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PCUICSubstitution.v
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PCUICSubstitution.v
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(* Distributed under the terms of the MIT license. *)
From MetaCoq.Template Require Import utils config.
From MetaCoq.PCUIC Require Import PCUICAst PCUICAstUtils PCUICInduction
PCUICLiftSubst PCUICEquality PCUICPosition PCUICSigmaCalculus
PCUICUnivSubst PCUICTyping PCUICWeakeningEnv PCUICClosed
PCUICReduction PCUICWeakening PCUICCumulativity PCUICUnivSubstitution.
Require Import ssreflect.
From Equations Require Import Equations.
(** * Substitution lemmas for typing derivations. *)
Local Set Keyed Unification.
Set Default Goal Selector "!".
Hint Rewrite @app_context_length : wf.
Generalizable Variables Σ Γ t T.
(** Well-typed substitution into a context with *no* let-ins *)
Inductive subs {cf:checker_flags} (Σ : global_env_ext) (Γ : context) : list term -> context -> Type :=
| emptys : subs Σ Γ [] []
| cons_ass Δ s na t T : subs Σ Γ s Δ -> Σ ;;; Γ |- t : subst0 s T -> subs Σ Γ (t :: s) (Δ ,, vass na T).
(** Linking a context (with let-ins), an instance (reversed substitution)
for its assumptions and a well-formed substitution for it. *)
Inductive context_subst : context -> list term -> list term -> Type :=
| context_subst_nil : context_subst [] [] []
| context_subst_ass Γ args s na t a :
context_subst Γ args s ->
context_subst (vass na t :: Γ) (args ++ [a]) (a :: s)
| context_subst_def Γ args s na b t :
context_subst Γ args s ->
context_subst (vdef na b t :: Γ) args (subst s 0 b :: s).
(** Promoting a substitution for the non-let declarations of ctx into a
substitution for the whole context *)
Fixpoint make_context_subst ctx args s :=
match ctx with
| [] => match args with
| [] => Some s
| a :: args => None
end
| d :: ctx =>
match d.(decl_body) with
| Some body => make_context_subst ctx args (subst0 s body :: s)
| None => match args with
| a :: args => make_context_subst ctx args (a :: s)
| [] => None
end
end
end.
(** Well-typed substitution into a context with let-ins *)
Inductive subslet {cf:checker_flags} Σ (Γ : context) : list term -> context -> Type :=
| emptyslet : subslet Σ Γ [] []
| cons_let_ass Δ s na t T : subslet Σ Γ s Δ ->
Σ ;;; Γ |- t : subst0 s T ->
subslet Σ Γ (t :: s) (Δ ,, vass na T)
| cons_let_def Δ s na t T :
subslet Σ Γ s Δ ->
Σ ;;; Γ |- subst0 s t : subst0 s T ->
subslet Σ Γ (subst0 s t :: s) (Δ ,, vdef na t T).
Lemma subslet_nth_error {cf:checker_flags} Σ Γ s Δ decl n t :
subslet Σ Γ s Δ ->
nth_error Δ n = Some decl ->
nth_error s n = Some t ->
match decl_body decl return Type with
| Some t' =>
let b := subst0 (skipn (S n) s) t' in
let ty := subst0 (skipn (S n) s) (decl_type decl) in
((t = b) * (Σ ;;; Γ |- b : ty))%type
| None =>
let ty := subst0 (skipn (S n) s) (decl_type decl) in
Σ ;;; Γ |- t : ty
end.
Proof.
induction 1 in n |- *; simpl; auto; destruct n; simpl; try congruence.
- intros [= <-]. intros [= ->].
simpl. exact t1.
- intros. destruct decl as [na' [b|] ty]; cbn in *.
+ specialize (IHX _ H H0). intuition auto.
+ now apply IHX.
- intros [= <-]. intros [= <-].
simpl. split; auto.
- apply IHX.
Qed.
Lemma subslet_length {cf:checker_flags} {Σ Γ s Δ} : subslet Σ Γ s Δ -> #|s| = #|Δ|.
Proof.
induction 1; simpl; auto with arith.
Qed.
Lemma subst_decl_closed n k d : closed_decl k d -> subst_decl n k d = d.
Proof.
case: d => na [body|] ty; rewrite /closed_decl /subst_decl /map_decl /=.
- move/andb_and => [cb cty]. rewrite !subst_closedn //.
- move=> cty; now rewrite !subst_closedn //.
Qed.
Lemma closed_ctx_subst n k ctx : closed_ctx ctx = true -> subst_context n k ctx = ctx.
Proof.
induction ctx in n, k |- *; auto.
unfold closed_ctx, id.
rewrite mapi_app forallb_app List.rev_length /= Nat.add_0_r.
move/andb_and => /= [Hctx /andb_and [Ha _]].
rewrite subst_context_snoc /snoc /= IHctx // subst_decl_closed //.
now apply: closed_decl_upwards.
Qed.
Lemma closed_tele_subst n k ctx :
closed_ctx ctx ->
mapi (fun (k' : nat) (decl : context_decl) => subst_decl n (k' + k) decl) (List.rev ctx) = List.rev ctx.
Proof.
rewrite /closedn_ctx /mapi. simpl. generalize 0.
induction ctx using rev_ind; try easy.
move=> n0.
rewrite /closedn_ctx !rev_app_distr /id /=.
move/andb_and => [closedx Hctx].
rewrite subst_decl_closed //.
- rewrite (closed_decl_upwards n0) //; lia.
- f_equal. now rewrite IHctx.
Qed.
Lemma map_vass_map_def_subst g l n k :
(mapi (fun i (d : def term) => vass (dname d) (lift0 i (dtype d)))
(map (map_def (subst n k) g) l)) =
(mapi (fun i d => map_decl (subst n (i + k)) d) (mapi (fun i (d : def term) => vass (dname d) (lift0 i (dtype d))) l)).
Proof.
rewrite mapi_mapi mapi_map. apply mapi_ext.
intros. unfold map_decl, vass; simpl; f_equal.
rewrite commut_lift_subst_rec. 1: lia. f_equal; lia.
Qed.
