Library MetaCoq.Common.Environment
(* Distributed under the terms of the MIT license. *)
From Coq Require Import ssreflect ssrbool ssrfun Morphisms Setoid.
From MetaCoq.Utils Require Import utils.
From MetaCoq.Common Require Import BasicAst Primitive Universes.
From Equations.Prop Require Import Classes.
Module Type Term.
Parameter Inline term : Type.
Parameter Inline tRel : nat → term.
Parameter Inline tSort : Universe.t → term.
Parameter Inline tProd : aname → term → term → term.
Parameter Inline tLambda : aname → term → term → term.
Parameter Inline tLetIn : aname → term → term → term → term.
Parameter Inline tInd : inductive → Instance.t → term.
Parameter Inline tProj : projection → term → term.
Parameter Inline mkApps : term → list term → term.
Parameter Inline lift : nat → nat → term → term.
Parameter Inline subst : list term → nat → term → term.
Parameter Inline closedn : nat → term → bool.
Parameter Inline noccur_between : nat → nat → term → bool.
Parameter Inline subst_instance_constr : UnivSubst term.
Notation lift0 n := (lift n 0).
End Term.
Module Type TermDecide (Import T : Term).
#[export] Declare Instance term_eq_dec : EqDec term.
End TermDecide.
Module Retroknowledge.
Record t := mk_retroknowledge {
retro_int63 : option kername;
retro_float64 : option kername;
}.
Definition empty := {| retro_int63 := None; retro_float64 := None |}.
Definition extends (x y : t) :=
option_extends x.(retro_int63) y.(retro_int63) ∧
option_extends x.(retro_float64) y.(retro_float64).
Existing Class extends.
#[global] Instance extends_refl x : extends x x.
Proof.
split; apply option_extends_refl.
Qed.
#[global] Instance extends_trans : RelationClasses.Transitive Retroknowledge.extends.
Proof.
intros x y z [] []; split; cbn; now etransitivity; tea.
Qed.
End Retroknowledge.
Module Environment (T : Term).
Import T.
#[global] Existing Instance subst_instance_constr.
Definition typ_or_sort := typ_or_sort_ term.
From Coq Require Import ssreflect ssrbool ssrfun Morphisms Setoid.
From MetaCoq.Utils Require Import utils.
From MetaCoq.Common Require Import BasicAst Primitive Universes.
From Equations.Prop Require Import Classes.
Module Type Term.
Parameter Inline term : Type.
Parameter Inline tRel : nat → term.
Parameter Inline tSort : Universe.t → term.
Parameter Inline tProd : aname → term → term → term.
Parameter Inline tLambda : aname → term → term → term.
Parameter Inline tLetIn : aname → term → term → term → term.
Parameter Inline tInd : inductive → Instance.t → term.
Parameter Inline tProj : projection → term → term.
Parameter Inline mkApps : term → list term → term.
Parameter Inline lift : nat → nat → term → term.
Parameter Inline subst : list term → nat → term → term.
Parameter Inline closedn : nat → term → bool.
Parameter Inline noccur_between : nat → nat → term → bool.
Parameter Inline subst_instance_constr : UnivSubst term.
Notation lift0 n := (lift n 0).
End Term.
Module Type TermDecide (Import T : Term).
#[export] Declare Instance term_eq_dec : EqDec term.
End TermDecide.
Module Retroknowledge.
Record t := mk_retroknowledge {
retro_int63 : option kername;
retro_float64 : option kername;
}.
Definition empty := {| retro_int63 := None; retro_float64 := None |}.
Definition extends (x y : t) :=
option_extends x.(retro_int63) y.(retro_int63) ∧
option_extends x.(retro_float64) y.(retro_float64).
Existing Class extends.
#[global] Instance extends_refl x : extends x x.
Proof.
split; apply option_extends_refl.
Qed.
#[global] Instance extends_trans : RelationClasses.Transitive Retroknowledge.extends.
Proof.
intros x y z [] []; split; cbn; now etransitivity; tea.
Qed.
End Retroknowledge.
Module Environment (T : Term).
Import T.
#[global] Existing Instance subst_instance_constr.
Definition typ_or_sort := typ_or_sort_ term.
Local (de Bruijn) variable binding
Local (de Bruijn) let-binding
Definition vdef x t A : context_decl :=
{| decl_name := x ; decl_body := Some t ; decl_type := A |}.
Local (de Bruijn) context
Last declaration first
Definition lift_decl n k d := (map_decl (lift n k) d).
Definition lift_context n k (Γ : context) : context :=
fold_context_k (fun k' ⇒ lift n (k' + k)) Γ.
Lemma lift_context_alt n k Γ :
lift_context n k Γ =
mapi (fun k' d ⇒ lift_decl n (Nat.pred #|Γ| - k' + k) d) Γ.
Proof.
unfold lift_context. apply: fold_context_k_alt.
Qed.
Lemma lift_context_length n k Γ : #|lift_context n k Γ| = #|Γ|.
Proof. now rewrite /lift_context; len. Qed.
#[global] Hint Rewrite lift_context_length : len.
Definition subst_context s k (Γ : context) : context :=
fold_context_k (fun k' ⇒ subst s (k' + k)) Γ.
Definition subst_decl s k (d : context_decl) := map_decl (subst s k) d.
Lemma subst_context_length s n Γ : #|subst_context s n Γ| = #|Γ|.
Proof. now rewrite /subst_context; len. Qed.
#[global] Hint Rewrite subst_context_length : len.
Lemma subst_context_nil s n : subst_context s n [] = [].
Proof. reflexivity. Qed.
Lemma subst_context_alt s k Γ :
subst_context s k Γ =
mapi (fun k' d ⇒ subst_decl s (Nat.pred #|Γ| - k' + k) d) Γ.
Proof.
unfold subst_context, fold_context_k. rewrite rev_mapi. rewrite List.rev_involutive.
apply mapi_ext. intros. f_equal. now rewrite List.rev_length.
Qed.
Lemma subst_context_snoc s k Γ d : subst_context s k (d :: Γ) = subst_context s k Γ ,, subst_decl s (#|Γ| + k) d.
Proof.
now rewrite /subst_context fold_context_k_snoc0.
Qed.
Definition subst_telescope s k (Γ : context) : context :=
mapi (fun k' decl ⇒ map_decl (subst s (k' + k)) decl) Γ.
Global Instance subst_instance_decl : UnivSubst context_decl
:= map_decl ∘ subst_instance.
Global Instance subst_instance_context : UnivSubst context
:= map_context ∘ subst_instance.
Lemma subst_instance_length u (ctx : context)
: #|subst_instance u ctx| = #|ctx|.
Proof. unfold subst_instance, subst_instance_context, map_context. now rewrite map_length. Qed.
#[global] Hint Rewrite subst_instance_length : len.
Definition set_binder_name (na : aname) (x : context_decl) : context_decl :=
{| decl_name := na;
decl_body := decl_body x;
decl_type := decl_type x |}.
Fixpoint context_assumptions (Γ : context) :=
match Γ with
| [] ⇒ 0
| d :: Γ ⇒
match d.(decl_body) with
| Some _ ⇒ context_assumptions Γ
| None ⇒ S (context_assumptions Γ)
end
end.
Fixpoint is_assumption_context (Γ : context) :=
match Γ with
| [] ⇒ true
| d :: Γ ⇒
match d.(decl_body) with
| Some _ ⇒ false
| None ⇒ is_assumption_context Γ
end
end.
Smashing a context produces an assumption context.
