(** * Typing Environment Definition and Function Manipulation.*) Require Import base ut_term. Require Import List. Require Import Peano_dec. Require Import Compare_dec. Require Import Lt Le Gt. Module Type ut_env_mod (X: term_sig) (TM: ut_term_mod X). Import TM. (** Since we use de Bruijn indexes, Environment (or Context) are simply lists of terms: Γ(x:A) is encoded as [A::Γ]. *) Definition Env := list Term. (** Some manipulation functions (mostly from Bruno Barras' PTS contrib): - how to find an item in the environment - how to truncate an environment - how to insert a new element (with correct de Bruijn update) - how to substitute something in the environment *) Set Implicit Arguments. Inductive item (A:Type) (x:A): list A ->nat->Prop := | item_hd: forall Γ :list A, (item x (cons x Γ) O) | item_tl: forall (Γ:list A)(n:nat)(y:A), item x Γ n -> item x (cons y Γ) (S n). Hint Constructors item. (** In the list [Γ], the [n]th item is syntacticaly [x]. *) Notation " x ↓ n ∈ Γ " := (item x Γ n) (at level 80, no associativity) : UT_scope. Lemma fun_item: forall T (A B:T)(Γ:list T)(n:nat), A ↓ n ∈ Γ -> B ↓ n ∈ Γ -> A = B. intros T A B Γ n;revert T A B Γ. induction n; intros. inversion H; inversion H0. rewrite <- H2 in H1; injection H1; trivial. inversion H; inversion H0; subst. injection H5; intros; subst. apply IHn with (Γ:=Γ0); trivial. Qed. Inductive trunc (A:Type) : nat->list A ->list A->Prop := trunc_O: forall (Γ:list A) , (trunc O Γ Γ) | trunc_S: forall (k:nat)(Γ Γ':list A)(x:A), trunc k Γ Γ' -> trunc (S k) (cons x Γ) Γ'. Hint Constructors trunc. Lemma item_trunc: forall (T:Type) (n:nat) (Γ:list T) (t:T), t ↓ n ∈ Γ -> exists Γ' , trunc (S n) Γ Γ'. intros T n; induction n; intros. inversion H. exists Γ0. apply trunc_S; apply trunc_O. inversion H; subst. destruct (IHn Γ0 t H2). exists x. apply trunc_S. apply H0. Qed. (** This type describe how do we add an element in an environment: no type checking is done, this is just the mecanic way to do it. *) Inductive ins_in_env (Γ:Env ) (d1:Term): nat->Env -> Env ->Prop := | ins_O: ins_in_env Γ d1 O Γ (d1::Γ) | ins_S: forall (n:nat)(Δ Δ':Env )(d:Term), (ins_in_env Γ d1 n Δ Δ') -> ins_in_env Γ d1 (S n) (d::Δ) ( (d ↑ 1 # n)::Δ' ). Hint Constructors ins_in_env. (** Some lemmas about inserting a new element. They explain how terms in the environment are lifted according to their original position and the position of insertion. *) Lemma ins_item_ge: forall (d':Term) (n:nat) (Γ Δ Δ':Env), ins_in_env Γ d' n Δ Δ' -> forall (v:nat), n<=v -> forall (d:Term), d ↓ v ∈ Δ -> d ↓ (S v) ∈ Δ'. induction n; intros. inversion H; subst. apply item_tl. exact H1. inversion H; subst. apply item_tl. destruct v. elim (le_Sn_O n H0). apply IHn with (Γ:=Γ) (Δ:=Δ0). trivial. apply le_S_n ; trivial. inversion H1. exact H4. Qed. Lemma gt_O: forall v, ~ 0 > v. intros; intro. unfold gt in H. apply lt_n_O in H; trivial. Qed. Lemma ins_item_lt: forall (d':Term)(n:nat)(Γ Δ Δ':Env), ins_in_env Γ d' n Δ Δ' -> forall (v:nat), n > v -> forall (d:Term), d ↓ v ∈ Δ -> (d ↑ 1 # (n-S v)) ↓ v ∈ Δ' . induction n; intros. elim (gt_O H0). inversion H; subst. destruct v. inversion H1; subst. replace (S n -1) with n by intuition. apply item_hd. apply item_tl. replace (S n - S (S v)) with (n -S v) by intuition. apply IHn with (Γ:=Γ) (Δ:=Δ0). exact H3. intuition. inversion H1. exact H4. Qed. (** In the list [Γ], [t] is [n]th element correctly lifted according to [Γ]: e.g.: if t ↓ n ⊂ Γ and we insert something in Γ, then we can still speak about t without think of the lift hidden by the insertion. *) Definition item_lift (t:Term) (Γ:Env) (n:nat) := exists u , t= u ↑ (S n) /\ u ↓ n ∈ Γ . Hint Unfold item_lift. Notation " t ↓ n ⊂ Γ " := (item_lift t Γ n) (at level 80, no associativity): UT_scope. (** Properties of the item_lift notation.*) Lemma ins_item_lift_lt: forall (d':Term)(n:nat)(Γ Δ Δ':Env ), ins_in_env Γ d' n Δ Δ' -> forall (v:nat), n>v -> forall (t:Term), t ↓ v ⊂ Δ -> (t ↑ 1 # n) ↓ v ⊂ Δ'. intros. destruct H1 as [u [ P Q]]. rewrite P. exists (u ↑ 1 # (n - S v) ); split. replace n with ( S v + (n - S v)) by intuition . rewrite liftP2. replace (S v+(n-S v)-S v) with (n-S v) by intuition. reflexivity. intuition. clear t P. inversion H; subst. elim (gt_O H0). inversion Q; subst. replace (S n0 -1) with n0 by intuition. apply item_hd. apply item_tl. replace (S n0 - S (S n)) with (n0 -S n) by intuition. apply ins_item_lt with d' Γ Δ0; trivial. intuition. Qed. (** This type describe how do we do substitution inside a context. As previously, no type checking is done at this point.*) Inductive sub_in_env (Γ : Env) (t T:Term): nat -> Env -> Env -> Prop := | sub_O : sub_in_env Γ t T 0 (T :: Γ) Γ | sub_S : forall Δ Δ' n B, sub_in_env Γ t T n Δ Δ' -> sub_in_env Γ t T (S n) (B :: Δ) ( B [n← t] :: Δ'). Hint Constructors sub_in_env. (** Some ins / subst related facts: what happens to term when we do a substitution in a context.*) Lemma nth_sub_sup : forall n Γ Δ Δ' t T, sub_in_env Γ t T n Δ Δ' -> forall v : nat, n <= v -> forall d , d ↓ (S v) ∈ Δ -> d ↓ v ∈ Δ'. intros n Γ Δ Δ' t T H; induction H; intros. inversion H0; subst. trivial. inversion H1; subst. destruct v. elim (le_Sn_O n H0). apply item_tl. apply le_S_n in H0. apply IHsub_in_env; trivial. Qed. Lemma nth_sub_eq : forall t T n Γ Δ Δ', sub_in_env Γ t T n Δ Δ' -> forall d , d↓ n ∈ Δ -> T = d. intros t T n Γ Δ Δ' H; induction H; intros. inversion H; trivial. inversion H0; subst. apply IHsub_in_env; trivial. Qed. Lemma nth_sub_inf : forall t T n Γ Δ Δ', sub_in_env Γ t T n Δ Δ' -> forall v : nat, n > v -> forall d , d ↓ v ∈ Δ -> ( d [n - S v ← t] )↓ v ∈ Δ' . intros t T n Γ Δ Δ' H; induction H; intros. elim (gt_O H). destruct v. inversion H1; subst. replace (S n -1) with n by intuition. apply item_hd. replace (S n - S (S v)) with (n - S v) by intuition. inversion H1; subst. apply item_tl. apply gt_S_n in H0. apply IHsub_in_env; trivial. Qed. Lemma nth_sub_item_inf : forall t T n g e f , sub_in_env g t T n e f -> forall v : nat, n > v -> forall u , item_lift u e v -> item_lift (subst_rec t u n) f v. intros. destruct H1 as [y [K L]]. exists (subst_rec t y (n-S v)); split. rewrite K; clear u K. pattern n at 1 . replace n with (S v + ( n - S v)) by intuition. apply substP2; intuition. apply nth_sub_inf with T g e; trivial. Qed. End ut_env_mod.