Library Fsub_LetSum_Infrastructure
Infrastructure lemmas and tactic definitions for Fsub.
Authors: Brian Aydemir and Arthur Charguéraud, with help from Aaron Bohannon, Jeffrey Vaughan, and Dimitrios Vytiniotis.
This file contains a number of definitions, tactics, and lemmas that are based only on the syntax of the language at hand. While the exact statements of everything here would change for a different language, the general structure of this file (i.e., the sequence of definitions, tactics, and lemmas) would remain the same.
Table of contents:
Authors: Brian Aydemir and Arthur Charguéraud, with help from Aaron Bohannon, Jeffrey Vaughan, and Dimitrios Vytiniotis.
This file contains a number of definitions, tactics, and lemmas that are based only on the syntax of the language at hand. While the exact statements of everything here would change for a different language, the general structure of this file (i.e., the sequence of definitions, tactics, and lemmas) would remain the same.
Table of contents:
Require Export Fsub_LetSum_Definitions.
In this section, we define free variable functions. The functions
fv_tt and fv_te calculate the set of atoms used as free type
variables in a type or expression, respectively. The function
fv_ee calculates the set of atoms used as free expression
variables in an expression. Cases involving binders are
straightforward since bound variables are indices, not names, in
locally nameless representation.
Fixpoint fv_tt (T : typ) {struct T} : atoms :=
match T with
| typ_top => {}
| typ_bvar J => {}
| typ_fvar X => singleton X
| typ_arrow T1 T2 => (fv_tt T1) `union` (fv_tt T2)
| typ_all T1 T2 => (fv_tt T1) `union` (fv_tt T2)
| typ_sum T1 T2 => (fv_tt T1) `union` (fv_tt T2)
end.
Fixpoint fv_te (e : exp) {struct e} : atoms :=
match e with
| exp_bvar i => {}
| exp_fvar x => {}
| exp_abs V e1 => (fv_tt V) `union` (fv_te e1)
| exp_app e1 e2 => (fv_te e1) `union` (fv_te e2)
| exp_tabs V e1 => (fv_tt V) `union` (fv_te e1)
| exp_tapp e1 V => (fv_tt V) `union` (fv_te e1)
| exp_let e1 e2 => (fv_te e1) `union` (fv_te e2)
| exp_inl e1 => (fv_te e1)
| exp_inr e1 => (fv_te e1)
| exp_case e1 e2 e3 => (fv_te e1) `union` (fv_te e2) `union` (fv_te e3)
end.
Fixpoint fv_ee (e : exp) {struct e} : atoms :=
match e with
| exp_bvar i => {}
| exp_fvar x => singleton x
| exp_abs V e1 => (fv_ee e1)
| exp_app e1 e2 => (fv_ee e1) `union` (fv_ee e2)
| exp_tabs V e1 => (fv_ee e1)
| exp_tapp e1 V => (fv_ee e1)
| exp_let e1 e2 => (fv_ee e1) `union` (fv_ee e2)
| exp_inl e1 => (fv_ee e1)
| exp_inr e1 => (fv_ee e1)
| exp_case e1 e2 e3 => (fv_ee e1) `union` (fv_ee e2) `union` (fv_ee e3)
end.
In this section, we define substitution for expression and type
variables appearing in types, expressions, and environments.
Substitution differs from opening because opening replaces indices
whereas substitution replaces free variables. The definitions
below are relatively simple for two reasons.
- We are using locally nameless representation, where bound variables are represented using indices. Thus, there is no need to rename variables to avoid capture.
- The definitions below assume that the term being substituted in, i.e., the second argument to each function, is locally closed. Thus, there is no need to shift indices when passing under a binder.
Fixpoint subst_tt (Z : atom) (U : typ) (T : typ) {struct T} : typ :=
match T with
| typ_top => typ_top
| typ_bvar J => typ_bvar J
| typ_fvar X => if X == Z then U else T
| typ_arrow T1 T2 => typ_arrow (subst_tt Z U T1) (subst_tt Z U T2)
| typ_all T1 T2 => typ_all (subst_tt Z U T1) (subst_tt Z U T2)
| typ_sum T1 T2 => typ_sum (subst_tt Z U T1) (subst_tt Z U T2)
end.
