Hi everyone, I have this goal and when I am using `simpl`

, it not simplifying. How can I simplify this expression to apply induction hypothesis?

```
cons (Vector.t R n) vhd m vtl =
finite_fun_to_matrix
(λ (i : t n) (j : t m.+1),
(\matrix_(i0, j0) matrix_to_finite_fun (cons (Vector.t R n) vhd m vtl)
(ord_to_finite i0) (ord_to_finite j0))%R
(finite_to_ord i) (finite_to_ord j))
Lemma matrix_back_and_forth_same :
forall (n m : nat) (mx : Matrix n m),
mx = matrix_to_Matrix (Matrix_to_matrix mx).
Proof.
intros ? ?.
revert n.
induction m.
+ intros ? ?.
unfold Matrix in mx.
pose proof (vector_inv_0 mx) as Hn.
subst; reflexivity.
+ intros ? ?.
unfold Matrix in mx.
destruct (vector_inv_S mx) as (vhd & vtl & vp).
subst.
unfold matrix_to_Matrix,
Matrix_to_matrix.
(* how to simplify *)
```

Complete Source Code:

```
Require Import Field_theory
Ring_theory List NArith
Ring Field Utf8 Lia
Coq.Arith.PeanoNat
Vector Fin.
From mathcomp Require Import
all_ssreflect algebra.matrix
algebra.ssralg.
Import ListNotations.
Set Implicit Arguments.
Section Mat.
Variable (R : Type).
Lemma vector_inv_0 (v : Vector.t R 0) :
v = @Vector.nil R.
Proof.
refine (match v with
| @Vector.nil _ => _
end).
reflexivity.
Defined.
Lemma vector_inv_S (n : nat) (v : Vector.t R (S n)) :
{x & {v' | v = @Vector.cons _ x _ v'}}.
Proof.
refine (match v with
| @Vector.cons _ x _ v' => _
end).
eauto.
Defined.
Lemma fin_inv_0 (i : Fin.t 0) : False.
Proof. refine (match i with end). Defined.
Lemma fin_inv_S (n : nat) (i : Fin.t (S n)) :
(i = Fin.F1) + {i' | i = Fin.FS i'}.
Proof.
refine (match i with
| Fin.F1 _ => _
| Fin.FS _ _ => _
end); eauto.
Defined.
Definition vector_to_finite_fun :
forall n, Vector.t R n -> (Fin.t n -> R).
Proof.
induction n.
+ intros v f.
apply fin_inv_0 in f.
refine (match f with end).
+ intros v f.
destruct (vector_inv_S v) as (vhd & vtl & vp).
destruct (fin_inv_S f) as [h | [t p]].
exact vhd.
exact (IHn vtl t).
Defined.
Definition finite_fun_to_vector :
forall n, (Fin.t n -> R) -> Vector.t R n.
Proof.
induction n.
+ intros f.
apply Vector.nil.
+ intros f.
apply Vector.cons.
apply f, Fin.F1.
apply IHn;
intro m.
apply f, Fin.FS, m.
Defined.
Lemma finite_fun_to_vector_correctness
(m : nat) (f : Fin.t m -> R) (i : Fin.t m) :
Vector.nth (finite_fun_to_vector f) i = f i.
Proof.
induction m.
- inversion i.
- pose proof fin_inv_S i as [-> | (i' & ->)].
+ reflexivity.
+ cbn.
now rewrite IHm.
Defined.
Lemma vector_to_finite_fun_correctness
(m : nat) (v : Vector.t R m) (i : Fin.t m) :
Vector.nth v i = (vector_to_finite_fun v) i.
Proof.
induction m.
- inversion i.
- pose proof fin_inv_S i as [-> | (i' & ->)].
destruct (vector_inv_S v) as (vhd & vtl & vp).
rewrite vp.
reflexivity.
destruct (vector_inv_S v) as (vhd & vtl & vp).
rewrite vp;
simpl;
now rewrite IHm.
Defined.
Lemma vector_finite_back_forth :
forall (n : nat) (v : Vector.t R n),
v = finite_fun_to_vector (vector_to_finite_fun v).
Proof.
induction n.
+ intros v.
pose proof (vector_inv_0 v) as Hv.
subst;
reflexivity.
+ intros v.
destruct (vector_inv_S v) as (vhd & vtl & vp).
subst; simpl; f_equal.
apply IHn.
Defined.
End Mat.
Section Mx.
Variable (R : Type).
Definition Matrix n m := Vector.t (Vector.t R n) m.
Definition finite_fun_to_matrix {n m}
