/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:
            first(0(),X) -> nil()
            first(s(X),cons(Y)) -> cons(Y)
            from(X) -> cons(X)
        - Signature:
            {first/2,from/1} / {0/0,cons/1,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {first,from} and constructors {0,cons,nil,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: ToInnermost.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            first(0(),X) -> nil()
            first(s(X),cons(Y)) -> cons(Y)
            from(X) -> cons(X)
        - Signature:
            {first/2,from/1} / {0/0,cons/1,nil/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {first,from} and constructors {0,cons,nil,s}
    + Applied Processor:
        ToInnermost
    + Details:
        switch to innermost, as the system is overlay and right linear and does not contain weak rules
* Step 3: DependencyPairs.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            first(0(),X) -> nil()
            first(s(X),cons(Y)) -> cons(Y)
            from(X) -> cons(X)
        - Signature:
            {first/2,from/1} / {0/0,cons/1,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {first,from} and constructors {0,cons,nil,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          first#(0(),X) -> c_1()
          first#(s(X),cons(Y)) -> c_2()
          from#(X) -> c_3()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 4: UsableRules.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            first#(0(),X) -> c_1()
            first#(s(X),cons(Y)) -> c_2()
            from#(X) -> c_3()
        - Weak TRS:
            first(0(),X) -> nil()
            first(s(X),cons(Y)) -> cons(Y)
            from(X) -> cons(X)
        - Signature:
            {first/2,from/1,first#/2,from#/1} / {0/0,cons/1,nil/0,s/1,c_1/0,c_2/0,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {first#,from#} and constructors {0,cons,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          first#(0(),X) -> c_1()
          first#(s(X),cons(Y)) -> c_2()
          from#(X) -> c_3()
* Step 5: Trivial.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            first#(0(),X) -> c_1()
            first#(s(X),cons(Y)) -> c_2()
            from#(X) -> c_3()
        - Signature:
            {first/2,from/1,first#/2,from#/1} / {0/0,cons/1,nil/0,s/1,c_1/0,c_2/0,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {first#,from#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:first#(0(),X) -> c_1()
             
          
          2:S:first#(s(X),cons(Y)) -> c_2()
             
          
          3:S:from#(X) -> c_3()
             
          
        The dependency graph contains no loops, we remove all dependency pairs.
* Step 6: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {first/2,from/1,first#/2,from#/1} / {0/0,cons/1,nil/0,s/1,c_1/0,c_2/0,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {first#,from#} and constructors {0,cons,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))