Fixpoint subst_te (Z : atom) (U : typ) (e : exp) {struct e} : exp :=
match e with
| exp_bvar i => exp_bvar i
| exp_fvar x => exp_fvar x
| exp_abs V e1 => exp_abs (subst_tt Z U V) (subst_te Z U e1)
| exp_app e1 e2 => exp_app (subst_te Z U e1) (subst_te Z U e2)
| exp_tabs V e1 => exp_tabs (subst_tt Z U V) (subst_te Z U e1)
| exp_tapp e1 V => exp_tapp (subst_te Z U e1) (subst_tt Z U V)
| exp_let e1 e2 => exp_let (subst_te Z U e1) (subst_te Z U e2)
| exp_inl e1 => exp_inl (subst_te Z U e1)
| exp_inr e1 => exp_inr (subst_te Z U e1)
| exp_case e1 e2 e3 => exp_case (subst_te Z U e1)
(subst_te Z U e2) (subst_te Z U e3)
end.
Fixpoint subst_ee (z : atom) (u : exp) (e : exp) {struct e} : exp :=
match e with
| exp_bvar i => exp_bvar i
| exp_fvar x => if x == z then u else e
| exp_abs V e1 => exp_abs V (subst_ee z u e1)
| exp_app e1 e2 => exp_app (subst_ee z u e1) (subst_ee z u e2)
| exp_tabs V e1 => exp_tabs V (subst_ee z u e1)
| exp_tapp e1 V => exp_tapp (subst_ee z u e1) V
| exp_let e1 e2 => exp_let (subst_ee z u e1) (subst_ee z u e2)
| exp_inl e1 => exp_inl (subst_ee z u e1)
| exp_inr e1 => exp_inr (subst_ee z u e1)
| exp_case e1 e2 e3 => exp_case (subst_ee z u e1)
(subst_ee z u e2) (subst_ee z u e3)
end.
Definition subst_tb (Z : atom) (P : typ) (b : binding) : binding :=
match b with
| bind_sub T => bind_sub (subst_tt Z P T)
| bind_typ T => bind_typ (subst_tt Z P T)
end.
The "
The first step is to define an auxiliary tactic
pick fresh" tactic introduces a fresh atom into the context.
We define it in two steps.
The first step is to define an auxiliary tactic
gather_atoms,
meant to be used in the definition of other tactics, which returns
a set of atoms in the current context. The definition of
gather_atoms follows a pattern based on repeated calls to
gather_atoms_with. The one argument to this tactic is a
function that takes an object of some particular type and returns
a set of atoms that appear in that argument. It is not necessary
to understand exactly how gather_atoms_with works. If we add a
new inductive datatype, say for kinds, to our language, then we
would need to modify gather_atoms. On the other hand, if we
merely add a new type, say products, then there is no need to
modify gather_atoms; the required changes would be made in
fv_tt.
Ltac gather_atoms :=
let A := gather_atoms_with (fun x : atoms => x) in
let B := gather_atoms_with (fun x : atom => singleton x) in
let C := gather_atoms_with (fun x : exp => fv_te x) in
let D := gather_atoms_with (fun x : exp => fv_ee x) in
let E := gather_atoms_with (fun x : typ => fv_tt x) in
let F := gather_atoms_with (fun x : env => dom x) in
constr:(A `union` B `union` C `union` D `union` E `union` F).
The second step in defining "
pick fresh" is to define the tactic
itself. It is based on the (pick fresh ... for ...) tactic
defined in the Atom library. Here, we use gather_atoms to
construct the set L rather than leaving it to the user to
provide. Thus, invoking (pick fresh x) introduces a new atom
x into the current context that is fresh for "everything" in the
context.
Tactic Notation "pick" "fresh" ident(x) :=
let L := gather_atoms in (pick fresh x for L).
This tactic is implementation specific only because of its
reliance on
gather_atoms, which is itself implementation
specific. The definition below may be copied between developments
without any changes, assuming that the other other developments
define an appropriate gather_atoms tactic. For documentation on
the tactic on which the one below is based, see the
Metatheory library.
Tactic Notation
"pick" "fresh" ident(atom_name) "and" "apply" constr(lemma) :=
let L := gather_atoms in
pick fresh atom_name excluding L and apply lemma.
The following lemmas provide useful structural properties of
substitution and opening. While the exact statements are language
specific, we have found that similar properties are needed in a
wide range of languages.
