## Stream: math-comp users

### Topic: ✔ Non-instantiated existential variables

#### Ricardo (Nov 14 2022 at 02:37):

I'm trying to define an `add_node` function in the following code.

``````From mathcomp
Require Import ssreflect ssrbool ssrnat ssrfun
eqtype choice fintype seq finfun finmap.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Module SiteGraphs.

Definition symmetric (T : finType) (R: rel T) :=
[forall x, forall y, R x y == R y x].
Definition relF (C : choiceType) (F : {fset C}) (_ _ : F) := false.
Lemma relF_sym (C : choiceType) (F : {fset C}) : symmetric (@relF C F).
Proof. apply/forallP => x. by apply/forallP. Qed.

Record sg (N S : choiceType) : Type :=
SG { nodes : {fset N}
; siteMap : {fmap S -> nodes}
; sites := domf siteMap (* : {fset S} *)
; edges : rel sites
; _ : symmetric edges
}.

Section SG_NS.
Variables (N S : choiceType).

Definition empty : sg N S :=
@SG _ _ fset0 fmap0 _ (relF_sym (domf fmap0)).

Variable (G : sg N S).

Definition add_node (n : N) : sg N S.
case: G => ns sm ss es es_sym.
refine (@SG _ _ (n |` ns)%fset _ _ _).
exact: es_sym.
``````

After the last tactic I get this warning.

``````No more goals, but there are non-instantiated existential variables:
Existential 1 =
...
You can use Unshelve.
``````

If I use Unshelve, nothing seems to happen. How can I transform the non-instantiated existential variables into subgoals here?

#### Julien Puydt (Nov 14 2022 at 06:17):

Shouldn't the `(foo: {someType})` actually be `{foo: someType}` ?

#### Julien Puydt (Nov 14 2022 at 06:24):

Ah, no that's part of the `{fset ...}` notation, not an implicit variable...

#### Julien Puydt (Nov 14 2022 at 06:26):

You know, if you drop your definition of `symmetric`, then your `relF_sym` lemma is just a `by []` away.

#### Julien Puydt (Nov 14 2022 at 06:29):

or is there a trick there too?

#### Julien Puydt (Nov 14 2022 at 06:55):

I modified your code slightly and obtained a goal which seems attainable: prove that `sm` which is a finite map from `S` to `ns` is also a finite map from `S` to `n | ns`, but since I never worked with `ffun`, I don't know how to do that yet...

#### Ricardo (Nov 14 2022 at 15:48):

Julien Puydt said:

You know, if you drop your definition of `symmetric`, then your `relF_sym` lemma is just a `by []` away.

I'm not sure I understand this. If I write the following

``````Lemma relF_sym (C : choiceType) (F : {fset C}) :
[forall x, forall y, (@relF C F) x y == (@relF C F) y x].
Proof. by []. Qed.
``````

I get `Error: No applicable tactic.`

#### Ricardo (Nov 14 2022 at 15:51):

Julien Puydt said:

I modified your code slightly and obtained a goal which seems attainable: prove that `sm` which is a finite map from `S` to `ns` is also a finite map from `S` to `n | ns`, but since I never worked with `ffun`, I don't know how to do that yet...

That's the goal I thought I'd have to prove, but I don't know how to get that as a goal from

``````Definition add_node (n : N) : sg N S.
``````

@Julien Puydt, would you share with us your modified code?

#### Julien Puydt (Nov 14 2022 at 15:56):

The modifications are:

• modify `relF` to simplify the lemma ;
• modify the record because I didn't like having `sites` if it's computable from the rest.

As you see, there are still two holes...

``````From mathcomp Require Import ssreflect ssrbool ssrnat ssrfun eqtype choice fintype seq finfun finmap.

Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Module SiteGraphs.

(* FIXME: both the definition and the lemma should be there already *)
Definition relF {F : Type} (_ _ : F) := false.

Lemma relF_sym {C : choiceType} (F : {fset C}) : symmetric (@relF F).
Proof.
by [].
Qed.

Record sg (N S : choiceType) : Type :=
SG {
nodes : {fset N};
siteMap : {fmap S -> nodes};
edges : rel (domf siteMap);
_ : symmetric edges
}.

Section SG_NS.

Variables (N S : choiceType).

Definition empty : sg N S := @SG N S fset0 fmap0 relF (@relF_sym _ _).

Variable (G : sg N S).

Definition add_node (n : N) : sg N S.
case: G => ns sm es es_sym.
have sm_plus: {fmap S -> (n |` ns)%fset}.
exists (domf sm).
refine [ffun x => _].
exists (val (sm x)). (* FIXME: suggestion of YB to understand *)
rewrite in_fsetU.
apply/orP.
right.
by case: (sm x) => v Hv.
have es_plus: rel (domf sm_plus).
have ->: domf sm_plus = domf sm ; last by [].
admit. (* FIXME: too many phantoms haunt the place *)
have symmetric_es_plus: symmetric es_plus.
admit. (* FIXME: should go like a breeze after the previous *)
exact (@SG _ _ (n |` ns)%fset sm_plus es_plus symmetric_es_plus).

End SG_NS.

End SiteGraphs.
``````