Fixpoint smash_context (Γ Γ' : context) : context :=
match Γ' with
| {| decl_body := Some b |} :: Γ' ⇒ smash_context (subst_context [b] 0 Γ) Γ'
| {| decl_body := None |} as d :: Γ' ⇒ smash_context (Γ ++ [d]) Γ'
| [] ⇒ Γ
end.
Lemma smash_context_length Γ Γ' : #|smash_context Γ Γ'| = #|Γ| + context_assumptions Γ'.
Proof.
induction Γ' as [|[na [body|] ty] tl] in Γ |- *; cbn; eauto.
- now rewrite IHtl subst_context_length.
- rewrite IHtl app_length. simpl. lia.
Qed.
#[global] Hint Rewrite smash_context_length : len.
(* Smashing a context Γ with Δ depending on it is the same as smashing Γ
and substituting all references to Γ in Δ by the expansions of let bindings. *)
Lemma smash_context_app Δ Γ Γ' :
smash_context Δ (Γ ++ Γ') = smash_context (smash_context Δ Γ) Γ'.
Proof.
revert Δ; induction Γ as [|[na [b|] ty]]; intros Δ; simpl; auto.
Qed.
Fixpoint extended_subst (Γ : context) (n : nat)
(* Δ, smash_context Γ, n |- extended_subst Γ n : Γ *) :=
match Γ with
| nil ⇒ nil
| cons d vs ⇒
match decl_body d with
| Some b ⇒
(* Δ , vs |- b *)
let s := extended_subst vs n in
(* Δ , smash_context vs , n |- s : vs *)
let b' := lift (context_assumptions vs + n) #|s| b in
(* Δ, smash_context vs, n , vs |- b' *)
let b' := subst s 0 b' in
(* Δ, smash_context vs , n |- b' *)
b' :: s
| None ⇒ tRel n :: extended_subst vs (S n)
end
end.
Lemma extended_subst_length Γ n : #|extended_subst Γ n| = #|Γ|.
Proof.
induction Γ in n |- *; simpl; auto.
now destruct a as [? [?|] ?] ⇒ /=; simpl; rewrite IHΓ.
Qed.
#[global] Hint Rewrite extended_subst_length : len.
Definition expand_lets_k Γ k t :=
(subst (extended_subst Γ 0) k (lift (context_assumptions Γ) (k + #|Γ|) t)).
Definition expand_lets Γ t := expand_lets_k Γ 0 t.
Definition expand_lets_k_ctx Γ k Δ :=
(subst_context (extended_subst Γ 0) k (lift_context (context_assumptions Γ) (k + #|Γ|) Δ)).
Definition expand_lets_ctx Γ Δ := expand_lets_k_ctx Γ 0 Δ.
Lemma expand_lets_k_ctx_length Γ k Δ : #|expand_lets_k_ctx Γ k Δ| = #|Δ|.
Proof. now rewrite /expand_lets_k_ctx; len. Qed.
#[global] Hint Rewrite expand_lets_k_ctx_length : len.
Lemma expand_lets_ctx_length Γ Δ : #|expand_lets_ctx Γ Δ| = #|Δ|.
Proof. now rewrite /expand_lets_ctx; len. Qed.
#[global] Hint Rewrite expand_lets_ctx_length : len.
Definition fix_context (m : mfixpoint term) : context :=
List.rev (mapi (fun i d ⇒ vass d.(dname) (lift i 0 d.(dtype))) m).
Record constructor_body := {
cstr_name : ident;
(* The arguments and indices are typeable under the context of
arities of the mutual inductive + parameters *)
cstr_args : context;
cstr_indices : list term;
cstr_type : term;
(* Closed type: on well-formed constructors: forall params, cstr_args, I params cstr_indices *)
cstr_arity : nat; (* arity, w/o lets, w/o parameters *)
}.
Record projection_body := {
proj_name : ident;
(* The arguments and indices are typeable under the context of
arities of the mutual inductive + parameters *)
proj_relevance : relevance;
proj_type : term; (* Type under context of params and inductive object *)
}.
Definition map_constructor_body npars arities f c :=
{| cstr_name := c.(cstr_name);
cstr_args := fold_context_k (fun x ⇒ f (x + npars + arities)) c.(cstr_args);
cstr_indices := map (f (npars + arities + #|c.(cstr_args)|)) c.(cstr_indices);
(* Note only after positivity checking we can ensure that the indices do not mention the
inductive type.. beware of lets! *)
cstr_type := f arities c.(cstr_type);
cstr_arity := c.(cstr_arity) |}.
(* Here npars should be the context_assumptions of the parameters context. *)
Definition map_projection_body npars f c :=
{| proj_name := c.(proj_name);
proj_relevance := c.(proj_relevance);
proj_type := f (S npars) c.(proj_type)
|}.
See one_inductive_body from declarations.ml.
Record one_inductive_body := {
ind_name : ident;
ind_indices : context; (* Indices of the inductive types, under params *)
ind_sort : Universe.t; (* Sort of the inductive. *)
ind_type : term; (* Closed arity = forall mind_params, ind_indices, tSort ind_sort *)
ind_kelim : allowed_eliminations; (* Allowed eliminations *)
ind_ctors : list constructor_body;
ind_projs : list projection_body; (* names and types of projections, if any. *)
ind_relevance : relevance (* relevance of the inductive definition *) }.
Definition map_one_inductive_body npars arities f m :=
match m with
| Build_one_inductive_body ind_name ind_indices ind_sort
ind_type ind_kelim ind_ctors ind_projs ind_relevance ⇒
Build_one_inductive_body
ind_name (fold_context_k (fun x ⇒ f (npars + x)) ind_indices) ind_sort
(f 0 ind_type) ind_kelim (map (map_constructor_body npars arities f) ind_ctors)
(map (map_projection_body npars f) ind_projs) ind_relevance
end.
ind_name : ident;
ind_indices : context; (* Indices of the inductive types, under params *)
ind_sort : Universe.t; (* Sort of the inductive. *)
ind_type : term; (* Closed arity = forall mind_params, ind_indices, tSort ind_sort *)
ind_kelim : allowed_eliminations; (* Allowed eliminations *)
ind_ctors : list constructor_body;
ind_projs : list projection_body; (* names and types of projections, if any. *)
ind_relevance : relevance (* relevance of the inductive definition *) }.
Definition map_one_inductive_body npars arities f m :=
match m with
| Build_one_inductive_body ind_name ind_indices ind_sort
ind_type ind_kelim ind_ctors ind_projs ind_relevance ⇒
Build_one_inductive_body
ind_name (fold_context_k (fun x ⇒ f (npars + x)) ind_indices) ind_sort
(f 0 ind_type) ind_kelim (map (map_constructor_body npars arities f) ind_ctors)
(map (map_projection_body npars f) ind_projs) ind_relevance
end.
See mutual_inductive_body from declarations.ml.
Record mutual_inductive_body := {
ind_finite : recursivity_kind;
ind_npars : nat;
ind_params : context;
ind_bodies : list one_inductive_body ;
ind_universes : universes_decl;
ind_variance : option (list Universes.Variance.t) }.
ind_finite : recursivity_kind;
ind_npars : nat;
ind_params : context;
ind_bodies : list one_inductive_body ;
ind_universes : universes_decl;
ind_variance : option (list Universes.Variance.t) }.
See constant_body from declarations.ml
Record constant_body := {
cst_type : term;
cst_body : option term;
cst_universes : universes_decl;
cst_relevance : relevance }.
Definition map_constant_body f decl :=
{| cst_type := f decl.(cst_type);
cst_body := option_map f decl.(cst_body);
cst_universes := decl.(cst_universes);
cst_relevance := decl.(cst_relevance) |}.