(f : Fin.t n -> Fin.t m -> R) : Matrix n m :=
@finite_fun_to_vector _ m (fun (x : Fin.t m) =>
@finite_fun_to_vector _ n (fun (y : Fin.t n) => f y x)).
Definition matrix_to_finite_fun {n m}
(mx : Matrix n m) : Fin.t n -> Fin.t m -> R :=
fun (x : Fin.t n) (y : Fin.t m) =>
@vector_to_finite_fun _ n
((@vector_to_finite_fun _ m mx) y) x.
Lemma matrix_to_finite_back_forth :
forall n m (mx : Matrix n m),
mx = finite_fun_to_matrix (matrix_to_finite_fun mx).
Proof.
intros ? ?.
revert n.
induction m.
+ intros ? ?.
unfold Matrix in mx.
pose proof (vector_inv_0 mx) as Hn.
subst; reflexivity.
+ intros ? ?.
unfold Matrix in mx.
destruct (vector_inv_S mx) as (vhd & vtl & vp).
subst.
unfold finite_fun_to_matrix,
matrix_to_finite_fun.
simpl; f_equal.
apply vector_finite_back_forth.
apply IHm.
Defined.
Definition finite_to_ord {n} (f : Fin.t n) : 'I_n.
have [m Hm] := (to_nat f).
apply (introT ltP) in Hm.
apply (Ordinal Hm).
Defined.
Definition ord_to_finite {n} (x : 'I_n) : Fin.t n.
have Hx: x < n by [].
apply (elimT ltP) in Hx.
apply (of_nat_lt Hx).
Defined.
Definition Matrix_to_matrix {n m}
(mx : Matrix n m) : 'M[R]_(n, m) :=
\matrix_(i < n, j < m)
(matrix_to_finite_fun
mx
(ord_to_finite i)
(ord_to_finite j)).
Definition matrix_to_Matrix {n m}
(mx : 'M[R]_(n, m)) : Matrix n m :=
finite_fun_to_matrix (fun (i : Fin.t n)
(j : Fin.t m) =>
mx (finite_to_ord i) (finite_to_ord j)).
Lemma matrix_to_Matrix_correctness :
forall n m (i : Fin.t n) (j : Fin.t m)
(mx : 'M[R]_(n, m)),
mx (finite_to_ord i) (finite_to_ord j) =
Vector.nth (Vector.nth (matrix_to_Matrix mx) j) i.
Proof.
intros *.
unfold matrix_to_Matrix,
finite_fun_to_matrix.
rewrite finite_fun_to_vector_correctness.
rewrite finite_fun_to_vector_correctness.
reflexivity.
Defined.
Lemma matrix_back_and_forth_same :
forall (n m : nat) (mx : Matrix n m),
mx = matrix_to_Matrix (Matrix_to_matrix mx).
Proof.
intros ? ?.
revert n.
induction m.
+ intros ? ?.
unfold Matrix in mx.
pose proof (vector_inv_0 mx) as Hn.
subst; reflexivity.
+ intros ? ?.
unfold Matrix in mx.
destruct (vector_inv_S mx) as (vhd & vtl & vp).
subst.
unfold matrix_to_Matrix,
Matrix_to_matrix.
(* This simp
simpl.
```

the standard answer is to try other evaluation approaches, like `cbn`

, `compute`

/`cbv`

, etc.

Thanks @Karl Palmskog . cbn worked but it's not simplifying the terms to the point that I can apply reflexivity or induction hypothesis. compute ran for 8 minutes and did nothing before I killed it. For example, the right hand side of the below expression should reduct to `vhd`

but apparently, it's not happening.

```
vhd =
finite_fun_to_vector
(λ y : t n,
(\matrix_(i, j) matrix_to_finite_fun (cons (Vector.t R n) vhd m vtl)
(ord_to_finite i) (ord_to_finite j))%R
(finite_to_ord y) (Ordinal (n:=m.+1) (m:=0) (erefl true)))
```

if it's slow, you can try `vm_compute`

, or even `native_compute`

if you have `coq-native`

in opam (recommend different opam switch)

some definitions will not reduce because they are "locked" (by library maintainers), not sure if this is happening here

I figured out that `introT`

, and similarly elimT, are opaque so I wrote one that reduces, using Defined (I also feel something like this happening, and hopefully I will figure it out).

