Re: [Hol-info] Improve EXISTS_TAC with by taking hints from the assumption list

[Michael N. <mic...@ni...>]

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 <ra...@me...> 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" <vin...@gm...> 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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