logic - Is it okay to decide the validity of formula by gathering the validity of subformulas?

Question

Problem

Formally, I want to know the relation between

forall I. I ? (f1∧f2∧f3)

and

(forall I. I ? f1) and (forall I. I ? f2) and (forall I. I ? f3).

Will the first holds true if the second holds true? Or vice versa?

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Details

I want to check the validity of the First Order Logic formula (f1∧f2∧f3), using SMT solver, encode it as is ?(f1∧f2∧f3) UNSAT?.

However, it takes too long time. So I consider to check the validity of each formulas f1, f2, and f3, encode it as (is (?f1) UNSAT) and (is (?f2) UNSAT) and (is (?f3) UNSAT)?. This calls the SMT solver 3 times, but the program can escape earlier if one of the SMT query returns "unknown" or "invalid".

Is it safe (and efficient) to use this method instead?

question from:https://stackoverflow.com/questions/65867370/is-it-okay-to-decide-the-validity-of-formula-by-gathering-the-validity-of-subfor

Answer 1

Waitting for answers