Below, we indicate which lemmas depend on which other lemmas. Since
The lemmas are split into three sections, one each for the
We keep the sections as uniform in structure as possible. In particular, we state explicitly strengthened induction hypotheses even when there are more concise ways of proving the lemmas of interest.
Below, we indicate which lemmas depend on which other lemmas. Since
te functions depend on their tt counterparts, a similar
dependency can be found in the lemmas.
The lemmas are split into three sections, one each for the
tt,
te, and ee functions. The most important lemmas are the
following:
- Substitution and opening commute with each other, e.g.,
subst_tt_open_tt_var. - Opening a term is equivalent to opening the term with a fresh
name and then substituting for that name, e.g.,
subst_tt_intro.
We keep the sections as uniform in structure as possible. In particular, we state explicitly strengthened induction hypotheses even when there are more concise ways of proving the lemmas of interest.
The next lemma is the strengthened induction hypothesis for the
lemma that follows, which states that opening a locally closed
term is the identity. This lemma is not otherwise independently
useful.
Lemma open_tt_rec_type_aux : forall T j V i U,
i <> j ->
open_tt_rec j V T = open_tt_rec i U (open_tt_rec j V T) ->
T = open_tt_rec i U T.
Proof with eauto*.
induction T; intros j V i U Neq H; simpl in *; inversion H; f_equal...
Case "typ_bvar".
destruct (j === n)... destruct (i === n)...
Qed.
Opening a locally closed term is the identity. This lemma depends
on the immediately preceding lemma.
Lemma open_tt_rec_type : forall T U k,
type T ->
T = open_tt_rec k U T.
Proof with auto*.
intros T U k Htyp. revert k.
induction Htyp; intros k; simpl; f_equal...
Case "typ_all".
unfold open_tt in *.
pick fresh X.
apply (open_tt_rec_type_aux T2 0 (typ_fvar X))...
Qed.
If a name is fresh for a term, then substituting for it is the
identity.
Lemma subst_tt_fresh : forall Z U T,
Z `notin` fv_tt T ->
T = subst_tt Z U T.
Proof with auto*.
induction T; simpl; intro H; f_equal...
Case "typ_fvar".
destruct (a == Z)...
absurd_hyp H; fsetdec.
Qed.
Substitution commutes with opening under certain conditions. This
lemma depends on the fact that opening a locally closed term is
the identity.
Lemma subst_tt_open_tt_rec : forall T1 T2 X P k,
type P ->
subst_tt X P (open_tt_rec k T2 T1) =
open_tt_rec k (subst_tt X P T2) (subst_tt X P T1).
Proof with auto*.
intros T1 T2 X P k WP. revert k.
induction T1; intros k; simpl; f_equal...
Case "typ_bvar".
destruct (k === n); subst...
Case "typ_fvar".
destruct (a == X); subst... apply open_tt_rec_type...
Qed.
The next lemma is a direct corollary of the immediately preceding
lemma---the index is specialized to zero.
Lemma subst_tt_open_tt : forall T1 T2 (X:atom) P,
type P ->
subst_tt X P (open_tt T1 T2) = open_tt (subst_tt X P T1) (subst_tt X P T2).
Proof with auto*.
intros.
unfold open_tt.
apply subst_tt_open_tt_rec...
Qed.
The next lemma is a direct corollary of the immediately preceding
lemma---here, we're opening the term with a variable. In
practice, this lemma seems to be needed as a left-to-right rewrite
rule, when stated in its current form.
Lemma subst_tt_open_tt_var : forall (X Y:atom) P T,
Y <> X ->
type P ->
open_tt (subst_tt X P T) Y = subst_tt X P (open_tt T Y).
Proof with auto*.
intros X Y P T Neq Wu.
unfold open_tt.
rewrite subst_tt_open_tt_rec...
simpl.
destruct (Y == X)...
Qed.
The next lemma states that opening a term is equivalent to first
opening the term with a fresh name and then substituting for the
name. This is actually the strengthened induction hypothesis for
the version we use in practice.
Lemma subst_tt_intro_rec : forall X T2 U k,
X `notin` fv_tt T2 ->
open_tt_rec k U T2 = subst_tt X U (open_tt_rec k (typ_fvar X) T2).
Proof with auto*.
induction T2; intros U k Fr; simpl in *; f_equal...
Case "typ_bvar".
destruct (k === n)... simpl. destruct (X == X)...