Lemma map_cst_type f decl :
f (cst_type decl) = cst_type (map_constant_body f decl).
Proof. destruct decl; reflexivity. Qed.
Lemma map_cst_body f decl :
option_map f (cst_body decl) = cst_body (map_constant_body f decl).
Proof. destruct decl; reflexivity. Qed.
Inductive global_decl :=
| ConstantDecl : constant_body → global_decl
| InductiveDecl : mutual_inductive_body → global_decl.
Derive NoConfusion for global_decl.
Definition global_declarations := list (kername × global_decl).
Record global_env := mk_global_env
{ universes : ContextSet.t;
declarations : global_declarations;
retroknowledge : Retroknowledge.t }.
Coercion universes : global_env >-> ContextSet.t.
Definition empty_global_env :=
{| universes := ContextSet.empty;
declarations := [];
retroknowledge := Retroknowledge.empty |}.
Definition add_global_decl Σ decl :=
{| universes := Σ.(universes);
declarations := decl :: Σ.(declarations);
retroknowledge := Σ.(retroknowledge) |}.
Lemma eta_global_env Σ : Σ = {| universes := Σ.(universes); declarations := Σ.(declarations);
retroknowledge := Σ.(retroknowledge) |}.
Proof. now destruct Σ. Qed.
Definition set_declarations Σ decls :=
{| universes := Σ.(universes);
declarations := decls;
retroknowledge := Σ.(retroknowledge) |}.
Fixpoint lookup_global (Σ : global_declarations) (kn : kername) : option global_decl :=
match Σ with
| nil ⇒ None
| d :: tl ⇒
if kn == d.1 then Some d.2
else lookup_global tl kn
end.
Definition lookup_env (Σ : global_env) (kn : kername) := lookup_global Σ.(declarations) kn.
(* version for possibly duplicative environments *)
Fixpoint lookup_globals (Σ : global_declarations) (kn : kername) : list global_decl :=
match Σ with
| nil ⇒ nil
| d :: tl ⇒
let tl := lookup_globals tl kn in
if kn == d.1 then d.2 :: tl else tl
end.
Definition lookup_envs (Σ : global_env) (kn : kername) := lookup_globals Σ.(declarations) kn.
cst_type : term;
cst_body : option term;
cst_universes : universes_decl;
cst_relevance : relevance }.
Definition map_constant_body f decl :=
{| cst_type := f decl.(cst_type);
cst_body := option_map f decl.(cst_body);
cst_universes := decl.(cst_universes);
cst_relevance := decl.(cst_relevance) |}.
Lemma map_cst_type f decl :
f (cst_type decl) = cst_type (map_constant_body f decl).
Proof. destruct decl; reflexivity. Qed.
Lemma map_cst_body f decl :
option_map f (cst_body decl) = cst_body (map_constant_body f decl).
Proof. destruct decl; reflexivity. Qed.
Inductive global_decl :=
| ConstantDecl : constant_body → global_decl
| InductiveDecl : mutual_inductive_body → global_decl.
Derive NoConfusion for global_decl.
Definition global_declarations := list (kername × global_decl).
Record global_env := mk_global_env
{ universes : ContextSet.t;
declarations : global_declarations;
retroknowledge : Retroknowledge.t }.
Coercion universes : global_env >-> ContextSet.t.
Definition empty_global_env :=
{| universes := ContextSet.empty;
declarations := [];
retroknowledge := Retroknowledge.empty |}.
Definition add_global_decl Σ decl :=
{| universes := Σ.(universes);
declarations := decl :: Σ.(declarations);
retroknowledge := Σ.(retroknowledge) |}.
Lemma eta_global_env Σ : Σ = {| universes := Σ.(universes); declarations := Σ.(declarations);
retroknowledge := Σ.(retroknowledge) |}.
Proof. now destruct Σ. Qed.
Definition set_declarations Σ decls :=
{| universes := Σ.(universes);
declarations := decls;
retroknowledge := Σ.(retroknowledge) |}.
Fixpoint lookup_global (Σ : global_declarations) (kn : kername) : option global_decl :=
match Σ with
| nil ⇒ None
| d :: tl ⇒
if kn == d.1 then Some d.2
else lookup_global tl kn
end.
Definition lookup_env (Σ : global_env) (kn : kername) := lookup_global Σ.(declarations) kn.
(* version for possibly duplicative environments *)
Fixpoint lookup_globals (Σ : global_declarations) (kn : kername) : list global_decl :=
match Σ with
| nil ⇒ nil
| d :: tl ⇒
let tl := lookup_globals tl kn in
if kn == d.1 then d.2 :: tl else tl
end.
Definition lookup_envs (Σ : global_env) (kn : kername) := lookup_globals Σ.(declarations) kn.
We define four notions of environment extension. The two
configurable bits are: is universe and retroknowledge extension
strict (Logic.eq) or loose (⊂_cs / Retroknowledge.extends);
and is declaration extension strict (fully order-preserving,
i.e., new declarations are added only at the front) or lax
(declarations may be reordered and added freely, as long as all
new declarations with existing names come before the old ones
with the same names, and the relative order of declarations with
identical names is preserved).
In most cases, we are actually interested only in duplicate-free
environments, where the lax construction is equivalent to merely
requiring that the lookup function agrees on all existing
delcarations. However, we formulate the property in a way that
makes sense for duplicative environments so that strict
declaration extension will always imply lax declaration
extension. (The lookup function prefers earlier / newer
declarations over older ones.)
We thus have the following implication structure:
<
┌-----------------------------┬------------------------┬---------------------------┐
| univ/retro extension is: | strict | lax |
├ declaration exension is ----┼------------------------┼---------------------------┤
| lax | extends_decls → extends |
| | ↑ ↗ ↑ |
| strict | strictly_extends_decls → extends_strictly_on_decls |
└-----------------------------┴------------------------┴---------------------------┘
>>>
Definition extends (Σ Σ' : global_env) :=
[× Σ.(universes) ⊂_cs Σ'.(universes),
∀ c, ∑ decls, lookup_envs Σ' c = decls ++ lookup_envs Σ c &
Retroknowledge.extends Σ.(retroknowledge) Σ'.(retroknowledge)].
Definition extends_decls (Σ Σ' : global_env) :=
[× Σ.(universes) = Σ'.(universes),
∀ c, ∑ decls, lookup_envs Σ' c = decls ++ lookup_envs Σ c &
Σ.(retroknowledge) = Σ'.(retroknowledge)].
Definition extends_strictly_on_decls (Σ Σ' : global_env) :=
[× Σ.(universes) ⊂_cs Σ'.(universes),
∑ Σ'', Σ'.(declarations) = Σ'' ++ Σ.(declarations) &
Retroknowledge.extends Σ.(retroknowledge) Σ'.(retroknowledge)].
Definition strictly_extends_decls (Σ Σ' : global_env) :=
[× Σ.(universes) = Σ'.(universes),
∑ Σ'', Σ'.(declarations) = Σ'' ++ Σ.(declarations) &
Σ.(retroknowledge) = Σ'.(retroknowledge)].
Existing Class extends.
Existing Class extends_decls.
Existing Class extends_strictly_on_decls.
Existing Class strictly_extends_decls.
Lemma lookup_global_None Σ kn : ¬In kn (List.map fst Σ) ↔ lookup_global Σ kn = None.
Proof.
move: Σ; elim ⇒ //=; try tauto.
move ⇒ ??; case: eqb_spec; intuition congruence.
Qed.
Lemma hd_error_lookup_globals Σ kn : hd_error (lookup_globals Σ kn) = lookup_global Σ kn.
