HOL4’s SATISFY_ss (from SatisfySimps) should solve this problem (particularly
now that Thomas Türk has fixed a bug in its code).
Michael
On 27/12/2012, at 11:42, Ramana Kumar <[email protected]> wrote:
> For what it's worth, my usual move in this situation is to do
>
> qmatch_assum_abbrev_tac 'P t' >>
> qexists_tac 't' >>
> simp[Abbr'X']
>
> Note that P is a metavariable, i.e. I have to type it out, but I avoid typing
> the large term abbreviated by t. The X stands for pieces of P I want
> unabbreviated after.
>
> HINT_EXISTS_TAC might still be an improvement.
>
> Sorry for no proper backquotes, using my phone.
> On Dec 26, 2012 10:57 PM, "Vincent Aravantinos"
> <[email protected]> wrote:
>> Hi list,
>>
>> here is another situation which I don't like and often meet (both in
>> HOL-Light and HOL4), and a potential solution.
>> Please tell me if you also often meet the situation, if you agree that
>> it is annoying, and if there is already a solution which I don't know of
>> (I'm pretty sure there is no solution in HOL-Light, but I'm not familiar
>> with all its extensions over there).
>>
>> SITUATION:
>>
>> goal of the form `?x. ... /\ P x /\ ...`
>> + one of the assumptions is of the form `P t` (t is a big a term)
>> + one wants to use t as the witness for x
>>
>>
>> USUAL MOVE:
>>
>> e (EXISTS_TAC `t`)
>> (*Then rewrite with the assumptions in order to remove the now
>> trivial P t:*)
>> e(ASM_REWRITE_TAC[])
>>
>>
>> PROBLEM WITH THIS:
>>
>> Annoying to write explicitly the big term t.
>> Plus the subsequent ASM_REWRITE_TAC is trivial and can thus be
>> systematized.
>> Not really annoying if it only appears from time to time, but I
>> personally often face this situation.
>>
>>
>> SOLUTION:
>>
>> A tactic HINT_EXISTS_TAC which looks for an assumption matching one
>> (or more) of the conjuncts in the conclusion and applies EXISTS_TAC with
>> the corresponding term.
>>
>>
>> EXAMPLE IN HOL-LIGHT:
>>
>> (* Consider the following goal:*)
>>
>> 0 [`P m`]
>> 1 [`!x. P x ==> x <= m`]
>>
>> `?x. P x`
>>
>> (* Usual move: *)
>> # e (EXISTS_TAC `m:num`);;
>> val it : goalstack = 1 subgoal (1 total)
>>
>> 0 [`P m`]
>> 1 [`!x. P x ==> x <= m`]
>>
>> `P m`
>>
>> # e (ASM_REWRITE_TAC[]);;
>> val it : goalstack = No subgoals
>>
>> (* New solution, which finds the witness automatically and removes
>> the trivial conjunct : *)
>>
>> # b (); b (); e HINT_EXISTS_TAC;;
>> val it : goalstack = No subgoals
>>
>> (* Notes:
>> * - The use case gets more interesting when m is actually a big term.
>> * - Though, in this example, the tactic allows to conclude the goal,
>> it can also be used just to make progress in the proof without necessary
>> concluding.
>> *)
>>
>> A HOL-Light implementation for HINT_EXISTS_TAC is provided below the
>> signature.
>> One for HOL4 can easily be implemented if anyone expresses some interest
>> for it.
>>
>> Cheers,
>> V.
>>
>> --
>> Vincent ARAVANTINOS - PostDoctoral Fellow - Concordia University, Hardware
>> Verification Group
>> http://users.encs.concordia.ca/~vincent/
>>
>>
>> let HINT_EXISTS_TAC (hs,c as g) =
>> let hs = map snd hs in
>> let v,c' = dest_exists c in
>> let vs,c' = strip_exists c' in
>> let hyp_match c h =
>> ignore (check (not o exists (C mem vs) o frees) c);
>> term_match (subtract (frees c) [v]) c (concl h), h
>> in
>> let (_,subs,_),h = tryfind (C tryfind hs o hyp_match) (binops `/\` c') in
>> let witness =
>> match subs with
>> |[] -> v
>> |[t,u] when u = v -> t
>> |_ -> failwith "HINT_EXISTS_TAC not applicable"
>> in
>> (EXISTS_TAC witness THEN REWRITE_TAC hs) g;;
>>
>>
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