/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(?,O(1))
* Step 1: Sum WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            f(X,X) -> c(X)
            f(X,c(X)) -> f(s(X),X)
            f(s(X),X) -> f(X,a(X))
        - Signature:
            {f/2} / {a/1,c/1,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f} and constructors {a,c,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            f(X,X) -> c(X)
            f(X,c(X)) -> f(s(X),X)
            f(s(X),X) -> f(X,a(X))
        - Signature:
            {f/2} / {a/1,c/1,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f} and constructors {a,c,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          f#(X,X) -> c_1(X)
          f#(X,c(X)) -> c_2(f#(s(X),X))
          f#(s(X),X) -> c_3(f#(X,a(X)))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(X,X) -> c_1(X)
            f#(X,c(X)) -> c_2(f#(s(X),X))
            f#(s(X),X) -> c_3(f#(X,a(X)))
        - Strict TRS:
            f(X,X) -> c(X)
            f(X,c(X)) -> f(s(X),X)
            f(s(X),X) -> f(X,a(X))
        - Signature:
            {f/2,f#/2} / {a/1,c/1,s/1,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#} and constructors {a,c,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(X,X) -> c_1(X)
          f#(X,c(X)) -> c_2(f#(s(X),X))
          f#(s(X),X) -> c_3(f#(X,a(X)))
* Step 4: WeightGap WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(X,X) -> c_1(X)
            f#(X,c(X)) -> c_2(f#(s(X),X))
            f#(s(X),X) -> c_3(f#(X,a(X)))
        - Signature:
            {f/2,f#/2} / {a/1,c/1,s/1,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#} and constructors {a,c,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          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:
            uargs(c_2) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(a) = [2]         
              p(c) = [1]         
              p(f) = [1] x2 + [1]
              p(s) = [1]         
             p(f#) = [4]         
            p(c_1) = [1]         
            p(c_2) = [1] x1 + [3]
            p(c_3) = [0]         
          
          Following rules are strictly oriented:
             f#(X,X) = [4]            
                     > [1]            
                     = c_1(X)         
          
          f#(s(X),X) = [4]            
                     > [0]            
                     = c_3(f#(X,a(X)))
          
          
          Following rules are (at-least) weakly oriented:
          f#(X,c(X)) =  [4]            
                     >= [7]            
                     =  c_2(f#(s(X),X))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: WeightGap WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(X,c(X)) -> c_2(f#(s(X),X))
        - Weak DPs:
            f#(X,X) -> c_1(X)
            f#(s(X),X) -> c_3(f#(X,a(X)))
        - Signature:
            {f/2,f#/2} / {a/1,c/1,s/1,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#} and constructors {a,c,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          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:
            uargs(c_2) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(a) = [1]                  
              p(c) = [8]                  
              p(f) = [0]                  
              p(s) = [1]                  
             p(f#) = [2] x1 + [2] x2 + [3]
            p(c_1) = [1]                  
            p(c_2) = [1] x1 + [8]         
            p(c_3) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          f#(X,c(X)) = [2] X + [19]   
                     > [2] X + [13]   
                     = c_2(f#(s(X),X))
          
          
          Following rules are (at-least) weakly oriented:
             f#(X,X) =  [4] X + [3]    
                     >= [1]            
                     =  c_1(X)         
          
          f#(s(X),X) =  [2] X + [5]    
                     >= [2] X + [5]    
                     =  c_3(f#(X,a(X)))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(X,X) -> c_1(X)
            f#(X,c(X)) -> c_2(f#(s(X),X))
            f#(s(X),X) -> c_3(f#(X,a(X)))
        - Signature:
            {f/2,f#/2} / {a/1,c/1,s/1,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#} and constructors {a,c,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))