Proof.
move: Σ; elim ⇒ //= ?? <-.
case: eqb_spec ⇒ //=.
Qed.
Lemma lookup_globals_nil Σ kn : ¬In kn (List.map fst Σ) ↔ lookup_globals Σ kn = nil.
Proof.
rewrite lookup_global_None-hd_error_lookup_globals.
case: lookup_globals ⇒ //.
Qed.
Lemma NoDup_length_lookup_globals Σ
: NoDup (List.map fst Σ)
→ ∀ kn, List.length (lookup_globals Σ kn) = match lookup_global Σ kn with
| Some _ ⇒ 1
| None ⇒ 0
end.
Proof.
move ⇒ H kn.
move: Σ H; elim ⇒ //=; try lia.
move ⇒ ?? H. inversion 1; subst.
move: (H ltac:(assumption)).
case: eqb_spec ⇒ //= →.
rewrite (proj1 (@lookup_global_None _ _)) ⇒ //= → //=.
Qed.
Lemma lookup_globals_In Σ kn decl
: In (kn, decl) Σ ↔ In decl (lookup_globals Σ kn).
Proof.
move: Σ; elim ⇒ //=; try tauto.
move ⇒ [??]?; case: eqb_spec ⇒ ? //=; subst ⇒ <-; cbn in *; firstorder (subst; auto).
all: (idtac + constructor); congruence.
Qed.
Lemma lookup_global_Some_if_In Σ kn decl
: lookup_global Σ kn = Some decl → In (kn, decl) Σ.
Proof.
move: Σ; elim ⇒ //=; try tauto.
move ⇒ [??]?; case: eqb_spec ⇒ ? IH; inversion 1; subst; try rewrite <- IH by assumption.
all: intuition try congruence; subst.
Qed.
Lemma lookup_global_Some_iff_In_NoDup Σ kn decl (H : NoDup (List.map fst Σ))
: In (kn, decl) Σ ↔ lookup_global Σ kn = Some decl.
Proof.
rewrite -hd_error_lookup_globals lookup_globals_In.
apply NoDup_length_lookup_globals with (kn:=kn) in H; move: H.
case: lookup_global; case: lookup_globals ⇒ [|?[]]; cbn.
all: try lia.
all: intuition congruence.
Qed.
Lemma lookup_env_extends_NoDup Σ Σ' k d :
NoDup (List.map fst Σ'.(declarations)) →
lookup_env Σ k = Some d →
extends Σ Σ' → lookup_env Σ' k = Some d.
Proof.
rewrite /lookup_env /= -!hd_error_lookup_globals ⇒ Hnd.
move: (@NoDup_length_lookup_globals _ Hnd k); clear Hnd.
rewrite -hd_error_lookup_globals.
move⇒ H Hd [? eq].
move: (eq k); clear eq.
rewrite /lookup_envs.
case ⇒ ls eq.
move: eq Hd H ⇒ →.
case: ls ⇒ //= ?.
case ⇒ //=.
case: lookup_globals ⇒ //=.
Qed.
Lemma lookup_globals_app Σ Σ' kn :
lookup_globals (Σ ++ Σ') kn = lookup_globals Σ kn ++ lookup_globals Σ' kn.
Proof.
move: Σ.
elim ⇒ //= ??.
case: eqb_spec ⇒ //= → → //=.
Qed.
#[global] Instance strictly_extends_decls_extends_decls Σ Σ' : strictly_extends_decls Σ Σ' → extends_decls Σ Σ'.
Proof.
destruct Σ, Σ'; case ⇒ //= → [Σ'' ->] →.
rewrite /extends_decls/lookup_envs //=; split ⇒ //= ?.
rewrite lookup_globals_app.
eexists; reflexivity.
Qed.
#[global] Instance strictly_extends_decls_extends_strictly_on_decls Σ Σ' : strictly_extends_decls Σ Σ' → extends_strictly_on_decls Σ Σ'.
Proof.
destruct Σ, Σ'; intros []. cbn in *; subst. split ⇒ //=.
split; [lsets|csets]. apply Retroknowledge.extends_refl.
Qed.
#[global] Instance extends_decls_extends Σ Σ' : extends_decls Σ Σ' → extends Σ Σ'.
Proof.
destruct Σ, Σ'; intros []. cbn in *; subst. split ⇒ //=.
split; [lsets|csets]. apply Retroknowledge.extends_refl.
Qed.
#[global] Instance extends_strictly_on_decls_extends Σ Σ' : extends_strictly_on_decls Σ Σ' → extends Σ Σ'.
Proof.
destruct Σ, Σ'; case ⇒ //= ? [Σ'' ->] ?.
rewrite /extends/lookup_envs //=; split ⇒ //= ?.
rewrite lookup_globals_app.
eexists; reflexivity.
Qed.
#[global] Instance strictly_extends_decls_extends_decls_subrel : CRelationClasses.subrelation strictly_extends_decls extends_decls := strictly_extends_decls_extends_decls.
#[global] Instance strictly_extends_decls_extends_strictly_on_decls_subrel : CRelationClasses.subrelation strictly_extends_decls extends_strictly_on_decls := strictly_extends_decls_extends_strictly_on_decls.
#[global] Instance extends_decls_extends_subrel : CRelationClasses.subrelation extends_decls extends := extends_decls_extends.
#[global] Instance extends_strictly_on_decls_extends_subrel : CRelationClasses.subrelation extends_strictly_on_decls extends := extends_strictly_on_decls_extends.
#[global] Instance strictly_extends_decls_extends_subrel : CRelationClasses.subrelation strictly_extends_decls extends := fun _ ⇒ _.
#[global] Instance strictly_extends_decls_refl : CRelationClasses.Reflexive strictly_extends_decls.
Proof. red. intros x. split ⇒ //; try ∃ [] ⇒ //. Qed.
#[global] Instance extends_decls_refl : CRelationClasses.Reflexive extends_decls.
Proof. red. intros x. split ⇒ //; try ∃ [] ⇒ //. Qed.
Lemma extends_strictly_on_decls_refl : CRelationClasses.Reflexive extends_strictly_on_decls.
Proof. red. intros x. split; [apply incl_cs_refl | try ∃ [] ⇒ // | apply Retroknowledge.extends_refl]. Qed.
Lemma extends_refl : CRelationClasses.Reflexive extends.
Proof. red. intros x. split; [apply incl_cs_refl | try ∃ [] ⇒ // | apply Retroknowledge.extends_refl]. Qed.
(* easy prefers this to the local hypotheses, which is annoying
[global] Instance extends_refl : CRelationClasses.Reflexive extends. Proof. apply extends_refl. Qed. *)
Local Ltac extends_trans_t :=
intros [?] [?] [?] [?] [?]; red; cbn in *; split;
try solve [ etransitivity; eassumption | eapply incl_cs_trans; eassumption ];
repeat first [ progress subst
| match goal with
| [ H : ∑ x : _, _ |- _ ] ⇒ destruct H
| [ H : ∀ c : kername, _, kn : kername |- _ ] ⇒ specialize (H kn)
| [ H : ?x = _ |- context[?x] ] ⇒ rewrite H
end
| split
| intro
| now eexists; rewrite app_assoc ].
#[global] Instance strictly_extends_decls_trans : CRelationClasses.Transitive strictly_extends_decls.
Proof. extends_trans_t. Qed.
#[global] Instance extends_decls_trans : CRelationClasses.Transitive extends_decls.
Proof. extends_trans_t. Qed.
#[global] Instance extends_strictly_on_decls_trans : CRelationClasses.Transitive extends_strictly_on_decls.