Case "typ_fvar".
destruct (a == X)... absurd_hyp Fr; fsetdec.
Qed.
The next lemma is a direct corollary of the immediately preceding
lemma---the index is specialized to zero.
Lemma subst_tt_intro : forall X T2 U,
X `notin` fv_tt T2 ->
open_tt T2 U = subst_tt X U (open_tt T2 X).
Proof with auto*.
intros.
unfold open_tt.
apply subst_tt_intro_rec...
Qed.
This section follows the structure of the previous section. The
one notable difference is that we require two auxiliary lemmas to
show that substituting a type in a locally-closed expression is
the identity.
Lemma open_te_rec_expr_aux : forall e j u i P ,
open_ee_rec j u e = open_te_rec i P (open_ee_rec j u e) ->
e = open_te_rec i P e.
Proof with eauto*.
induction e; intros j u i P H; simpl in *; inversion H; f_equal...
Qed.
Lemma open_te_rec_type_aux : forall e j Q i P,
i <> j ->
open_te_rec j Q e = open_te_rec i P (open_te_rec j Q e) ->
e = open_te_rec i P e.
Proof.
induction e; intros j Q i P Neq Heq; simpl in *; inversion Heq;
f_equal; eauto using open_tt_rec_type_aux.
Qed.
Lemma open_te_rec_expr : forall e U k,
expr e ->
e = open_te_rec k U e.
Proof with auto*.
intros e U k WF. revert k.
induction WF; intros k; simpl; f_equal; auto using open_tt_rec_type;
try solve [
unfold open_ee in *;
pick fresh x;
eapply open_te_rec_expr_aux with (j := 0) (u := exp_fvar x);
auto*
| unfold open_te in *;
pick fresh X;
eapply open_te_rec_type_aux with (j := 0) (Q := typ_fvar X);
auto*
].
Qed.
Lemma subst_te_fresh : forall X U e,
X `notin` fv_te e ->
e = subst_te X U e.
Proof.
induction e; simpl; intros; f_equal; auto using subst_tt_fresh.
Qed.
Lemma subst_te_open_te_rec : forall e T X U k,
type U ->
subst_te X U (open_te_rec k T e) =
open_te_rec k (subst_tt X U T) (subst_te X U e).
Proof.
intros e T X U k WU. revert k.
induction e; intros k; simpl; f_equal; auto using subst_tt_open_tt_rec.
Qed.
Lemma subst_te_open_te : forall e T X U,
type U ->
subst_te X U (open_te e T) = open_te (subst_te X U e) (subst_tt X U T).
Proof with auto*.
intros.
unfold open_te.
apply subst_te_open_te_rec...
Qed.
Lemma subst_te_open_te_var : forall (X Y:atom) U e,
Y <> X ->
type U ->
open_te (subst_te X U e) Y = subst_te X U (open_te e Y).
Proof with auto*.
intros X Y U e Neq WU.
unfold open_te.
rewrite subst_te_open_te_rec...
simpl.
destruct (Y == X)...
Qed.
Lemma subst_te_intro_rec : forall X e U k,
X `notin` fv_te e ->
open_te_rec k U e = subst_te X U (open_te_rec k (typ_fvar X) e).
Proof.
induction e; intros U k Fr; simpl in *; f_equal;
auto using subst_tt_intro_rec.
Qed.
Lemma subst_te_intro : forall X e U,
X `notin` fv_te e ->
open_te e U = subst_te X U (open_te e X).
Proof with auto*.
intros.
unfold open_te.
apply subst_te_intro_rec...
Qed.
This section follows the structure of the previous two sections.
Lemma open_ee_rec_expr_aux : forall e j v u i,
i <> j ->
open_ee_rec j v e = open_ee_rec i u (open_ee_rec j v e) ->
e = open_ee_rec i u e.
Proof with eauto*.
induction e; intros j v u i Neq H; simpl in *; inversion H; f_equal...
Case "exp_bvar".
destruct (j===n)... destruct (i===n)...
Qed.
Lemma open_ee_rec_type_aux : forall e j V u i,
open_te_rec j V e = open_ee_rec i u (open_te_rec j V e) ->
e = open_ee_rec i u e.
Proof.
induction e; intros j V u i H; simpl; inversion H; f_equal; eauto.
Qed.
Lemma open_ee_rec_expr : forall u e k,
expr e ->
e = open_ee_rec k u e.