Lemma All_local_env_subst {cf:checker_flags} (P Q : context -> term -> option term -> Type) c n k :
All_local_env Q c ->
(forall Γ t T,
Q Γ t T ->
P (subst_context n k Γ) (subst n (#|Γ| + k) t)
(option_map (subst n (#|Γ| + k)) T)
) ->
All_local_env P (subst_context n k c).
Proof.
intros Hq Hf.
induction Hq in |- *; try econstructor; eauto;
simpl; unfold snoc; rewrite subst_context_snoc; econstructor; eauto.
- simpl. eapply (Hf _ _ None). eauto.
- simpl. eapply (Hf _ _ None). eauto.
- simpl. eapply (Hf _ _ (Some t)). eauto.
Qed.
Lemma subst_length {cf:checker_flags} Σ Γ s Γ' : subs Σ Γ s Γ' -> #|s| = #|Γ'|.
Proof.
induction 1; simpl; auto with arith.
Qed.
Lemma subs_nth_error_ge {cf:checker_flags} Σ Γ Γ' Γ'' v s :
subs Σ Γ s Γ' ->
#|Γ' ,,, Γ''| <= v ->
nth_error (Γ ,,, Γ' ,,, Γ'') v =
nth_error (Γ ,,, subst_context s 0 Γ'') (v - #|Γ'|).
Proof.
simpl.
intros. rewrite app_context_length in H.
rewrite !nth_error_app_ge; autorewrite with wf; f_equal; try lia.
Qed.
Lemma nth_error_subst_context (Γ' : context) s (v : nat) k :
nth_error (subst_context s k Γ') v =
option_map (subst_decl s (#|Γ'| - S v + k)) (nth_error Γ' v).
Proof.
induction Γ' in v |- *; intros.
- simpl. unfold subst_context, fold_context; simpl; rewrite nth_error_nil. easy.
- simpl. destruct v; rewrite subst_context_snoc.
+ simpl. repeat f_equal; try lia.
+ simpl. rewrite IHΓ'; simpl in *; (lia || congruence).
Qed.
Lemma subs_nth_error_lt {cf:checker_flags} Σ Γ Γ' Γ'' v s :
subs Σ Γ s Γ' ->
v < #|Γ''| ->
nth_error (Γ ,,, subst_context s 0 Γ'') v =
option_map (map_decl (subst s (#|Γ''| - S v))) (nth_error (Γ ,,, Γ' ,,, Γ'') v).
Proof.
simpl. intros Hs Hv.
rewrite !nth_error_app_lt; autorewrite with wf; f_equal; try lia.
erewrite nth_error_subst_context. f_equal. unfold subst_decl. rewrite Nat.add_0_r. reflexivity.
Qed.
Lemma subslet_nth_error_lt {cf:checker_flags} Σ Γ Γ' Γ'' v s :
subslet Σ Γ s Γ' ->
v < #|Γ''| ->
nth_error (Γ ,,, subst_context s 0 Γ'') v =
option_map (map_decl (subst s (#|Γ''| - S v))) (nth_error (Γ ,,, Γ' ,,, Γ'') v).
Proof.
simpl. intros Hs Hv.
rewrite !nth_error_app_lt; autorewrite with wf; f_equal; try lia.
erewrite nth_error_subst_context. f_equal. unfold subst_decl. rewrite Nat.add_0_r. reflexivity.
Qed.
Lemma subst_iota_red s k pars c args brs :
subst s k (iota_red pars c args brs) =
iota_red pars c (List.map (subst s k) args) (List.map (on_snd (subst s k)) brs).
Proof.
unfold iota_red. rewrite !subst_mkApps. f_equal; auto using map_skipn.
rewrite nth_map; simpl; auto.
Qed.
Lemma subst_unfold_fix n k mfix idx narg fn :
unfold_fix mfix idx = Some (narg, fn) ->
unfold_fix (map (map_def (subst n k) (subst n (#|mfix| + k))) mfix) idx = Some (narg, subst n k fn).
Proof.
unfold unfold_fix.
rewrite nth_error_map. destruct (nth_error mfix idx) eqn:Hdef; try congruence.
move=> [= <- <-] /=. f_equal. f_equal.
solve_all.
erewrite (distr_subst_rec _ _ _ k 0).
rewrite fix_subst_length. simpl. f_equal.
unfold fix_subst. rewrite !map_length.
generalize #|mfix| at 2 3. induction n0; auto. simpl.
f_equal. apply IHn0.
Qed.
#[global]
Hint Resolve subst_unfold_fix : core.
Lemma subst_unfold_cofix n k mfix idx narg fn :
unfold_cofix mfix idx = Some (narg, fn) ->
unfold_cofix (map (map_def (subst n k) (subst n (#|mfix| + k))) mfix) idx = Some (narg, subst n k fn).
Proof.
unfold unfold_cofix.
rewrite nth_error_map. destruct (nth_error mfix idx) eqn:Hdef; try congruence.
intros [= <- <-]. simpl. do 2 f_equal. solve_all.
erewrite (distr_subst_rec _ _ _ k 0).
rewrite cofix_subst_length. simpl. f_equal.
unfold cofix_subst. rewrite !map_length.
generalize #|mfix| at 2 3. induction n0; auto. simpl.
f_equal. apply IHn0.
Qed.
#[global]
Hint Resolve subst_unfold_cofix : core.
Lemma decompose_app_rec_subst n k t l :
let (f, a) := decompose_app_rec t l in
subst n k f = f ->
decompose_app_rec (subst n k t) (map (subst n k) l) = (f, map (subst n k) a).
Proof.
induction t in k, l |- *; simpl; auto; try congruence.
- destruct Nat.leb; try reflexivity.
destruct nth_error.
+ simpl. intros ->. simpl. reflexivity.
+ intros ->. simpl. reflexivity.
- specialize (IHt1 k (t2 :: l)).
destruct decompose_app_rec. intros H. rewrite IHt1; auto.
Qed.
Lemma decompose_app_subst n k t f a :
decompose_app t = (f, a) -> subst n k f = f ->
decompose_app (subst n k t) = (subst n k f, map (subst n k) a).