#### Ricardo (Nov 14 2022 at 16:11):

I see. You're using the `Prop`-based `symmetric` instead of the `bool`-based I defined. I thought it'd be more aligned with ssreflect's spirit to write and use a `bool`-based `symmetric`.

#### Ricardo (Nov 14 2022 at 16:12):

Thank you! With these ideas I'll try to continue.

#### Julien Puydt (Nov 14 2022 at 16:25):

I'm not sure that's the right track... I'm a beginner too!

#### Ricardo (Nov 14 2022 at 16:42):

I'm not getting why the line `exists (domf sm)` after the first `have` transforms `{fmap S -> (n |` ns)%fset}` into `{ffun domf sm -> (n |` ns)%fset}`. I can replicate the transformation with `apply: (@FinMap S ns' ss).` where `ns' := (n |` ns)%fset` and `ss := (domf sm)`. Why does `exists` help here? What is the existential variable that we are instantiating with `exists`? I tried unfolding everything in sight but didn't find any existential variables.

#### Ricardo (Nov 14 2022 at 16:58):

I'm also not getting the second `exists`.

#### Julien Puydt (Nov 14 2022 at 17:44):

The first exist is because an fmap is first a domain ; the second I haven't grasped yet

#### Ricardo (Nov 14 2022 at 18:12):

What does it mean that fmap is first a domain? How's that related to existential variables?

#### Ricardo (Nov 14 2022 at 18:15):

You mean that the first field of the `fmap` record is `domf`?

#### Ricardo (Nov 14 2022 at 18:45):

I mean, the second `exists` makes sense but I don't understand what it's doing mechanically.

#### Alexander Gryzlov (Nov 14 2022 at 19:06):

https://github.com/math-comp/finmap/blob/master/finmap.v#L488

https://github.com/math-comp/finmap/blob/master/finmap.v#L2885

#### Ricardo (Nov 14 2022 at 19:13):

I'm trying to understand how to work with the goal we reach after the `refine` in your code.

``````Goal forall (N S : choiceType) (G : sg N S), (nodes G).
(* move=> N S G. exact: (nodes G). *)
move=> N S.
case=> ns sm ss es es_sym.
(* case: ns. *)
case: (enum_fset ns) => [| n ns' ].
- (* exact: fset0. *) admit.
- exists n.
``````

Why doesn't the first commented line work? How can I prove this goal? Is it provable?

#### Alexander Gryzlov (Nov 14 2022 at 19:23):

Oh nevermind, it's actually constructing this with the second `exists`: https://github.com/math-comp/finmap/blob/master/finmap.v#L527

#### Ricardo (Nov 14 2022 at 19:31):

@Alexander Gryzlov, how can an `exists` construct an `FSetSub`?

#### Alexander Gryzlov (Nov 14 2022 at 19:34):

It's a form of `constructor` tactic, see https://coq.inria.fr/refman/proofs/writing-proofs/reasoning-inductives.html?highlight=exists#coq:tacn.exists

#### Ricardo (Nov 14 2022 at 19:36):

To prove `es_plus` it'd be nice if we could remember how `sm_plus` was constructed, since `domf sm_plus = domf sm` by construction. Am I understading correctly that `have` is designed not to remember the computationally relevant part?

#### Ricardo (Nov 14 2022 at 19:38):

Alexander Gryzlov said:

It's a form of `constructor` tactic, see https://coq.inria.fr/refman/proofs/writing-proofs/reasoning-inductives.html?highlight=exists#coq:tacn.exists

I see. Thank you! That helps me a lot :smile:

#### Ricardo (Nov 14 2022 at 20:03):

With the line `case: (sm x)` it seems we are doing case-analysis on something of type `FSetSub`. Why is that so? Because the codomain of `sm` is an `fset`? Or more precisely because `sm x` is an element of `ns`? Or because there is a coercion from `fset` to `fset_sub_type`?