Proof. extends_trans_t. Qed.
#[global] Instance extends_trans : CRelationClasses.Transitive extends.
Proof. extends_trans_t. Qed.
Definition primitive_constant (Σ : global_env) (p : prim_tag) : option kername :=
match p with
| primInt ⇒ Σ.(retroknowledge).(Retroknowledge.retro_int63)
| primFloat ⇒ Σ.(retroknowledge).(Retroknowledge.retro_float64)
end.
Definition primitive_invariants (cdecl : constant_body) :=
∑ s, [/\ cdecl.(cst_type) = tSort s, cdecl.(cst_body) = None &
cdecl.(cst_universes) = Monomorphic_ctx].
[× Σ.(universes) ⊂_cs Σ'.(universes),
∀ c, ∑ decls, lookup_envs Σ' c = decls ++ lookup_envs Σ c &
Retroknowledge.extends Σ.(retroknowledge) Σ'.(retroknowledge)].
Definition extends_decls (Σ Σ' : global_env) :=
[× Σ.(universes) = Σ'.(universes),
∀ c, ∑ decls, lookup_envs Σ' c = decls ++ lookup_envs Σ c &
Σ.(retroknowledge) = Σ'.(retroknowledge)].
Definition extends_strictly_on_decls (Σ Σ' : global_env) :=
[× Σ.(universes) ⊂_cs Σ'.(universes),
∑ Σ'', Σ'.(declarations) = Σ'' ++ Σ.(declarations) &
Retroknowledge.extends Σ.(retroknowledge) Σ'.(retroknowledge)].
Definition strictly_extends_decls (Σ Σ' : global_env) :=
[× Σ.(universes) = Σ'.(universes),
∑ Σ'', Σ'.(declarations) = Σ'' ++ Σ.(declarations) &
Σ.(retroknowledge) = Σ'.(retroknowledge)].
Existing Class extends.
Existing Class extends_decls.
Existing Class extends_strictly_on_decls.
Existing Class strictly_extends_decls.
Lemma lookup_global_None Σ kn : ¬In kn (List.map fst Σ) ↔ lookup_global Σ kn = None.
Proof.
move: Σ; elim ⇒ //=; try tauto.
move ⇒ ??; case: eqb_spec; intuition congruence.
Qed.
Lemma hd_error_lookup_globals Σ kn : hd_error (lookup_globals Σ kn) = lookup_global Σ kn.
Proof.
move: Σ; elim ⇒ //= ?? <-.
case: eqb_spec ⇒ //=.
Qed.
Lemma lookup_globals_nil Σ kn : ¬In kn (List.map fst Σ) ↔ lookup_globals Σ kn = nil.
Proof.
rewrite lookup_global_None-hd_error_lookup_globals.
case: lookup_globals ⇒ //.
Qed.
Lemma NoDup_length_lookup_globals Σ
: NoDup (List.map fst Σ)
→ ∀ kn, List.length (lookup_globals Σ kn) = match lookup_global Σ kn with
| Some _ ⇒ 1
| None ⇒ 0
end.
Proof.
move ⇒ H kn.
move: Σ H; elim ⇒ //=; try lia.
move ⇒ ?? H. inversion 1; subst.
move: (H ltac:(assumption)).
case: eqb_spec ⇒ //= →.
rewrite (proj1 (@lookup_global_None _ _)) ⇒ //= → //=.
Qed.
Lemma lookup_globals_In Σ kn decl
: In (kn, decl) Σ ↔ In decl (lookup_globals Σ kn).
Proof.
move: Σ; elim ⇒ //=; try tauto.
move ⇒ [??]?; case: eqb_spec ⇒ ? //=; subst ⇒ <-; cbn in *; firstorder (subst; auto).
all: (idtac + constructor); congruence.
Qed.
Lemma lookup_global_Some_if_In Σ kn decl
: lookup_global Σ kn = Some decl → In (kn, decl) Σ.
Proof.
move: Σ; elim ⇒ //=; try tauto.
move ⇒ [??]?; case: eqb_spec ⇒ ? IH; inversion 1; subst; try rewrite <- IH by assumption.
all: intuition try congruence; subst.
Qed.
Lemma lookup_global_Some_iff_In_NoDup Σ kn decl (H : NoDup (List.map fst Σ))
: In (kn, decl) Σ ↔ lookup_global Σ kn = Some decl.
Proof.
rewrite -hd_error_lookup_globals lookup_globals_In.
apply NoDup_length_lookup_globals with (kn:=kn) in H; move: H.
case: lookup_global; case: lookup_globals ⇒ [|?[]]; cbn.
all: try lia.
all: intuition congruence.
Qed.
Lemma lookup_env_extends_NoDup Σ Σ' k d :
NoDup (List.map fst Σ'.(declarations)) →
lookup_env Σ k = Some d →
extends Σ Σ' → lookup_env Σ' k = Some d.
Proof.
rewrite /lookup_env /= -!hd_error_lookup_globals ⇒ Hnd.
move: (@NoDup_length_lookup_globals _ Hnd k); clear Hnd.
rewrite -hd_error_lookup_globals.
move⇒ H Hd [? eq].
move: (eq k); clear eq.
rewrite /lookup_envs.
case ⇒ ls eq.
move: eq Hd H ⇒ →.
case: ls ⇒ //= ?.
case ⇒ //=.
case: lookup_globals ⇒ //=.
Qed.
Lemma lookup_globals_app Σ Σ' kn :
lookup_globals (Σ ++ Σ') kn = lookup_globals Σ kn ++ lookup_globals Σ' kn.
Proof.
move: Σ.
elim ⇒ //= ??.
case: eqb_spec ⇒ //= → → //=.
Qed.
#[global] Instance strictly_extends_decls_extends_decls Σ Σ' : strictly_extends_decls Σ Σ' → extends_decls Σ Σ'.
Proof.
destruct Σ, Σ'; case ⇒ //= → [Σ'' ->] →.
rewrite /extends_decls/lookup_envs //=; split ⇒ //= ?.
rewrite lookup_globals_app.
eexists; reflexivity.
Qed.
#[global] Instance strictly_extends_decls_extends_strictly_on_decls Σ Σ' : strictly_extends_decls Σ Σ' → extends_strictly_on_decls Σ Σ'.
Proof.
destruct Σ, Σ'; intros []. cbn in *; subst. split ⇒ //=.
split; [lsets|csets]. apply Retroknowledge.extends_refl.
Qed.
#[global] Instance extends_decls_extends Σ Σ' : extends_decls Σ Σ' → extends Σ Σ'.
Proof.
destruct Σ, Σ'; intros []. cbn in *; subst. split ⇒ //=.
split; [lsets|csets]. apply Retroknowledge.extends_refl.
Qed.
#[global] Instance extends_strictly_on_decls_extends Σ Σ' : extends_strictly_on_decls Σ Σ' → extends Σ Σ'.
Proof.
destruct Σ, Σ'; case ⇒ //= ? [Σ'' ->] ?.
rewrite /extends/lookup_envs //=; split ⇒ //= ?.
rewrite lookup_globals_app.
eexists; reflexivity.
Qed.
#[global] Instance strictly_extends_decls_extends_decls_subrel : CRelationClasses.subrelation strictly_extends_decls extends_decls := strictly_extends_decls_extends_decls.
#[global] Instance strictly_extends_decls_extends_strictly_on_decls_subrel : CRelationClasses.subrelation strictly_extends_decls extends_strictly_on_decls := strictly_extends_decls_extends_strictly_on_decls.
#[global] Instance extends_decls_extends_subrel : CRelationClasses.subrelation extends_decls extends := extends_decls_extends.