Proof with auto*.
intros u e k Hexpr. revert k.
induction Hexpr; intro k; simpl; f_equal; auto*;
try solve [
unfold open_ee in *;
pick fresh x;
eapply open_ee_rec_expr_aux with (j := 0) (v := exp_fvar x);
auto*
| unfold open_te in *;
pick fresh X;
eapply open_ee_rec_type_aux with (j := 0) (V := typ_fvar X);
auto*
].
Qed.
Lemma subst_ee_fresh : forall (x: atom) u e,
x `notin` fv_ee e ->
e = subst_ee x u e.
Proof with auto*.
intros x u e; induction e; simpl; intro H; f_equal...
Case "exp_fvar".
destruct (a==x)...
absurd_hyp H; fsetdec.
Qed.
Lemma subst_ee_open_ee_rec : forall e1 e2 x u k,
expr u ->
subst_ee x u (open_ee_rec k e2 e1) =
open_ee_rec k (subst_ee x u e2) (subst_ee x u e1).
Proof with auto*.
intros e1 e2 x u k WP. revert k.
induction e1; intros k; simpl; f_equal...
Case "exp_bvar".
destruct (k === n); subst...
Case "exp_fvar".
destruct (a == x); subst... apply open_ee_rec_expr...
Qed.
Lemma subst_ee_open_ee : forall e1 e2 x u,
expr u ->
subst_ee x u (open_ee e1 e2) =
open_ee (subst_ee x u e1) (subst_ee x u e2).
Proof with auto*.
intros.
unfold open_ee.
apply subst_ee_open_ee_rec...
Qed.
Lemma subst_ee_open_ee_var : forall (x y:atom) u e,
y <> x ->
expr u ->
open_ee (subst_ee x u e) y = subst_ee x u (open_ee e y).
Proof with auto*.
intros x y u e Neq Wu.
unfold open_ee.
rewrite subst_ee_open_ee_rec...
simpl.
destruct (y == x)...
Qed.
Lemma subst_te_open_ee_rec : forall e1 e2 Z P k,
subst_te Z P (open_ee_rec k e2 e1) =
open_ee_rec k (subst_te Z P e2) (subst_te Z P e1).
Proof with auto*.
induction e1; intros e2 Z P k; simpl; f_equal...
Case "exp_bvar".
destruct (k === n)...
Qed.
Lemma subst_te_open_ee : forall e1 e2 Z P,
subst_te Z P (open_ee e1 e2) = open_ee (subst_te Z P e1) (subst_te Z P e2).
Proof with auto*.
intros.
unfold open_ee.
apply subst_te_open_ee_rec...
Qed.
Lemma subst_te_open_ee_var : forall Z (x:atom) P e,
open_ee (subst_te Z P e) x = subst_te Z P (open_ee e x).
Proof with auto*.
intros.
rewrite subst_te_open_ee...
Qed.
Lemma subst_ee_open_te_rec : forall e P z u k,
expr u ->
subst_ee z u (open_te_rec k P e) = open_te_rec k P (subst_ee z u e).
Proof with auto*.
induction e; intros P z u k H; simpl; f_equal...
Case "exp_fvar".
destruct (a == z)... apply open_te_rec_expr...
Qed.
Lemma subst_ee_open_te : forall e P z u,
expr u ->
subst_ee z u (open_te e P) = open_te (subst_ee z u e) P.
Proof with auto*.
intros.
unfold open_te.
apply subst_ee_open_te_rec...
Qed.
Lemma subst_ee_open_te_var : forall z (X:atom) u e,
expr u ->
open_te (subst_ee z u e) X = subst_ee z u (open_te e X).
Proof with auto*.
intros z X u e H.
rewrite subst_ee_open_te...
Qed.
Lemma subst_ee_intro_rec : forall x e u k,
x `notin` fv_ee e ->
open_ee_rec k u e = subst_ee x u (open_ee_rec k (exp_fvar x) e).
Proof with auto*.
induction e; intros u k Fr; simpl in *; f_equal...
Case "exp_bvar".
destruct (k === n)... simpl. destruct (x == x)...
Case "exp_fvar".
destruct (a == x)... absurd_hyp Fr; fsetdec.
Qed.
Lemma subst_ee_intro : forall x e u,
x `notin` fv_ee e ->
open_ee e u = subst_ee x u (open_ee e x).