An orthogonal question: is there any rationale behind using `Qed`

in proofs?

`Qed`

is used for several reasons, not least:

- proof terms are not persisted in memory after
`Qed`

, but only written to disk - this saves a lot of resources `Qed`

is an abstraction barrier that guarantees that other constants can't depend on the body of the defined constant, only its type. This makes it possible to prove theorems asynchronously, and also to skip doing`Qed`

-proofs when convenient (`.vos`

/`.vio`

toolchains)

Do you think it's a good idea to have same theorem with two versions: one with Qed and one with Defined?

if you know you need to access (compute with) the body of the constant somewhere else, go with `Defined`

. Otherwise, I'd go with `Qed`

until you find out you need something different

even though it's a classic problem in Coq that some constants won't unfold, I think the problem of over-unfolding can be a serious issue (proof context becomes a mess). `Qed`

can be a way to manage over-unfolding (but for more advanced unfold mitigation, locking is probably the way to go)

Is this what you meant by `locked`

@Karl Palmskog ?

```
mx =
finite_fun_to_vector
(λ x : t m,
finite_fun_to_vector
(λ y : t n,
locked_with matrix_key (* SEE HERE *)
(matrix_of_fun_def (m:=n) (n:=m))
(λ (i : ordinal_finType n)
(j : ordinal_finType m),
vector_to_finite_fun
(vector_to_finite_fun mx
(ord_to_finite j))
(ord_to_finite i))
(finite_to_ord y) (finite_to_ord x)))
```

right, yes, that looks like a library locking. This was a library designer choice so that one can't unfold, only use lemmas

you could search for "locked definition" or similar in this stream or the MathComp stream, I know people talk about it form time to time. I don't know the technicalities

https://github.com/math-comp/hierarchy-builder/wiki/Locking says you can try `rewrite unlock`

Thanks @Li-yao and @Karl Palmskog .

`rewrite unlock`

did not work, and these are the relevant definitions in matrix. However, I could not find any lemma but I am not expert in math-comp so I might be missing something obvious.

```
Definition matrix_of_fun k := locked_with k matrix_of_fun_def.
Canonical matrix_unlockable k := [unlockable fun matrix_of_fun k].
```

Nevermind, I just unfolded the locked definition and get rid of it by destructing it.

```
∀ (n m : nat) (mx : Matrix n m),
mx =
finite_fun_to_vector
(λ x : t m,
finite_fun_to_vector
(λ y : t n,
(let 'tt := matrix_key in λ (T : Type) (x0 : T), x0)
((ordinal_finType n → ordinal_finType m → R) → 'M_(n, m))
(matrix_of_fun_def (m:=n) (n:=m))
(λ (i : ordinal_finType n) (j : ordinal_finType m),
vector_to_finite_fun
(vector_to_finite_fun mx (ord_to_finite j))
(ord_to_finite i)) (finite_to_ord y)
(finite_to_ord x)))
```

Very interesting way to not let definition unfold.

Karl Palmskog said:

`Qed`

is used for several reasons, not least:

- proof terms are not persisted in memory after
`Qed`

, but only written to disk - this saves a lot of resources

Is that actually true? I thought it was only that the ones from other files don't get loaded from disk

I've heard it was true at some point in Coq history, but you may be right that nowadays it only affects other files

hence you get these absurdly huge files which take 10GB memory to run `coqc`

on, but then you don't care once they are compiled

I don't think that Qed proofs were ever removed from the memory of a coqc process. But you're right that their loading is deferred when requiring a vo, which is an important feature.

but you *could* theoretically make the proof term inaccessible (and thus marked for GC) as soon as Qed is reached, right?

theoretically, yes

unfortunately the STM is a mess beyond repair

Its not entirely correct that Qed can be GCed safely tho right? For example, extraction is able to peek behind Qed's (which is another design issue in itself). https://github.com/coq/coq/issues/15874

the idea is that it can be swapped to disk (but not by the OS which doesn't have the info)

not gc'd completely away

Last updated: Jan 29 2023 at 06:02 UTC