#[global] Instance extends_strictly_on_decls_extends_subrel : CRelationClasses.subrelation extends_strictly_on_decls extends := extends_strictly_on_decls_extends.
#[global] Instance strictly_extends_decls_extends_subrel : CRelationClasses.subrelation strictly_extends_decls extends := fun _ ⇒ _.
#[global] Instance strictly_extends_decls_refl : CRelationClasses.Reflexive strictly_extends_decls.
Proof. red. intros x. split ⇒ //; try ∃ [] ⇒ //. Qed.
#[global] Instance extends_decls_refl : CRelationClasses.Reflexive extends_decls.
Proof. red. intros x. split ⇒ //; try ∃ [] ⇒ //. Qed.
Lemma extends_strictly_on_decls_refl : CRelationClasses.Reflexive extends_strictly_on_decls.
Proof. red. intros x. split; [apply incl_cs_refl | try ∃ [] ⇒ // | apply Retroknowledge.extends_refl]. Qed.
Lemma extends_refl : CRelationClasses.Reflexive extends.
Proof. red. intros x. split; [apply incl_cs_refl | try ∃ [] ⇒ // | apply Retroknowledge.extends_refl]. Qed.
(* easy prefers this to the local hypotheses, which is annoying
[global] Instance extends_refl : CRelationClasses.Reflexive extends. Proof. apply extends_refl. Qed. *)
Local Ltac extends_trans_t :=
intros [?] [?] [?] [?] [?]; red; cbn in *; split;
try solve [ etransitivity; eassumption | eapply incl_cs_trans; eassumption ];
repeat first [ progress subst
| match goal with
| [ H : ∑ x : _, _ |- _ ] ⇒ destruct H
| [ H : ∀ c : kername, _, kn : kername |- _ ] ⇒ specialize (H kn)
| [ H : ?x = _ |- context[?x] ] ⇒ rewrite H
end
| split
| intro
| now eexists; rewrite app_assoc ].
#[global] Instance strictly_extends_decls_trans : CRelationClasses.Transitive strictly_extends_decls.
Proof. extends_trans_t. Qed.
#[global] Instance extends_decls_trans : CRelationClasses.Transitive extends_decls.
Proof. extends_trans_t. Qed.
#[global] Instance extends_strictly_on_decls_trans : CRelationClasses.Transitive extends_strictly_on_decls.
Proof. extends_trans_t. Qed.
#[global] Instance extends_trans : CRelationClasses.Transitive extends.
Proof. extends_trans_t. Qed.
Definition primitive_constant (Σ : global_env) (p : prim_tag) : option kername :=
match p with
| primInt ⇒ Σ.(retroknowledge).(Retroknowledge.retro_int63)
| primFloat ⇒ Σ.(retroknowledge).(Retroknowledge.retro_float64)
end.
Definition primitive_invariants (cdecl : constant_body) :=
∑ s, [/\ cdecl.(cst_type) = tSort s, cdecl.(cst_body) = None &
cdecl.(cst_universes) = Monomorphic_ctx].
A context of global declarations + global universe constraints,
i.e. a global environment
Use a coercion for this common projection of the global context.
Definition fst_ctx : global_env_ext → global_env := fst.
Coercion fst_ctx : global_env_ext >-> global_env.
Definition empty_ext (Σ : global_env) : global_env_ext
:= (Σ, Monomorphic_ctx).
Coercion fst_ctx : global_env_ext >-> global_env.
Definition empty_ext (Σ : global_env) : global_env_ext
:= (Σ, Monomorphic_ctx).
Definition program : Type := global_env × term.
(* TODO MOVE AstUtils factorisation *)
Definition app_context (Γ Γ' : context) : context := Γ' ++ Γ.
Notation "Γ ,,, Γ'" :=
(app_context Γ Γ') (at level 25, Γ' at next level, left associativity).
Make a lambda/let-in string of abstractions from a context Γ, ending with term t.
Definition mkLambda_or_LetIn d t :=
match d.(decl_body) with
| None ⇒ tLambda d.(decl_name) d.(decl_type) t
| Some b ⇒ tLetIn d.(decl_name) b d.(decl_type) t
end.
Definition it_mkLambda_or_LetIn (l : context) (t : term) :=
List.fold_left (fun acc d ⇒ mkLambda_or_LetIn d acc) l t.
Make a prod/let-in string of abstractions from a context Γ, ending with term t.
Definition mkProd_or_LetIn d t :=
match d.(decl_body) with
| None ⇒ tProd d.(decl_name) d.(decl_type) t
| Some b ⇒ tLetIn d.(decl_name) b d.(decl_type) t
end.
Definition it_mkProd_or_LetIn (l : context) (t : term) :=
List.fold_left (fun acc d ⇒ mkProd_or_LetIn d acc) l t.
Lemma it_mkProd_or_LetIn_app l l' t :
it_mkProd_or_LetIn (l ++ l') t = it_mkProd_or_LetIn l' (it_mkProd_or_LetIn l t).
Proof. induction l in l', t |- *; simpl; auto. Qed.
Fixpoint reln (l : list term) (p : nat) (Γ0 : list context_decl) {struct Γ0} : list term :=
match Γ0 with
| [] ⇒ l
| {| decl_body := Some _ |} :: hyps ⇒ reln l (p + 1) hyps
| {| decl_body := None |} :: hyps ⇒ reln (tRel p :: l) (p + 1) hyps
end.
Definition to_extended_list_k Γ k := reln [] k Γ.
Definition to_extended_list Γ := to_extended_list_k Γ 0.
Lemma reln_fold f ctx n acc :
reln acc n (fold_context_k f ctx) =
reln acc n ctx.
Proof.
induction ctx as [|[na [b|] ty] ctx] in n, acc |- *; simpl; auto;
rewrite fold_context_k_snoc0 /=; apply IHctx.
Qed.
Lemma reln_list_lift_above l p Γ :
Forall (fun x ⇒ ∃ n, x = tRel n ∧ p ≤ n ∧ n < p + length Γ) l →
Forall (fun x ⇒ ∃ n, x = tRel n ∧ p ≤ n ∧ n < p + length Γ) (reln l p Γ).
Proof.
generalize (Nat.le_refl p).
generalize p at 1 3 5.
induction Γ in p, l |- ×. simpl. auto.
intros. destruct a. destruct decl_body. simpl.
assert(p0 ≤ S p) by lia.
specialize (IHΓ l (S p) p0 H1). rewrite <- Nat.add_succ_comm, Nat.add_1_r.
simpl in ×. rewrite <- Nat.add_succ_comm in H0. eauto.
simpl in ×.
specialize (IHΓ (tRel p :: l) (S p) p0 ltac:(lia)).
rewrite <- Nat.add_succ_comm, Nat.add_1_r.
eapply IHΓ. simpl in ×. rewrite <- Nat.add_succ_comm in H0. auto.
simpl in ×.
constructor. ∃ p. intuition lia. auto.
Qed.
Lemma to_extended_list_k_spec Γ k :
Forall (fun x ⇒ ∃ n, x = tRel n ∧ k ≤ n ∧ n < k + length Γ)
(to_extended_list_k Γ k).
Proof.
pose (reln_list_lift_above [] k Γ).
unfold to_extended_list_k.
forward f. constructor. apply f.
Qed.
Lemma to_extended_list_lift_above Γ :
Forall (fun x ⇒ ∃ n, x = tRel n ∧ n < length Γ) (to_extended_list Γ).
Proof.
pose (reln_list_lift_above [] 0 Γ).
unfold to_extended_list.
forward f. constructor. eapply Forall_impl; eauto. intros.
destruct H; eexists; intuition eauto.
