Proving MStar' in Logical Foundations (IndProp.v)

Question

In Logical Foundations' chapter on Inductive Propositions, the exercise exp_match_ex1 involves the following definitions:

Inductive reg_exp (T : Type) : Type :=
  | EmptySet
  | EmptyStr
  | Char (t : T)
  | App (r1 r2 : reg_exp T)
  | Union (r1 r2 : reg_exp T)
  | Star (r : reg_exp T).

Arguments EmptySet {T}.
Arguments EmptyStr {T}.
Arguments Char {T} _.
Arguments App {T} _ _.
Arguments Union {T} _ _.
Arguments Star {T} _.

Inductive exp_match {T} : list T -> reg_exp T -> Prop :=
  | MEmpty : [] =~ EmptyStr
  | MChar x : [x] =~ (Char x)
  | MApp s1 re1 s2 re2
             (H1 : s1 =~ re1)
             (H2 : s2 =~ re2)
           : (s1 ++ s2) =~ (App re1 re2)
  | MUnionL s1 re1 re2
                (H1 : s1 =~ re1)
              : s1 =~ (Union re1 re2)
  | MUnionR re1 s2 re2
                (H2 : s2 =~ re2)
              : s2 =~ (Union re1 re2)
  | MStar0 re : [] =~ (Star re)
  | MStarApp s1 s2 re
                 (H1 : s1 =~ re)
                 (H2 : s2 =~ (Star re))
               : (s1 ++ s2) =~ (Star re)
  where "s =~ re" := (exp_match s re).

I'm stuck trying to prove the following lemma:

Lemma MStar' : forall T (ss : list (list T)) (re : reg_exp T),
  (forall s, In s ss -> s =~ re) ->
  fold app ss [] =~ Star re.
Proof.
  intros. induction ss.
    - simpl. apply MStar0.
    - simpl. pose proof (H x). assert (Hx: In x (x :: ss)). {
        simpl. left. reflexivity.
    } pose proof (H0 Hx).
    (* stuck *)

Which results in:

T: Type
x: list T
ss: list (list T)
re: reg_exp T
H: forall s : list T, In s (x :: ss) -> s =~ re
IHss: (forall s : list T, In s ss -> s =~ re) -> fold app ss [ ] =~ Star re
H0: In x (x :: ss) -> x =~ re
Hx: In x (x :: ss)
H1: x =~ re
====================================
1/1
x ++ fold app ss [ ] =~ Star re

Initially it looked like trying to proceed by induction on ss would allow me to make progress but I can't find any way to transform the hypothesis forall s : list T, In s (x :: ss) -> s =~ re so that I can prove fold app ss [ ] =~ Star re from the inductive hypothesis (forall s : list T, In s ss -> s =~ re) -> fold app ss [ ] =~ Star re.

score 1 · Accepted Answer · answered Sep 11 '21 at 16:27

1

I think the thing is that you do not need to apply induction hypothesis yet. Just try to see again on your constructors right when you have situation you've described (so, right on your (* stuck *) step).

answered Sep 11 '21 at 16:27

Andrey

712
6
16

Thank you for your pointer. Unfortunately I'm struggling to find something else to do than trying to use the inductive hypothesis. Among the `exp_match` constructors, I only see `MStarApp` and `Mapp` being potentially useful. The former will surely help me finish the proof, but only when I have proven that `ss =~ Star re` (since I have already established that `x ~= re`). The latter can't be applied either as I don't have `ss ~= re`. If I try leveraging the previous lemmas, I could get `x ~= Star re` from `Lemma MStar1`, but I don't see the point. `Lemma MUnion` seems similarly unhelpful... – user566206 Sep 11 '21 at 17:55
2

@user566206, you move in right way. But it may be you need a small hint about order. First, right on your (* stuck *) step you need apply MStarApp. Then, after solving first trivial goal, on the second goal you will be able to apply IH. And then you will see a way to move forward I think. – Andrey Sep 11 '21 at 20:09
1

An heartfelt thank you @Andrey. By letting me know that I was on the right track, you encouraged me to persevere and I finally was able to solve the proof through a few `assert`s. By the way, it turns out it is, in fact, possible to use `MStarApp` at the end rather than at my "stuck" step. However I tried both orders and I think your way is more elegant. Thanks again! – user566206 Sep 12 '21 at 12:59
@user566206, you are welcome, happy to hear that you've solved it :) – Andrey Sep 12 '21 at 17:07

Proving MStar' in Logical Foundations (IndProp.v)

1 Answers1