Proof.
generalize (decompose_app_rec_subst n k t []).
unfold decompose_app. destruct decompose_app_rec.
move=> Heq [= <- <-] Heq'. now rewrite Heq' (Heq Heq').
Qed.
Hint Rewrite decompose_app_subst using auto : lift.
Lemma subst_is_constructor:
forall (args : list term) (narg : nat) n k,
is_constructor narg args = true -> is_constructor narg (map (subst n k) args) = true.
Proof.
intros args narg.
unfold is_constructor; intros.
rewrite nth_error_map. destruct nth_error; try discriminate. simpl. intros.
unfold isConstruct_app in *.
destruct (decompose_app t) eqn:Heq. eapply decompose_app_subst in Heq as ->.
- destruct t0; try discriminate || reflexivity.
- destruct t0; try discriminate || reflexivity.
Qed.
#[global]
Hint Resolve subst_is_constructor : core.
#[global]
Hint Constructors All_local_env : core.
Lemma typed_subst `{checker_flags} Σ Γ t T n k :
wf Σ.1 -> k >= #|Γ| ->
Σ ;;; Γ |- t : T -> subst n k T = T /\ subst n k t = t.
Proof.
intros wfΣ Hk Hty.
pose proof (typing_wf_local Hty).
apply typecheck_closed in Hty; eauto.
destruct Hty as [_ Hcl].
rewrite -> andb_and in Hcl. destruct Hcl as [clb clty].
pose proof (closed_upwards k clb).
simpl in *. forward H0 by lia.
pose proof (closed_upwards k clty).
simpl in *. forward H1 by lia.
apply (subst_closedn n) in H0; apply (subst_closedn n) in H1; auto.
Qed.
Lemma subst_wf_local `{checker_flags} Σ Γ n k :
wf Σ.1 ->
wf_local Σ Γ ->
subst_context n k Γ = Γ.
Proof.
intros wfΣ.
induction 1; auto; unfold subst_context, snoc; rewrite fold_context_snoc0;
auto; unfold snoc;
f_equal; auto; unfold map_decl; simpl.
- destruct t0 as [s Hs]. unfold vass. simpl. f_equal.
eapply typed_subst; eauto. lia.
- unfold vdef.
f_equal.
+ f_equal. eapply typed_subst; eauto. lia.
+ eapply typed_subst in t1 as [Ht HT]; eauto. lia.
Qed.
Lemma subst_declared_constant `{H:checker_flags} Σ cst decl n k u :
wf Σ ->
declared_constant Σ cst decl ->
map_constant_body (subst n k) (map_constant_body (subst_instance_constr u) decl) =
map_constant_body (subst_instance_constr u) decl.
Proof.
intros.
eapply declared_decl_closed in H0; eauto.
unfold map_constant_body.
do 2 red in H0. destruct decl as [ty [body|] univs]; simpl in *.
- rewrite -> andb_and in H0. intuition.
rewrite <- (closedn_subst_instance_constr 0 body u) in H1.
rewrite <- (closedn_subst_instance_constr 0 ty u) in H2.
f_equal.
+ apply subst_closedn; eauto using closed_upwards with arith wf.
+ f_equal. apply subst_closedn; eauto using closed_upwards with arith wf.
- red in H0. f_equal.
intuition. simpl in *.
rewrite <- (closedn_subst_instance_constr 0 ty u) in H0.
rewrite andb_true_r in H0.
eapply subst_closedn; eauto using closed_upwards with arith wf.
Qed.
Definition subst_mutual_inductive_body n k m :=
map_mutual_inductive_body (fun k' => subst n (k' + k)) m.
Lemma subst_declared_minductive {cf:checker_flags} Σ cst decl n k :
wf Σ ->
declared_minductive Σ cst decl ->
subst_mutual_inductive_body n k decl = decl.
Proof.
intros wfΣ Hdecl.
pose proof (on_declared_minductive wfΣ Hdecl). apply onNpars in X.
apply (declared_inductive_closed (Σ:=(empty_ext Σ))) in Hdecl; auto.
move: Hdecl.
rewrite /closed_inductive_decl /lift_mutual_inductive_body.
destruct decl; simpl.
move/andb_and => [clpar clbodies]. f_equal.
- now rewrite [fold_context _ _]closed_ctx_subst.
- eapply forallb_All in clbodies.
eapply Alli_mapi_id.
* eapply (All_Alli clbodies). intros; eauto.
intros. eapply H.
* simpl; intros.
move: H. rewrite /closed_inductive_body.
destruct x; simpl. move=> /andb_and[/andb_and [ci ct] cp].
f_equal.
+ rewrite subst_closedn; eauto.
eapply closed_upwards; eauto; lia.
+ eapply All_map_id. eapply forallb_All in ct.
eapply (All_impl ct). intros x.
destruct x as [[id ty] arg]; unfold on_pi2; intros c; simpl; repeat f_equal.
apply subst_closedn. unfold cdecl_type in c; simpl in c.
eapply closed_upwards; eauto; lia.
+ simpl in X. rewrite -X in cp.
eapply forallb_All in cp. eapply All_map_id; eauto.
eapply (All_impl cp); pcuicfo.
destruct x; unfold on_snd; simpl; f_equal.
apply subst_closedn. rewrite context_assumptions_fold.
eapply closed_upwards; eauto; lia.
Qed.
Lemma subst_declared_inductive {cf:checker_flags} Σ ind mdecl idecl n k :
wf Σ ->
declared_inductive Σ mdecl ind idecl ->
map_one_inductive_body (context_assumptions mdecl.(ind_params))
(length (arities_context mdecl.(ind_bodies)))
(fun k' => subst n (k' + k)) (inductive_ind ind) idecl = idecl.
Proof.
unfold declared_inductive. intros wfΣ [Hmdecl Hidecl].
eapply (subst_declared_minductive _ _ _ n k) in Hmdecl. 2: auto.
unfold subst_mutual_inductive_body in Hmdecl.
destruct mdecl. simpl in *.
injection Hmdecl. intros Heq.
clear Hmdecl.
pose proof Hidecl as Hidecl'.
rewrite <- Heq in Hidecl'.
rewrite nth_error_mapi in Hidecl'.
clear Heq.
unfold option_map in Hidecl'. rewrite Hidecl in Hidecl'.
congruence.