Qed.
Fixpoint reln_alt p (Γ : context) :=
match Γ with
| [] ⇒ []
| {| decl_body := Some _ |} :: Γ ⇒ reln_alt (p + 1) Γ
| {| decl_body := None |} :: Γ ⇒ tRel p :: reln_alt (p + 1) Γ
end.
Lemma reln_alt_eq l Γ k : reln l k Γ = List.rev (reln_alt k Γ) ++ l.
Proof.
induction Γ in l, k |- *; simpl; auto.
destruct a as [na [body|] ty]; simpl.
now rewrite IHΓ.
now rewrite IHΓ -app_assoc.
Qed.
Lemma to_extended_list_k_cons d Γ k :
to_extended_list_k (d :: Γ) k =
match d.(decl_body) with
| None ⇒ to_extended_list_k Γ (S k) ++ [tRel k]
| Some b ⇒ to_extended_list_k Γ (S k)
end.
Proof.
unfold to_extended_list_k.
rewrite reln_alt_eq. simpl.
destruct d as [na [body|] ty]. simpl.
now rewrite reln_alt_eq Nat.add_1_r.
simpl. rewrite reln_alt_eq.
now rewrite <- app_assoc, !app_nil_r, Nat.add_1_r.
Qed.
Definition arities_context (l : list one_inductive_body) :=
rev_map (fun ind ⇒ vass (mkBindAnn (nNamed ind.(ind_name))
(ind.(ind_relevance))) ind.(ind_type)) l.
Lemma arities_context_length l : #|arities_context l| = #|l|.
Proof. unfold arities_context. now rewrite rev_map_length. Qed.
#[global] Hint Rewrite arities_context_length : len.
Lemma app_context_nil_l Γ : [] ,,, Γ = Γ.
Proof.
unfold app_context. rewrite app_nil_r. reflexivity.
Qed.
Lemma app_context_assoc Γ Γ' Γ'' : Γ ,,, (Γ' ,,, Γ'') = Γ ,,, Γ' ,,, Γ''.
Proof. unfold app_context; now rewrite app_assoc. Qed.
Lemma app_context_cons Γ Γ' A : Γ ,,, (Γ' ,, A) = (Γ ,,, Γ') ,, A.
Proof. exact (app_context_assoc _ _ [A]). Qed.
Lemma app_context_length Γ Γ' : #|Γ ,,, Γ'| = #|Γ'| + #|Γ|.
Proof. unfold app_context. now rewrite app_length. Qed.
#[global] Hint Rewrite app_context_length : len.
Lemma nth_error_app_context_ge v Γ Γ' :
#|Γ'| ≤ v → nth_error (Γ ,,, Γ') v = nth_error Γ (v - #|Γ'|).
Proof. apply nth_error_app_ge. Qed.
Lemma nth_error_app_context_lt v Γ Γ' :
v < #|Γ'| → nth_error (Γ ,,, Γ') v = nth_error Γ' v.
Proof. apply nth_error_app_lt. Qed.
Definition map_mutual_inductive_body f m :=
match m with
| Build_mutual_inductive_body finite ind_npars ind_pars ind_bodies ind_universes ind_variance ⇒
let arities := arities_context ind_bodies in
let pars := fold_context_k f ind_pars in
Build_mutual_inductive_body finite ind_npars pars
(map (map_one_inductive_body (context_assumptions pars) (length arities) f) ind_bodies)
ind_universes ind_variance
end.
Lemma ind_type_map f npars_ass arities oib :
ind_type (map_one_inductive_body npars_ass arities f oib) = f 0 (ind_type oib).
Proof. destruct oib. reflexivity. Qed.
Lemma ind_ctors_map f npars_ass arities oib :
ind_ctors (map_one_inductive_body npars_ass arities f oib) =
map (map_constructor_body npars_ass arities f) (ind_ctors oib).
Proof. destruct oib; simpl; reflexivity. Qed.
Lemma ind_pars_map f m :
ind_params (map_mutual_inductive_body f m) =
fold_context_k f (ind_params m).
Proof. destruct m; simpl; reflexivity. Qed.
Lemma ind_projs_map f npars_ass arities oib :
ind_projs (map_one_inductive_body npars_ass arities f oib) =
map (map_projection_body npars_ass f) (ind_projs oib).
Proof. destruct oib; simpl. reflexivity. Qed.
Fixpoint projs ind npars k :=
match k with
| 0 ⇒ []
| S k' ⇒ (tProj (mkProjection ind npars k') (tRel 0)) :: projs ind npars k'
end.
Lemma projs_length ind npars k : #|projs ind npars k| = k.
Proof. induction k; simpl; auto. Qed.
Lemma context_assumptions_fold Γ f : context_assumptions (fold_context_k f Γ) = context_assumptions Γ.
Proof.
rewrite fold_context_k_alt.
unfold mapi. generalize 0 (Nat.pred #|Γ|).
induction Γ as [|[na [body|] ty] tl]; cbn; intros; eauto.
Qed.
#[global] Hint Rewrite context_assumptions_fold : len.
Lemma nth_error_fold_context_k (f : nat → term → term):
∀ (Γ' Γ'' : context) (v : nat),
v < length Γ' → ∀ nth,
nth_error Γ' v = Some nth →
nth_error (fold_context_k f Γ') v = Some (map_decl (f (length Γ' - S v)) nth).
Proof.
induction Γ'; intros.
- easy.
- simpl. destruct v; rewrite fold_context_k_snoc0.
+ simpl. repeat f_equal; try lia. simpl in ×. congruence.
+ simpl. apply IHΓ'; simpl in *; (lia || congruence).
Qed.
Lemma nth_error_fold_context_k_eq:
∀ (Γ' : context) (v : nat) (f : nat → term → term),
nth_error (fold_context_k f Γ') v =
option_map (map_decl (f (length Γ' - S v))) (nth_error Γ' v).
Proof.
induction Γ'; intros.
- simpl. unfold fold_context_k; simpl. now rewrite nth_error_nil.
- simpl. destruct v; rewrite fold_context_k_snoc0.
+ simpl. repeat f_equal; try lia.
+ simpl. apply IHΓ'; simpl in *; (lia || congruence).
Qed.
Lemma nth_error_ge {Γ Γ' v Γ''} (f : nat → term → term) :
length Γ' ≤ v →
nth_error (Γ' ++ Γ) v =
nth_error (fold_context_k f Γ' ++ Γ'' ++ Γ) (length Γ'' + v).
Proof.
intros Hv.
rewrite → !nth_error_app_ge, ?fold_context_k_length. f_equal. lia.
rewrite fold_context_k_length. lia.
rewrite fold_context_k_length. lia. auto.
Qed.
Lemma nth_error_lt {Γ Γ' Γ'' v} (f : nat → term → term) :
v < length Γ' →
nth_error (fold_context_k f Γ' ++ Γ'' ++ Γ) v =
option_map (map_decl (f (length Γ' - S v))) (nth_error (Γ' ++ Γ) v).
Proof.
simpl. intros Hv.
rewrite → !nth_error_app_lt.
rewrite nth_error_fold_context_k_eq.
do 2 f_equal. lia. now rewrite fold_context_k_length.
Qed.
Lemma context_assumptions_length_bound Γ : context_assumptions Γ ≤ #|Γ|.
Proof.
induction Γ; simpl; auto. destruct a as [? [?|] ?]; simpl; auto.
lia.
Qed.
Lemma context_assumptions_map f Γ : context_assumptions (map_context f Γ) = context_assumptions Γ.
