0]                       
                            =  u21(ackin(s(X),Y),X)      
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            ackin#(s(X),s(Y)) -> c_1(u21#(ackin(s(X),Y),X))
        - Strict TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
        - Weak DPs:
            u21#(ackout(X),Y) -> c_2(ackin#(Y,X))
        - Weak TRS:
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2,ackin#/2,u21#/2} / {ackout/1,s/1,u22/1,c_1/1,c_2/1}
        - Obligation:
             runtime complexity wrt. defined symbols {ackin#,u21#} and constructors {ackout,s,u22}
    + 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(u21) = {1},
            uargs(u22) = {1},
            uargs(u21#) = {1},
            uargs(c_1) = {1},
            uargs(c_2) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
             p(ackin) = [1] x2 + [0]
            p(ackout) = [1] x1 + [0]
                 p(s) = [1] x1 + [9]
               p(u21) = [1] x1 + [8]
               p(u22) = [1] x1 + [8]
            p(ackin#) = [1] x2 + [0]
              p(u21#) = [1] x1 + [0]
               p(c_1) = [1] x1 + [0]
               p(c_2) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          ackin#(s(X),s(Y)) = [1] Y + [9]               
                            > [1] Y + [0]               
                            = c_1(u21#(ackin(s(X),Y),X))
          
           ackin(s(X),s(Y)) = [1] Y + [9]               
                            > [1] Y + [8]               
                            = u21(ackin(s(X),Y),X)      
          
          
          Following rules are (at-least) weakly oriented:
          u21#(ackout(X),Y) =  [1] X + [0]     
                            >= [1] X + [0]     
                            =  c_2(ackin#(Y,X))
          
           u21(ackout(X),Y) =  [1] X + [8]     
                            >= [1] X + [8]     
                            =  u22(ackin(Y,X)) 
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            ackin#(s(X),s(Y)) -> c_1(u21#(ackin(s(X),Y),X))
            u21#(ackout(X),Y) -> c_2(ackin#(Y,X))
        - Weak TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2,ackin#/2,u21#/2} / {ackout/1,s/1,u22/1,c_1/1,c_2/1}
        - Obligation:
             runtime complexity wrt. defined symbols {ackin#,u21#} and constructors {ackout,s,u22}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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