Qed.
Lemma subst0_inds_subst ind u mdecl n k t :
(subst0 (inds (inductive_mind ind) u (ind_bodies mdecl))
(subst n (#|arities_context (ind_bodies mdecl)| + k) t)) =
subst n k (subst0 (inds (inductive_mind ind) u (ind_bodies mdecl)) t).
Proof.
pose proof (distr_subst_rec t (inds (inductive_mind ind) u (ind_bodies mdecl)) n k 0).
simpl in H. rewrite H.
unfold arities_context. rewrite rev_map_length inds_length.
f_equal. generalize (ind_bodies mdecl).
clear. intros.
induction l; unfold inds; simpl; auto. f_equal. auto.
Qed.
Lemma subst_declared_constructor {cf:checker_flags} Σ c u mdecl idecl cdecl n k :
wf Σ -> declared_constructor Σ mdecl idecl c cdecl ->
subst (map (subst_instance_constr u) n) k (type_of_constructor mdecl cdecl c u) = (type_of_constructor mdecl cdecl c u).
Proof.
unfold declared_constructor. destruct c as [i ci]. intros wfΣ [Hidecl Hcdecl].
eapply (subst_declared_inductive _ _ _ _ n k) in Hidecl; eauto.
unfold type_of_constructor. destruct cdecl as [[id t'] arity].
destruct idecl; simpl in *.
injection Hidecl. intros.
pose Hcdecl as Hcdecl'.
rewrite <- H0 in Hcdecl'.
rewrite nth_error_map in Hcdecl'. rewrite Hcdecl in Hcdecl'.
simpl in Hcdecl'. injection Hcdecl'.
intros.
rewrite <- H2 at 2.
rewrite <- subst0_inds_subst. f_equal.
now rewrite <- subst_subst_instance_constr.
Qed.
Lemma subst_declared_projection {cf:checker_flags} Σ c mdecl idecl pdecl n k :
wf Σ ->
declared_projection Σ mdecl idecl c pdecl ->
on_snd (subst n (S (ind_npars mdecl + k))) pdecl = pdecl.
Proof.
intros.
eapply (declared_projection_closed (Σ:=empty_ext Σ)) in H; auto.
unfold on_snd. simpl.
rewrite subst_closedn.
- eapply closed_upwards; eauto; try lia.
- destruct pdecl; reflexivity.
Qed.
Lemma subst_fix_context:
forall (mfix : list (def term)) n (k : nat),
fix_context (map (map_def (subst n k) (subst n (#|mfix| + k))) mfix) =
subst_context n k (fix_context mfix).
Proof.
intros mfix n k. unfold fix_context.
rewrite map_vass_map_def_subst rev_mapi.
fold (fix_context mfix).
rewrite (subst_context_alt n k (fix_context mfix)).
now rewrite /subst_decl mapi_length fix_context_length.
Qed.
Lemma subst_destArity ctx t n k :
match destArity ctx t with
| Some (args, s) =>
destArity (subst_context n k ctx) (subst n (#|ctx| + k) t) = Some (subst_context n k args, s)
| None => True
end.
Proof.
revert ctx.
induction t in n, k |- * using term_forall_list_ind; intros ctx; simpl; trivial.
- move: (IHt2 n k (ctx,, vass n0 t1)).
now rewrite subst_context_snoc /= /subst_decl /map_decl /vass /=.
- move: (IHt3 n k (ctx,, vdef n0 t1 t2)).
now rewrite subst_context_snoc /= /subst_decl /map_decl /vass /=.
Qed.
Lemma decompose_prod_n_assum0 ctx t : decompose_prod_n_assum ctx 0 t = Some (ctx, t).
Proof. destruct t; simpl; reflexivity. Qed.
Lemma subst_instantiate_params_subst n k params args s t :
forall s' t',
instantiate_params_subst params args s t = Some (s', t') ->
instantiate_params_subst
(mapi_rec (fun k' decl => subst_decl n (k' + k) decl) params #|s|)
(map (subst n k) args) (map (subst n k) s) (subst n (#|s| + k) t) =
Some (map (subst n k) s', subst n (#|s| + k + #|params|) t').
Proof.
induction params in args, t, n, k, s |- *; intros ctx' t'.
- destruct args; simpl; rewrite ?Nat.add_0_r; try congruence.
- simpl.
destruct a as [na [body|] ty]; simpl; try congruence;
destruct t; try congruence.
-- intros Ht'.
specialize (IHparams n k _ _ _ _ _ Ht').
simpl in IHparams.
replace (#|s| + k + S #|params|) with (S (#|s| + k + #|params|)) by lia.
rewrite <- IHparams. f_equal.
now rewrite distr_subst.
-- intros Ht'. destruct args; try congruence. simpl.
specialize (IHparams n k _ _ _ _ _ Ht').
simpl in IHparams.
replace (#|s| + k + S #|params|) with (S (#|s| + k + #|params|)) by lia.
now rewrite <- IHparams.
Qed.
Lemma decompose_prod_n_assum_extend_ctx {ctx n t ctx' t'} ctx'' :
decompose_prod_n_assum ctx n t = Some (ctx', t') ->
decompose_prod_n_assum (ctx ++ ctx'') n t = Some (ctx' ++ ctx'', t').
Proof.
induction n in ctx, t, ctx', t', ctx'' |- *.
- simpl. intros [= -> ->]. eauto.
- simpl.
destruct t; simpl; try congruence.
+ intros H. eapply (IHn _ _ _ _ ctx'' H).
+ intros H. eapply (IHn _ _ _ _ ctx'' H).
Qed.
Lemma subst_it_mkProd_or_LetIn n k ctx t :
subst n k (it_mkProd_or_LetIn ctx t) =
it_mkProd_or_LetIn (subst_context n k ctx) (subst n (length ctx + k) t).