Proof.
induction Γ as [|[? [?|] ?] ?]; simpl; auto.
Qed.
Lemma context_assumptions_app Γ Δ : context_assumptions (Γ ++ Δ) =
context_assumptions Γ + context_assumptions Δ.
Proof.
induction Γ as [|[? [] ?] ?]; simpl; auto.
Qed.
Lemma context_assumptions_rev Γ : context_assumptions (List.rev Γ) = context_assumptions Γ.
Proof using Type.
induction Γ; simpl; auto. rewrite context_assumptions_app IHΓ /=.
destruct (decl_body a); simpl; lia.
Qed.
Lemma context_assumptions_mapi f Γ : context_assumptions (mapi (fun i ⇒ map_decl (f i)) Γ) =
context_assumptions Γ.
Proof.
rewrite /mapi; generalize 0.
induction Γ; simpl; intros; eauto.
destruct a as [? [b|] ?]; simpl; auto.
Qed.
#[global] Hint Rewrite context_assumptions_map context_assumptions_mapi context_assumptions_app : len.
Lemma context_assumptions_subst_instance u Γ :
context_assumptions (subst_instance u Γ) =
context_assumptions Γ.
Proof. apply context_assumptions_map. Qed.
Lemma context_assumptions_subst_context s k Γ :
context_assumptions (subst_context s k Γ) =
context_assumptions Γ.
Proof. apply context_assumptions_fold. Qed.
Lemma context_assumptions_lift_context n k Γ :
context_assumptions (lift_context n k Γ) =
context_assumptions Γ.
Proof. apply context_assumptions_fold. Qed.
#[global] Hint Rewrite context_assumptions_subst_instance
context_assumptions_subst_context context_assumptions_lift_context : len.
Lifting a relation to declarations, without alpha renaming.
Inductive All_decls (P : term → term → Type) : context_decl → context_decl → Type :=
| on_vass na t t' :
P t t' →
All_decls P (vass na t) (vass na t')
| on_vdef na b t b' t' :
P b b' →
P t t' →
All_decls P (vdef na b t) (vdef na b' t').
Derive Signature NoConfusion for All_decls.
| on_vass na t t' :
P t t' →
All_decls P (vass na t) (vass na t')
| on_vdef na b t b' t' :
P b b' →
P t t' →
All_decls P (vdef na b t) (vdef na b' t').
Derive Signature NoConfusion for All_decls.
Allow alpha-renaming of binders
Inductive All_decls_alpha (P : term → term → Type) : context_decl → context_decl → Type :=
| on_vass_alpha na na' t t' :
eq_binder_annot na na' →
P t t' →
All_decls_alpha P (vass na t) (vass na' t')
| on_vdef_alpha na na' b t b' t' :
eq_binder_annot na na' →
P b b' →
P t t' →
All_decls_alpha P (vdef na b t) (vdef na' b' t').
Derive Signature NoConfusion for All_decls_alpha.
Lemma All_decls_impl (P Q : term → term → Type) d d' :
All_decls P d d' →
(∀ t t', P t t' → Q t t') →
All_decls Q d d'.
Proof.
intros ond H; destruct ond; constructor; auto.
Qed.
Lemma All_decls_alpha_impl (P Q : term → term → Type) d d' :
All_decls_alpha P d d' →
(∀ t t', P t t' → Q t t') →
All_decls_alpha Q d d'.
Proof.
intros ond H; destruct ond; constructor; auto.
Qed.
Lemma All_decls_to_alpha (P : term → term → Type) d d' :
All_decls P d d' →
All_decls_alpha P d d'.
Proof.
intros []; constructor; auto; reflexivity.
Qed.
Definition All2_fold_over (P : context → context → context_decl → context_decl → Type) Γ Γ' :=
All2_fold (All_over P Γ Γ').
Notation on_decls P := (fun Γ Γ' ⇒ All_decls (P Γ Γ')).
Notation on_contexts P := (All2_fold (on_decls P)).
Notation on_contexts_over P Γ Γ' := (All2_fold (All_over (on_decls P) Γ Γ')).
End Environment.
Module Type EnvironmentSig (T : Term).
Include Environment T.
End EnvironmentSig.
Module Type EnvironmentDecide (T : Term) (Import E : EnvironmentSig T).
#[export] Declare Instance context_eq_dec : EqDec context.
#[export] Declare Instance constructor_body_eq_dec : EqDec constructor_body.
#[export] Declare Instance projection_body_eq_dec : EqDec projection_body.
#[export] Declare Instance one_inductive_body_eq_dec : EqDec one_inductive_body.
#[export] Declare Instance mutual_inductive_body_eq_dec : EqDec mutual_inductive_body.
#[export] Declare Instance constant_body_eq_dec : EqDec constant_body.
#[export] Declare Instance global_decl_eq_dec : EqDec global_decl.
End EnvironmentDecide.
Module Type TermUtils (T: Term) (E: EnvironmentSig T).
Import T E.
Parameter Inline destArity : context → term → option (context × Universe.t).
Parameter Inline inds : kername → Instance.t → list one_inductive_body → list term.
End TermUtils.
| on_vass_alpha na na' t t' :
eq_binder_annot na na' →
P t t' →
All_decls_alpha P (vass na t) (vass na' t')
| on_vdef_alpha na na' b t b' t' :
eq_binder_annot na na' →
P b b' →
P t t' →
All_decls_alpha P (vdef na b t) (vdef na' b' t').
Derive Signature NoConfusion for All_decls_alpha.
Lemma All_decls_impl (P Q : term → term → Type) d d' :
All_decls P d d' →
(∀ t t', P t t' → Q t t') →
All_decls Q d d'.
Proof.
intros ond H; destruct ond; constructor; auto.
Qed.
Lemma All_decls_alpha_impl (P Q : term → term → Type) d d' :
All_decls_alpha P d d' →
(∀ t t', P t t' → Q t t') →
All_decls_alpha Q d d'.
Proof.
intros ond H; destruct ond; constructor; auto.
Qed.
Lemma All_decls_to_alpha (P : term → term → Type) d d' :
All_decls P d d' →
All_decls_alpha P d d'.
Proof.
intros []; constructor; auto; reflexivity.
Qed.
Definition All2_fold_over (P : context → context → context_decl → context_decl → Type) Γ Γ' :=
All2_fold (All_over P Γ Γ').
Notation on_decls P := (fun Γ Γ' ⇒ All_decls (P Γ Γ')).
Notation on_contexts P := (All2_fold (on_decls P)).
Notation on_contexts_over P Γ Γ' := (All2_fold (All_over (on_decls P) Γ Γ')).
End Environment.
Module Type EnvironmentSig (T : Term).
Include Environment T.
End EnvironmentSig.
Module Type EnvironmentDecide (T : Term) (Import E : EnvironmentSig T).
#[export] Declare Instance context_eq_dec : EqDec context.
#[export] Declare Instance constructor_body_eq_dec : EqDec constructor_body.
#[export] Declare Instance projection_body_eq_dec : EqDec projection_body.
#[export] Declare Instance one_inductive_body_eq_dec : EqDec one_inductive_body.
#[export] Declare Instance mutual_inductive_body_eq_dec : EqDec mutual_inductive_body.
#[export] Declare Instance constant_body_eq_dec : EqDec constant_body.
#[export] Declare Instance global_decl_eq_dec : EqDec global_decl.
End EnvironmentDecide.
Module Type TermUtils (T: Term) (E: EnvironmentSig T).
Import T E.
Parameter Inline destArity : context → term → option (context × Universe.t).
Parameter Inline inds : kername → Instance.t → list one_inductive_body → list term.
End TermUtils.