#### Ricardo (Nov 14 2022 at 20:22):

Ricardo said:

With the line `case: (sm x)` it seems we are doing case-analysis on something of type `FSetSub`. Why is that so? Because the codomain of `sm` is an `fset`? Or more precisely because `sm x` is an element of `ns`? Or because there is a coercion from `fset` to `fset_sub_type`?

It seems we can do case-analysis on any element of an `fset` as if it was an element of `fset_sub_type`. Is that due to a coercion?

#### Ricardo (Nov 14 2022 at 20:57):

I used `Pring Graph` and searched for `> fset` to find coercions into `fset_sub_type` but didin't find anything. There are lots of coercion paths that mention `fset_sub_type` though. For example,

``````[fset_sub_type] : finSet >-> predArgType
``````

Is this the coercion I'm looking for? If so, I'm not getting why `case: (sm x)` is case-analysing on something of the form `FSetSub` when `predArgType = Type`.

#### Julien Puydt (Nov 14 2022 at 22:38):

I simplified things a bit:

``````Definition add_node (n : N) : sg N S.
case: G => ns sm es es_sym.
have sm_plus: {fmap S -> (n |` ns)%fset}.
refine [fmap x: domf sm => fincl _ (sm x)].
by rewrite /fsubset (fsetKUC _ _).
have same_dom: domf sm_plus = domf sm.
admit. (* FIXME: sm_plus is opaque *)
have es_plus: rel (domf sm_plus).
rewrite same_dom.
exact es.
have symmetric_es_plus: symmetric es_plus.
admit. (* FIXME: es_plus is opaque! *)
exact (@SG _ _ (n |` ns)%fset sm_plus es_plus symmetric_es_plus).
``````

and now I just need to find out how to replace `have`: it only gives opaque values...

#### Ricardo (Nov 14 2022 at 22:39):

By setting `Printing Coercions` I see that `sm x` is transformed into `fun_of_fin (ffun_of_fmap sm) x` of type `fset_sub_type ns`. By checking with `Check sm` I see that the type of `sm` is actually `{fmap Choice.sort S -> fset_sub_type ns}` instead of `{fmap Choice.sort S -> ns}`. By printing with `Print sg` I see that the type of `sg` does have `fset_sub_type nodes` in the type of `siteMap` already.

That solves the puzzle. Thank you everyone :smile:

#### Julien Puydt (Nov 14 2022 at 22:40):

Here is some sample code showing off what I understand from finmap:

``````From mathcomp Require Import ssreflect ssrbool ssrnat ssrfun eqtype choice fintype seq finfun.
From mathcomp.finmap Require Import finmap.

Section Foo.

Local Open Scope fset.
Local Open Scope fmap.

Variable D T: choiceType. (* Definition and Target *)

Variable A: {fset D}.
Variable B C: {fset T}.
Hypothesis BsubC: B `<=` C.

Variable f: {fmap A -> B}.

Definition ftilde: {fmap A -> C}.
Proof.
refine [fmap x: domf f => fincl _ (f x)].
by rewrite /fsubset.
Defined.

Lemma same_dom: domf f = domf ftilde.
Proof.
by [].
Qed.

(* FIXME: not really satisfying *)
Lemma same_values: forall x, ftilde.[? x] = [ffun x0 => (fincl BsubC) (f x0)].[? x].
Proof.
by rewrite /ftilde.
Qed.

End Foo.
``````

#### Julien Puydt (Nov 14 2022 at 22:42):

@Ricardo If you have working code, I'd like to see it, please.

#### Ricardo (Nov 14 2022 at 22:50):

I haven't solved the puzzle of how to write `add_node`. I only solved the puzzle of why we can do case analysis on `sm x`.

#### Ricardo (Nov 14 2022 at 23:04):

Julien Puydt said:

and now I just need to find out how to replace `have`: it only gives opaque values...

Yeah that's exactly what I was thinking. Your last code works because `ftilde` is defined transparently. We need a transparent `have` for this.