Proof.
induction ctx in n, k, t |- *; simpl; try congruence.
pose (subst_context_snoc n k ctx a). unfold snoc in e. rewrite e. clear e.
simpl. rewrite -> IHctx.
pose (subst_context_snoc n k ctx a). simpl. now destruct a as [na [b|] ty].
Qed.
Lemma to_extended_list_k_subst n k c k' :
to_extended_list_k (subst_context n k c) k' = to_extended_list_k c k'.
Proof.
unfold to_extended_list_k. revert k'.
unf_term. generalize (@nil term) at 1 2.
induction c in n, k |- *; simpl; intros. 1: reflexivity.
rewrite subst_context_snoc. unfold snoc. simpl.
destruct a. destruct decl_body.
- unfold subst_decl, map_decl. simpl.
now rewrite IHc.
- simpl. apply IHc.
Qed.
Lemma to_extended_list_k_map_subst:
forall n (k : nat) (c : context) k',
#|c| + k' <= k ->
to_extended_list_k c k' = map (subst n k) (to_extended_list_k c k').
Proof.
intros n k c k'.
pose proof (to_extended_list_k_spec c k').
symmetry. solve_all.
destruct H as [x' [-> Hx']]. intuition. simpl.
destruct (leb_spec_Set k x').
- lia.
- reflexivity.
Qed.
Lemma subst_instantiate_params n k params args t ty :
closed_ctx params ->
instantiate_params params args t = Some ty ->
instantiate_params params (map (subst n k) args) (subst n k t) = Some (subst n k ty).
Proof.
unfold instantiate_params.
move/(closed_tele_subst n k params)=> Heq.
rewrite -{2}Heq //.
specialize (subst_instantiate_params_subst n k (List.rev params) args [] t).
move=> /= Heq'.
case E: (instantiate_params_subst (List.rev params) args)=> /= [[l' t']|] /= //.
specialize (Heq' _ _ E). rewrite Heq'. move=> [= <-]. f_equal.
rewrite distr_subst //.
move/instantiate_params_subst_length: E => -> /=. do 2 f_equal. lia.
Qed.
Hint Rewrite subst_instantiate_params : lift.
Lemma wf_arities_context' {cf:checker_flags}:
forall (Σ : global_env_ext) mind (mdecl : mutual_inductive_body),
wf Σ ->
on_inductive (lift_typing typing) Σ mind mdecl ->
wf_local Σ (arities_context (ind_bodies mdecl)).
Proof.
intros Σ mind mdecl wfΣ Hdecl.
apply onInductives in Hdecl.
unfold arities_context.
revert Hdecl.
induction (ind_bodies mdecl) using rev_ind. 1: constructor.
intros Ha.
rewrite rev_map_app.
simpl. apply Alli_app in Ha as [Hl Hx].
inv Hx. clear X0.
apply onArity in X as [s Hs].
specialize (IHl Hl).
econstructor; eauto.
fold (arities_context l) in *.
unshelve epose proof (weakening Σ [] (arities_context l) _ _ wfΣ _ Hs).
1: now rewrite app_context_nil_l.
simpl in X.
eapply (env_prop_typing _ _ typecheck_closed) in Hs; eauto.
rewrite -> andb_and in Hs. destruct Hs as [Hs Ht].
simpl in Hs. apply (lift_closed #|arities_context l|) in Hs.
rewrite -> Hs, app_context_nil_l in X. simpl. exists s.
apply X.
Qed.
Lemma wf_arities_context {cf:checker_flags} Σ mind mdecl : wf Σ ->
declared_minductive Σ mind mdecl -> wf_local (Σ, ind_universes mdecl) (arities_context mdecl.(ind_bodies)).
Proof.
intros wfΣ Hdecl.
eapply declared_minductive_inv in Hdecl. 2:apply weaken_env_prop_typing. all:eauto.
eapply wf_arities_context'; eauto.
Qed.
Lemma on_constructor_closed {cf:checker_flags} {Σ mind mdecl u idecl indices cdecl cs} :
wf Σ ->
on_constructor (lift_typing typing) (Σ, ind_universes mdecl) mdecl (inductive_ind mind) indices idecl cdecl cs ->
let cty := subst0 (inds (inductive_mind mind) u (ind_bodies mdecl))
(subst_instance_constr u (snd (fst cdecl)))
in closed cty.
Proof.
intros wfΣ [? ? ? [s Hs] Hparams].
pose proof (typing_wf_local Hs).
destruct cdecl as [[id cty] car].
apply subject_closed in Hs; eauto.
rewrite arities_context_length in Hs.
simpl in *.
eapply (closedn_subst _ 0 0).
- clear. unfold inds.
induction #|ind_bodies mdecl|; simpl; try constructor; auto.
- simpl. now rewrite -> inds_length, closedn_subst_instance_constr.
Qed.
Lemma context_subst_length {Γ a s} : context_subst Γ a s -> #|Γ| = #|s|.
Proof. induction 1; simpl; congruence. Qed.
Lemma context_subst_assumptions_length {Γ a s} : context_subst Γ a s -> context_assumptions Γ = #|a|.
Proof. induction 1; simpl; try congruence. rewrite app_length /=. lia. Qed.
