/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),?)
* Step 1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            and(false(),false()) -> false()
            and(false(),true()) -> false()
            and(true(),false()) -> false()
            and(true(),true()) -> true()
            eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
            eq(apply(T,S),lambda(X,Tp)) -> false()
            eq(apply(T,S),var(L)) -> false()
            eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
            eq(cons(T,L),nil()) -> false()
            eq(lambda(X,T),apply(Tp,Sp)) -> false()
            eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
            eq(lambda(X,T),var(L)) -> false()
            eq(nil(),cons(T,L)) -> false()
            eq(nil(),nil()) -> true()
            eq(var(L),apply(T,S)) -> false()
            eq(var(L),lambda(X,T)) -> false()
            eq(var(L),var(Lp)) -> eq(L,Lp)
            if(false(),var(K),var(L)) -> var(L)
            if(true(),var(K),var(L)) -> var(K)
            ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
            ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil()))))
                                          ,ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))
            ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
        - Signature:
            {and/2,eq/2,if/3,ren/3} / {apply/2,cons/2,false/0,lambda/2,nil/0,true/0,var/1}
        - Obligation:
             runtime complexity wrt. defined symbols {and,eq,if,ren} and constructors {apply,cons,false,lambda,nil,true
            ,var}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            and(false(),false()) -> false()
            and(false(),true()) -> false()
            and(true(),false()) -> false()
            and(true(),true()) -> true()
            eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
            eq(apply(T,S),lambda(X,Tp)) -> false()
            eq(apply(T,S),var(L)) -> false()
            eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
            eq(cons(T,L),nil()) -> false()
            eq(lambda(X,T),apply(Tp,Sp)) -> false()
            eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
            eq(lambda(X,T),var(L)) -> false()
            eq(nil(),cons(T,L)) -> false()
            eq(nil(),nil()) -> true()
            eq(var(L),apply(T,S)) -> false()
            eq(var(L),lambda(X,T)) -> false()
            eq(var(L),var(Lp)) -> eq(L,Lp)
            if(false(),var(K),var(L)) -> var(L)
            if(true(),var(K),var(L)) -> var(K)
            ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
            ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil()))))
                                          ,ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))
            ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
        - Signature:
            {and/2,eq/2,if/3,ren/3} / {apply/2,cons/2,false/0,lambda/2,nil/0,true/0,var/1}
        - Obligation:
             runtime complexity wrt. defined symbols {and,eq,if,ren} and constructors {apply,cons,false,lambda,nil,true
            ,var}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 3: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            and(false(),false()) -> false()
            and(false(),true()) -> false()
            and(true(),false()) -> false()
            and(true(),true()) -> true()
            eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
            eq(apply(T,S),lambda(X,Tp)) -> false()
            eq(apply(T,S),var(L)) -> false()
            eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
            eq(cons(T,L),nil()) -> false()
            eq(lambda(X,T),apply(Tp,Sp)) -> false()
            eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
            eq(lambda(X,T),var(L)) -> false()
            eq(nil(),cons(T,L)) -> false()
            eq(nil(),nil()) -> true()
            eq(var(L),apply(T,S)) -> false()
            eq(var(L),lambda(X,T)) -> false()
            eq(var(L),var(Lp)) -> eq(L,Lp)
            if(false(),var(K),var(L)) -> var(L)
            if(true(),var(K),var(L)) -> var(K)
            ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
            ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil()))))
                                          ,ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))
            ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
        - Signature:
            {and/2,eq/2,if/3,ren/3} / {apply/2,cons/2,false/0,lambda/2,nil/0,true/0,var/1}
        - Obligation:
             runtime complexity wrt. defined symbols {and,eq,if,ren} and constructors {apply,cons,false,lambda,nil,true
            ,var}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          eq(x,z){x -> apply(x,y),z -> apply(z,u)} =
            eq(apply(x,y),apply(z,u)) ->^+ and(eq(x,z),eq(y,u))
              = C[eq(x,z) = eq(x,z){}]

WORST_CASE(Omega(n^1),?)