#### Ricardo (Nov 14 2022 at 23:47):

Wow I made it. Thank you @Julien Puydt! Your help was very useful to get here. I mean, you did it really.

``````From mathcomp Require Import ssreflect ssrbool ssrnat ssrfun eqtype choice fintype seq finfun finmap.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Module SiteGraphs.
Local Open Scope fset.
Local Open Scope fmap.

(* symmetric relations *)
Definition symmetric (T : finType) (R: rel T) :=
[forall x, forall y, R x y == R y x].
Definition rel0 (C : choiceType) (F : {fset C}) (_ _ : F) := false.
Lemma rel0_sym (C : choiceType) (F : {fset C}) : symmetric (@rel0 C F).
Proof. apply/forallP => x. by apply/forallP. Qed.

(* site graphs *)
Record sg (N S : choiceType) : Type :=
SG { nodes : {fset N}
; siteMap : {fmap S -> nodes}
; sites := domf siteMap (* : {fset S} *)
; edges : rel sites
; _ : symmetric edges
}.

Section SG_NS.
Variables (N S : choiceType).

Definition empty : sg N S :=
@SG _ _ fset0 fmap0 _ (rel0_sym (domf fmap0)).

Variable (G : sg N S).

Definition add_node (n : N) : sg N S.
case: G => ns sm ss es es_sym.
set ns' := (n |` ns). About fsetKUC.
have ns_sub_ns' : ns `<=` ns' by rewrite /fsubset fsetKUC.
set sm' : {fmap S -> ns'} :=
[fmap x: ss => fincl ns_sub_ns' (sm x)].
have eq_sites : domf sm' = ss by [].
pose es' : rel (domf sm') := fun x y => es x y.
have es'_sym : symmetric es' by [].
exact: (@SG _ _ ns' sm' es' es'_sym).
Defined.

End SG_NS.

End SiteGraphs.
``````

#### Ricardo (Nov 14 2022 at 23:48):

Still, it'd be great to know if there's a variant of `have` that is transparent.

#### Ricardo (Nov 14 2022 at 23:57):

Oh, it was easier than we thought. You just need to put an `@` in front of the name to make the definition transparent according to the docs, like in

``````have @sm' : {fmap S -> ns'}.
refine [fmap x: ss => fincl _ (sm x)].
by rewrite /fsubset fsetKUC.
``````

#### Notification Bot (Nov 14 2022 at 23:58):

Ricardo has marked this topic as resolved.

#### Cyril Cohen (Nov 14 2022 at 23:59):

Direct solution

``````Local Open Scope fset_scope.
Definition add_node (n : N) : sg N S :=
@SG N S (n |` nodes G)
(FinMap (finfun (fincl (fsubsetUr [fset n] (nodes G)) \o siteMap G)))
(@edges N S G) (@edges_sym N S G).
``````

#### Ricardo (Nov 15 2022 at 00:03):

Nice! Thanks @Cyril Cohen.

#### Julien Puydt (Nov 15 2022 at 06:04):

I didn't know and didn't find the `have @` trick! I'm still not entirely satisfied with what I have. Changes: I mostly put some arguments in `{ }` to turn them optional, use `g` instead of `G` because it's a specific object and not a type and use `'` as suffix instead of `_plus`:

``````(* Ricardo @ 'math-comp users' *)

From mathcomp Require Import ssreflect ssrbool ssrnat ssrfun eqtype choice fintype seq finfun finmap.

Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Module SiteGraphs.

(* FIXME: both the definition and the lemma should be there already *)

Definition relF {F : Type} (_ _ : F) := false.

Lemma relF_sym {C : choiceType} {F : {fset C}} : symmetric (@relF F).
Proof.
by [].
Qed.

Record sg {N S : choiceType} : Type :=
SG {
nodes : {fset N};
siteMap : {fmap S -> nodes};
edges : rel (domf siteMap);
edges_sym : symmetric edges
}.

Section SG_NS.

Local Open Scope fset.

Variables (N S : choiceType).

Definition empty : sg := @SG N S fset0 fmap0 relF relF_sym.

Variable (g : @sg N S).

(* after a discussion with Ricardo, this is the result: *)
Definition add_node (n : N) : @sg N S.
case: g => ns sm es es_sym.
set ns' := (n |` ns).
have @sm': {fmap S -> ns'}.
refine [fmap x: domf sm => fincl _ (sm x)].
by rewrite /fsubset (fsetKUC _ _).
have @es': rel (domf sm') := es.
have symmetric_es': symmetric es' by [].
exact (SG symmetric_es').
Qed.

Check g.
Check edges.
Fail Check edges g. (* FIXME: why? consequence: below @edges ... *)

Check g.
Check edges_sym.
Fail Check edges_sym g. (* FIXME: same issue... *)

(* and Cyril Cohen proposes: *)
Definition add_node_bis (n: N): sg :=
@SG _ _ (n |` nodes g)
(FinMap (finfun (fincl (fsubsetUr [fset n] (nodes g)) \o siteMap g)))
(@edges _ _ g) (@edges_sym _ _ g).

End SG_NS.

End SiteGraphs.
``````

Last updated: Jul 23 2024 at 21:01 UTC