(* Lemma context_subst_app {cf:checker_flags} Γ Γ' a s : *)
(* context_subst (Γ' ++ Γ) a s -> *)
(* { a0 & { a1 & { s0 & { s1 & (context_subst Γ a0 s0 * context_subst (subst_context s0 0 Γ') a1 s1 *)
(* * (a = a0 ++ a1) * (s = s1 ++ s0))%type } } } }. *)
(* Proof. *)
(* induction Γ' in Γ, a, s |- *. simpl. *)
(* exists a, [], s, []. rewrite app_nil_r; intuition. constructor. *)
(* simpl. intros Hs. *)
(* inv Hs. *)
(* - specialize (IHΓ' _ _ _ H). *)
(* destruct IHΓ' as (a0' & a1' & s1 & s2 & ((sa0 & sa1) & eqargs) & eqs0). *)
(* subst. exists a0', (a1' ++ [a1]), s1, (a1 :: s2). intuition eauto. *)
(* rewrite subst_context_snoc. constructor. auto. now rewrite app_assoc. *)
(* - specialize (IHΓ' _ _ _ H). *)
(* destruct IHΓ' as (a0' & a1' & s1 & s2 & ((sa0 & sa1) & eqargs) & eqs0). *)
(* subst. exists a0', a1', s1, (subst s2 0 (subst s1 #|Γ'| b) :: s2). intuition eauto. *)
(* rewrite -> subst_context_snoc, Nat.add_0_r. *)
(* unfold subst_decl; simpl. unfold map_decl. simpl. *)
(* econstructor. auto. simpl. f_equal. *)
(* rewrite -> subst_app_simpl; auto. simpl. *)
(* pose proof(context_subst_length _ _ _ sa1) as Hs1. *)
(* rewrite subst_context_length in Hs1. rewrite -> Hs1. auto. *)
(* Qed. *)
Lemma make_context_subst_rec_spec ctx args s tele args' s' :
context_subst ctx args s ->
make_context_subst tele args' s = Some s' ->
context_subst (List.rev tele ++ ctx) (args ++ args') s'.
Proof.
induction tele in ctx, args, s, args', s' |- *.
- move=> /= Hc. case: args'.
+ move => [= <-].
now rewrite app_nil_r.
+ move=> a l //.
- move=> Hc /=. case: a => [na [body|] ty] /=.
-- specialize (IHtele (vdef na body ty :: ctx) args (subst0 s body :: s) args' s').
move=> /=. rewrite <- app_assoc.
move/(IHtele _). move=> H /=. apply H.
constructor. auto.
-- case: args' => [|a args']; try congruence.
specialize (IHtele (vass na ty :: ctx) (args ++ [a]) (a :: s) args' s').
move=> /=. rewrite <- app_assoc.
move/(IHtele _). move=> H /=. simpl in H. rewrite <- app_assoc in H. apply H.
constructor. auto.
Qed.
Lemma make_context_subst_spec tele args s' :
make_context_subst tele args [] = Some s' ->
context_subst (List.rev tele) args s'.
Proof.
move/(make_context_subst_rec_spec [] [] [] _ _ _ context_subst_nil).
rewrite app_nil_r /= //.
Qed.
Lemma subst_telescope_cons s k d Γ :
subst_telescope s k (d :: Γ) =
map_decl (subst s k) d :: subst_telescope s (S k) Γ.
Proof.
simpl.
unfold subst_telescope, mapi. simpl.
rewrite Nat.add_0_r; f_equal.
rewrite mapi_rec_Sk. apply mapi_rec_ext.
intros. simpl. now rewrite Nat.add_succ_r.
Qed.
Lemma subst_telescope_comm_rec s k s' k' Γ:
subst_telescope (map (subst s' k) s) k' (subst_telescope s' (#|s| + k' + k) Γ) =
subst_telescope s' (k' + k) (subst_telescope s k' Γ).
Proof.
induction Γ in k, k' |- *; rewrite ?subst_telescope_cons; simpl; auto.
f_equal.
* unfold map_decl. simpl.
f_equal.
+ destruct a as [na [b|] ty]; simpl; auto.
f_equal. now rewrite distr_subst_rec.
+ now rewrite distr_subst_rec.
* specialize (IHΓ k (S k')). now rewrite Nat.add_succ_r in IHΓ.
Qed.
Lemma subst_telescope_comm s k s' Γ:
subst_telescope (map (subst s' k) s) 0 (subst_telescope s' (#|s| + k) Γ) =
subst_telescope s' k (subst_telescope s 0 Γ).
Proof.
now rewrite -(subst_telescope_comm_rec _ _ _ 0) Nat.add_0_r.
Qed.
Lemma instantiate_params_subst_make_context_subst ctx args s ty s' ty' :
instantiate_params_subst ctx args s ty = Some (s', ty') ->
∑ ctx'',
make_context_subst ctx args s = Some s' /\
decompose_prod_n_assum [] (length ctx) ty = Some (ctx'', ty').
Proof.
induction ctx in args, s, ty, s' |- *; simpl.
- case: args => [|a args'] // [= <- <-]. exists []; intuition congruence.
- case: a => [na [body|] ty''] /=.
+ destruct ty; try congruence.
intros. move: (IHctx _ _ _ _ H) => [ctx'' [Hmake Hdecomp]].
eapply (decompose_prod_n_assum_extend_ctx [vdef na0 ty1 ty2]) in Hdecomp.
unfold snoc. eexists; intuition eauto.
+ destruct ty; try congruence.
case: args => [|a args']; try congruence.
move=> H. move: (IHctx _ _ _ _ H) => [ctx'' [Hmake Hdecomp]].
eapply (decompose_prod_n_assum_extend_ctx [vass na0 ty1]) in Hdecomp.
unfold snoc. eexists; intuition eauto.
Qed.
Lemma instantiate_params_make_context_subst ctx args ty ty' :
instantiate_params ctx args ty = Some ty' ->
∑ ctx' ty'' s',
decompose_prod_n_assum [] (length ctx) ty = Some (ctx', ty'') /\
make_context_subst (List.rev ctx) args [] = Some s' /\ ty' = subst0 s' ty''.
Proof.
unfold instantiate_params.
case E: instantiate_params_subst => // [[s ty'']].
move=> [= <-].
eapply instantiate_params_subst_make_context_subst in E.
destruct E as [ctx'' [Hs Hty'']].
exists ctx'', ty'', s. split; auto.
now rewrite -> List.rev_length in Hty''.
Qed.
Lemma subst_cstr_concl_head ind u mdecl (arity : context) args :
let head := tRel (#|ind_bodies mdecl| - S (inductive_ind ind) + #|ind_params mdecl| + #|arity|) in
let s := (inds (inductive_mind ind) u (ind_bodies mdecl)) in
inductive_ind ind < #|ind_bodies mdecl| ->
subst s (#|arity| + #|ind_params mdecl|)
(subst_instance_constr u (mkApps head (to_extended_list_k (ind_params mdecl) #|arity| ++ args)))
= mkApps (tInd ind u) (to_extended_list_k (ind_params mdecl) #|arity| ++
map (subst s (#|arity| + #|ind_params mdecl|)) (map (subst_instance_constr u) args)).