Proof with auto*.
intros.
unfold open_ee.
apply subst_ee_intro_rec...
Qed.
While these lemmas may be considered properties of substitution, we
separate them out due to the lemmas that they depend on.
The following lemma depends on
subst_tt_open_tt_var.
Lemma subst_tt_type : forall Z P T,
type T ->
type P ->
type (subst_tt Z P T).
Proof with auto.
intros Z P T HT HP.
induction HT; simpl...
Case "type_fvar".
destruct (X == Z)...
Case "type_all".
pick fresh Y and apply type_all...
rewrite subst_tt_open_tt_var...
Qed.
The following lemma depends on
subst_tt_type and
subst_te_open_ee_var.
Lemma subst_te_expr : forall Z P e,
expr e ->
type P ->
expr (subst_te Z P e).
Proof with eauto using subst_tt_type.
intros Z P e He Hp.
induction He; simpl; auto using subst_tt_type;
try solve [
econstructor;
try instantiate (1 := L `union` singleton Z);
intros;
try rewrite subst_te_open_ee_var;
try rewrite subst_te_open_te_var;
eauto using subst_tt_type
].
Qed.
The following lemma depends on
subst_ee_open_ee_var and
subst_ee_open_te_var.
Lemma subst_ee_expr : forall z e1 e2,
expr e1 ->
expr e2 ->
expr (subst_ee z e2 e1).
Proof with auto.
intros z e1 e2 He1 He2.
induction He1; simpl; auto;
try solve [
econstructor;
try instantiate (1 := L `union` singleton z);
intros;
try rewrite subst_ee_open_ee_var;
try rewrite subst_ee_open_te_var;
auto
].
Case "expr_var".
destruct (x == z)...
Qed.
The two kinds of facts we need about
Since we use it only in the context of
body_e are the following:
- How to use it to derive that terms are locally closed.
- How to derive it from the facts that terms are locally closed.
Since we use it only in the context of
exp_let and exp_sum
(see the definition of reduction), those two constructors are the
only ones we consider below.
Lemma expr_let_from_body : forall e1 e2,
expr e1 ->
body_e e2 ->
expr (exp_let e1 e2).
Proof.
intros e1 e2 H [J1 J2].
pick fresh y and apply expr_let; auto.
Qed.
Lemma body_from_expr_let : forall e1 e2,
expr (exp_let e1 e2) ->
body_e e2.
Proof.
intros e1 e2 H.
unfold body_e.
inversion H; eauto.
Qed.
Lemma expr_case_from_body : forall e1 e2 e3,
expr e1 ->
body_e e2 ->
body_e e3 ->
expr (exp_case e1 e2 e3).
Proof.
intros e1 e2 e3 H [J1 J2] [K1 K2].
pick fresh y and apply expr_case; auto.
Qed.
Lemma body_inl_from_expr_case : forall e1 e2 e3,
expr (exp_case e1 e2 e3) ->
body_e e2.
Proof.
intros e1 e2 e3 H.
unfold body_e.
inversion H; eauto.
Qed.
Lemma body_inr_from_expr_case : forall e1 e2 e3,
expr (exp_case e1 e2 e3) ->
body_e e3.
Proof.
intros e1 e2 e3 H.
unfold body_e.
inversion H; eauto.
Qed.
Lemma open_ee_body_e : forall e1 e2,
body_e e1 -> expr e2 -> expr (open_ee e1 e2).
Proof.
intros e1 e2 [L H] J.
pick fresh x.
rewrite (subst_ee_intro x); auto using subst_ee_expr.
Qed.
We add as hints the fact that local closure is preserved under
substitution. This is part of our strategy for automatically
discharging local-closure proof obligations.
Hint Resolve subst_tt_type subst_te_expr subst_ee_expr.
We also add as hints the lemmas concerning
body_e.
Hint Resolve expr_let_from_body body_from_expr_let.
Hint Resolve expr_case_from_body.
Hint Resolve body_inl_from_expr_case body_inr_from_expr_case.
Hint Resolve open_ee_body_e.
When reasoning about the
binds relation and map, we
occasionally encounter situations where the binding is
over-simplified. The following hint undoes that simplification,
thus enabling Hints from the Environment library.
Hint Extern 1 (binds _ (?F (subst_tt ?X ?U ?T)) _) =>
unsimpl (subst_tb X U (F T)).