/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(?,O(1))
* Step 1: Sum.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            app(cons(X),YS) -> cons(X)
            app(nil(),YS) -> YS
            from(X) -> cons(X)
            prefix(L) -> cons(nil())
            zWadr(XS,nil()) -> nil()
            zWadr(cons(X),cons(Y)) -> cons(app(Y,cons(X)))
            zWadr(nil(),YS) -> nil()
        - Signature:
            {app/2,from/1,prefix/1,zWadr/2} / {cons/1,nil/0}
        - Obligation:
             runtime complexity wrt. defined symbols {app,from,prefix,zWadr} and constructors {cons,nil}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: ToInnermost.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            app(cons(X),YS) -> cons(X)
            app(nil(),YS) -> YS
            from(X) -> cons(X)
            prefix(L) -> cons(nil())
            zWadr(XS,nil()) -> nil()
            zWadr(cons(X),cons(Y)) -> cons(app(Y,cons(X)))
            zWadr(nil(),YS) -> nil()
        - Signature:
            {app/2,from/1,prefix/1,zWadr/2} / {cons/1,nil/0}
        - Obligation:
             runtime complexity wrt. defined symbols {app,from,prefix,zWadr} and constructors {cons,nil}
    + 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:
            app(cons(X),YS) -> cons(X)
            app(nil(),YS) -> YS
            from(X) -> cons(X)
            prefix(L) -> cons(nil())
            zWadr(XS,nil()) -> nil()
            zWadr(cons(X),cons(Y)) -> cons(app(Y,cons(X)))
            zWadr(nil(),YS) -> nil()
        - Signature:
            {app/2,from/1,prefix/1,zWadr/2} / {cons/1,nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app,from,prefix,zWadr} and constructors {cons,nil}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          app#(cons(X),YS) -> c_1()
          app#(nil(),YS) -> c_2()
          from#(X) -> c_3()
          prefix#(L) -> c_4()
          zWadr#(XS,nil()) -> c_5()
          zWadr#(cons(X),cons(Y)) -> c_6(app#(Y,cons(X)))
          zWadr#(nil(),YS) -> c_7()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 4: UsableRules.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            app#(cons(X),YS) -> c_1()
            app#(nil(),YS) -> c_2()
            from#(X) -> c_3()
            prefix#(L) -> c_4()
            zWadr#(XS,nil()) -> c_5()
            zWadr#(cons(X),cons(Y)) -> c_6(app#(Y,cons(X)))
            zWadr#(nil(),YS) -> c_7()
        - Weak TRS:
            app(cons(X),YS) -> cons(X)
            app(nil(),YS) -> YS
            from(X) -> cons(X)
            prefix(L) -> cons(nil())
            zWadr(XS,nil()) -> nil()
            zWadr(cons(X),cons(Y)) -> cons(app(Y,cons(X)))
            zWadr(nil(),YS) -> nil()
        - Signature:
            {app/2,from/1,prefix/1,zWadr/2,app#/2,from#/1,prefix#/1,zWadr#/2} / {cons/1,nil/0,c_1/0,c_2/0,c_3/0,c_4/0
            ,c_5/0,c_6/1,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,from#,prefix#,zWadr#} and constructors {cons,nil}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          app#(cons(X),YS) -> c_1()
          app#(nil(),YS) -> c_2()
          from#(X) -> c_3()
          prefix#(L) -> c_4()
          zWadr#(XS,nil()) -> c_5()
          zWadr#(cons(X),cons(Y)) -> c_6(app#(Y,cons(X)))
          zWadr#(nil(),YS) -> c_7()
* Step 5: Trivial.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            app#(cons(X),YS) -> c_1()
            app#(nil(),YS) -> c_2()
            from#(X) -> c_3()
            prefix#(L) -> c_4()
            zWadr#(XS,nil()) -> c_5()
            zWadr#(cons(X),cons(Y)) -> c_6(app#(Y,cons(X)))
            zWadr#(nil(),YS) -> c_7()
        - Signature:
            {app/2,from/1,prefix/1,zWadr/2,app#/2,from#/1,prefix#/1,zWadr#/2} / {cons/1,nil/0,c_1/0,c_2/0,c_3/0,c_4/0
            ,c_5/0,c_6/1,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,from#,prefix#,zWadr#} and constructors {cons,nil}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:app#(cons(X),YS) -> c_1()
             
          
          2:S:app#(nil(),YS) -> c_2()
             
          
          3:S:from#(X) -> c_3()
             
          
          4:S:prefix#(L) -> c_4()
             
          
          5:S:zWadr#(XS,nil()) -> c_5()
             
          
          6:S:zWadr#(cons(X),cons(Y)) -> c_6(app#(Y,cons(X)))
             -->_1 app#(nil(),YS) -> c_2():2
             -->_1 app#(cons(X),YS) -> c_1():1
          
          7:S:zWadr#(nil(),YS) -> c_7()
             
          
        The dependency graph contains no loops, we remove all dependency pairs.
* Step 6: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {app/2,from/1,prefix/1,zWadr/2,app#/2,from#/1,prefix#/1,zWadr#/2} / {cons/1,nil/0,c_1/0,c_2/0,c_3/0,c_4/0
            ,c_5/0,c_6/1,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,from#,prefix#,zWadr#} and constructors {cons,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))