Proof.
intros.
rewrite subst_instance_constr_mkApps subst_mkApps.
f_equal.
- subst head. simpl subst_instance_constr.
rewrite (subst_rel_eq _ _ (#|ind_bodies mdecl| - S (inductive_ind ind)) (tInd ind u)) //; try lia.
subst s. rewrite inds_spec rev_mapi nth_error_mapi /=.
elim nth_error_spec.
+ intros. simpl.
f_equal. destruct ind; simpl. f_equal. f_equal. simpl in H. lia.
+ rewrite List.rev_length. lia.
- rewrite !map_app. f_equal.
rewrite map_subst_instance_constr_to_extended_list_k.
erewrite to_extended_list_k_map_subst at 2.
+ now rewrite Nat.add_comm.
+ lia.
Qed.
Lemma subst_to_extended_list_k s k args ctx :
make_context_subst (List.rev ctx) args [] = Some s ->
map (subst s k) (to_extended_list_k ctx k) = map (lift0 k) args.
Proof.
move/make_context_subst_spec. rewrite List.rev_involutive.
move=> H; induction H; simpl; auto.
- rewrite map_app -IHcontext_subst //.
rewrite to_extended_list_k_cons /= !map_app.
f_equal.
+ rewrite (lift_to_extended_list_k _ _ 1) map_map_compose.
pose proof (to_extended_list_k_spec Γ k).
solve_all. destruct H0 as [n [-> Hn]].
rewrite /lift (subst_app_decomp [a] s k); auto with wf.
rewrite subst_rel_gt.
* simpl. lia.
* unf_term. repeat (f_equal; simpl; try lia).
+ now rewrite /map (subst_rel_eq _ _ 0 a).
- rewrite -IHcontext_subst // to_extended_list_k_cons /=.
rewrite (lift_to_extended_list_k _ _ 1) map_map_compose.
pose proof (to_extended_list_k_spec Γ k).
solve_all. destruct H0 as [n [-> Hn]].
rewrite /lift (subst_app_decomp [subst0 s b] s k); auto with wf.
rewrite subst_rel_gt.
+ simpl; lia.
+ unf_term. repeat (f_equal; simpl; try lia).
Qed.
Lemma Alli_map_option_out_mapi_Some_spec' {A B} (f g : nat -> A -> option B) h l t P :
All P l ->
(forall i x t, P x -> f i x = Some t -> g i x = Some (h t)) ->
map_option_out (mapi f l) = Some t ->
map_option_out (mapi g l) = Some (map h t).
Proof.
unfold mapi. generalize 0.
move => n Ha Hfg. move: t.
induction Ha in n |- *; try constructor; auto.
- destruct t; cbnr; discriminate.
- move=> t /=. case E: (f n x) => [b|]; try congruence.
rewrite (Hfg n _ _ p E).
case E' : map_option_out => [b'|]; try congruence.
move=> [= <-]. now rewrite (IHHa _ _ E').
Qed.
Lemma map_subst_instance_constr_to_extended_list_k u ctx k :
to_extended_list_k (subst_instance_context u ctx) k
= to_extended_list_k ctx k.
Proof.
unfold to_extended_list_k.
cut (map (subst_instance_constr u) [] = []); [|reflexivity].
unf_term. generalize (@nil term); intros l Hl.
induction ctx in k, l, Hl |- *; cbnr.
destruct a as [? [] ?]; cbnr; eauto.
unf_term. eapply IHctx; cbn; congruence.
Qed.
Lemma subst_build_case_predicate_type ind mdecl idecl u params ps pty n k :
closed_ctx (subst_instance_context u (ind_params mdecl)) ->
closed (ind_type idecl) ->
build_case_predicate_type ind mdecl idecl params u ps = Some pty ->
build_case_predicate_type ind mdecl
(map_one_inductive_body (context_assumptions mdecl.(ind_params))
(length (arities_context (ind_bodies mdecl))) (fun k' => subst n (k' + k))
(inductive_ind ind) idecl)
(map (subst n k) params) u ps
= Some (subst n k pty).
Proof.
intros closedparams closedtype.
unfold build_case_predicate_type; simpl.
case_eq (instantiate_params (subst_instance_context u (ind_params mdecl)) params
(subst_instance_constr u (ind_type idecl)));
[|discriminate ].
intros ipars Hipars.
apply (subst_instantiate_params n k) in Hipars; tas.
rewrite ind_type_map. simpl.
rewrite subst_closedn in Hipars.
{ rewrite closedn_subst_instance_constr; eapply closed_upwards; tea; lia. }
rewrite subst_closedn; [eapply closed_upwards; tea; lia|].
rewrite Hipars.
specialize (subst_destArity [] ipars n k);
destruct (destArity [] ipars) as [[ctx s]|]; [|discriminate].
simpl. cbn. move => -> /some_inj-HH. simpl. f_equal.
etransitivity.
2: exact (f_equal (subst n k) HH).
rewrite subst_it_mkProd_or_LetIn. unf_term. simpl. f_equal. f_equal.
{ destruct idecl; reflexivity. }
rewrite subst_mkApps; simpl. f_equal.
rewrite map_app; f_equal.
- rewrite !map_map; apply map_ext; clear; intro.
rewrite subst_context_length.
rewrite -> commut_lift_subst_rec by lia.
f_equal. lia.
- rewrite /to_extended_list to_extended_list_k_subst.
rewrite <- to_extended_list_k_map_subst by lia. reflexivity.
Qed.
Lemma subst_build_branches_type {cf:checker_flags}
n k Σ ind mdecl idecl indices args u p brs cs :
wf Σ ->
declared_inductive Σ mdecl ind idecl ->
closed_ctx (subst_instance_context u (ind_params mdecl)) ->
on_inductive (lift_typing typing) (Σ, ind_universes mdecl)
(inductive_mind ind) mdecl ->
on_constructors (lift_typing typing) (Σ, ind_universes mdecl)
mdecl (inductive_ind ind) idecl indices (ind_ctors idecl) cs ->
map_option_out (build_branches_type ind mdecl idecl args u p) = Some brs ->
map_option_out (build_branches_type ind mdecl
idecl (map (subst n k) args) u (subst n k p)) =
Some (map (on_snd (subst n k)) brs).
Proof.
intros wfΣ wfidecl closedparams onmind Honcs.
rewrite !build_branches_type_. cbn.
eapply Alli_map_option_out_mapi_Some_spec'.
{ eapply All2_All_left; tea. intros x y u'; exact (y; u'). }
clear Honcs brs.
intros j [[id t] i] [t' k'] [cs' Honc].
case_eq (instantiate_params (subst_instance_context u (ind_params mdecl)) args
(subst0 (inds (inductive_mind ind) u
(ind_bodies mdecl))
(subst_instance_constr u t)));
[|discriminate].
intros ty Heq; cbn.
pose proof (on_constructor_closed wfΣ (u:=u) Honc) as clt; cbn in clt.
eapply (closed_upwards k) in clt; try lia.
remember (subst0 (inds (inductive_mind ind) u (ind_bodies mdecl))
(subst_instance_constr u t)) as ty'.
apply (subst_instantiate_params n k) in Heq as Heq'; tas.
erewrite subst_closedn in Heq'; tas.
rewrite Heq'.
eapply instantiate_params_make_context_subst in Heq.
destruct Heq as [ctx' [ty'' [s' [Hty [Hsubst Ht0]]]]].
subst ty; simpl.
rewrite Heqty' in Hty.
destruct Honc as [Hcshape_args ? cshape_eq Hty' _ _]; unfold cdecl_type in *; simpl in *.
rewrite cshape_eq in Hty.
rewrite -> !subst_instance_constr_it_mkProd_or_LetIn in Hty.
rewrite !subst_it_mkProd_or_LetIn in Hty.
assert (H0: #|subst_context (inds (inductive_mind ind) u (ind_bodies mdecl)) 0
(subst_instance_context u (ind_params mdecl))|
= #|subst_instance_context u (ind_params mdecl)|). {
now rewrite subst_context_length. }
rewrite <- H0 in Hty.
rewrite decompose_prod_n_assum_it_mkProd in Hty.
injection Hty. clear Hty.
intros Heqty'' <-. revert Heqty''.
rewrite !subst_instance_context_length Nat.add_0_r.
rewrite subst_context_length subst_instance_context_length.
rewrite (subst_cstr_concl_head ind u mdecl cs'.(cshape_args) cs'.(cshape_indices)).
{ destruct wfidecl as [Hmdecl Hnth].
now apply nth_error_Some_length in Hnth.
}
intros <-. rewrite !subst_it_mkProd_or_LetIn !subst_mkApps /=.
rewrite !decompose_prod_assum_it_mkProd /=;
try by rewrite is_ind_app_head_mkApps.
rewrite !decompose_app_mkApps; try by reflexivity.
simpl. rewrite !map_app !subst_context_length
!subst_instance_context_length Nat.add_0_r.
eapply subst_to_extended_list_k with (k:=#|cs'.(cshape_args)|) in Hsubst as XX.
rewrite map_subst_instance_constr_to_extended_list_k in XX.
rewrite !XX; clear XX.
apply make_context_subst_spec in Hsubst as Hsubst'.
rewrite rev_involutive in Hsubst'.
pose proof (context_subst_assumptions_length Hsubst') as H1.
case E: chop => [l l'].
have chopm := (chop_map _ _ _ _ _ E).
move: E chopm.
rewrite chop_n_app ?map_length. {
rewrite <- H1. apply onNpars in onmind.
now rewrite subst_instance_context_assumptions. }
move=> [= <- <-] chopm.
move: {chopm}(chopm _ (subst n (#|cs'.(cshape_args)| + k))).
rewrite map_app.
move=> chopm; rewrite {}chopm /=.
inversion 1; subst. f_equal. unfold on_snd; cbn; f_equal.
rewrite !app_nil_r /on_snd /=.
rewrite subst_it_mkProd_or_LetIn !subst_context_length !subst_mkApps
!subst_instance_context_length !map_app.
f_equal. f_equal.
-- rewrite -> commut_lift_subst_rec by lia. arith_congr.
-- f_equal. simpl. f_equal.
rewrite !subst_mkApps /= !map_app. f_equal.
f_equal.
rewrite /to_extended_list -to_extended_list_k_map_subst.
++ rewrite !subst_context_length subst_instance_context_length. lia.
++ now rewrite to_extended_list_k_subst.
Qed.
#[global]
Hint Unfold subst1 : subst.
Hint Rewrite subst_mkApps distr_subst: subst.
Inductive untyped_subslet (Γ : context) : list term -> context -> Type :=
| untyped_emptyslet : untyped_subslet Γ [] []
| untyped_cons_let_ass Δ s na t T :
untyped_subslet Γ s Δ ->
untyped_subslet Γ (t :: s) (Δ ,, vass na T)
| untyped_cons_let_def Δ s na t T :
untyped_subslet Γ s Δ ->
untyped_subslet Γ (subst0 s t :: s) (Δ ,, vdef na t T).
Lemma decompose_prod_assum_it_mkProd_or_LetIn ctx t ctx' t' :
decompose_prod_assum ctx t = (ctx', t') ->
it_mkProd_or_LetIn ctx t = it_mkProd_or_LetIn ctx' t'.
Proof.
induction t in ctx, ctx', t' |- *; simpl; try intros [= -> <-]; auto.
- intros H. now apply IHt2 in H.
- intros H. now apply IHt3 in H.
Qed.
Inductive decompose_prod_assum_graph : term -> (context * term) -> Type :=
| decompose_arity ctx t : decompose_prod_assum_graph (it_mkProd_or_LetIn ctx t) (ctx, t).
Lemma decompose_prod_assum_spec ctx t :
decompose_prod_assum_graph (it_mkProd_or_LetIn ctx t) (decompose_prod_assum ctx t).
Proof.
induction t in ctx |- *; simpl; try constructor 1.
- specialize (IHt2 (vass na t1 :: ctx)).