146 lines
3.1 KiB
Coq
146 lines
3.1 KiB
Coq
From Coq Require Import Lists.List.
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Require Import Atom.
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Require Import Environment.
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Require Import Metatheory.
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Require Import debruijn.
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Require Import subtype.
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Require Import env.
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Require Import morph.
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Require Import subst_lemmas.
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Require Import typing.
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Require Import typing_weakening.
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Require Import typing_regular.
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Require Import translate_morph.
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Lemma typing_inv_tabs : forall Γ t τ,
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(Γ |- [{ Λ t }] \is [< ∀ τ >]) ->
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forall L x, x `notin` L ->
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(Γ |- (expr_open_type (ty_fvar x) t) \is τ)
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.
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Proof.
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Admitted.
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Lemma typing_inv_abs : forall Γ σ t τ,
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(Γ |- [{ λ σ ↦ t }] \is [< σ -> τ >]) ->
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forall L x, x `notin` L ->
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((x,σ)::Γ |- (expr_open (ex_fvar x) t) \is τ)
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.
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Proof.
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Admitted.
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Lemma typing_inv_morph : forall Γ σ t τ,
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(Γ |- [{ λ σ ↦morph t }] \is [< σ ->morph τ >]) ->
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forall L x, x `notin` L ->
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((x,σ)::Γ |- (expr_open (ex_fvar x) t) \is τ)
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.
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Proof.
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intros.
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inversion H.
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subst.
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Admitted.
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Lemma typing_inv_let : forall Γ s σ t τ,
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(Γ |- s \is σ) ->
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(Γ |- [{ let s in t }] \is [< τ >]) ->
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forall L x, x `notin` L ->
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((x,σ)::Γ |- (expr_open (ex_fvar x) t) \is τ)
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.
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Proof.
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Admitted.
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(*
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* translated morphism path has valid typing
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*)
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Lemma transl_preservation : forall Γ e e' τ,
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(Γ |- e \is τ) ->
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(Γ |- [[ e \is τ ]] = e') ->
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(Γ |- e' \is τ)
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.
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Proof.
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intros Γ e e' τ Typing Transl.
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induction Transl.
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(* free var *)
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- assumption.
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(* let *)
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- apply T_Let with (L:=L) (σ:=σ).
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* apply IHTransl.
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assumption.
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* intros x Fr.
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apply H1.
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assumption.
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apply typing_inv_let with (L:=L) (s:=e).
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1-3:assumption.
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apply typing_inv_let with (L:=L) (s:=e).
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1-3:assumption.
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(* type abs *)
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- apply T_TypeAbs with (L:=L).
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intros x Fr.
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apply H0.
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assumption.
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apply typing_inv_tabs with (L:=L).
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1-2:assumption.
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apply typing_inv_tabs with (L:=L).
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1-2:assumption.
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(* type app *)
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- apply T_TypeApp.
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apply IHTransl.
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assumption.
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(* abs *)
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- apply T_Abs with (L:=L).
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intros x Fr.
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apply H1.
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assumption.
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apply typing_inv_abs with (L:=L).
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1-2:assumption.
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apply typing_inv_abs with (L:=L).
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1-2:assumption.
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(* morph abs *)
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- apply T_MorphAbs with (L:=L).
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intros x Fr.
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apply H1.
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assumption.
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apply typing_inv_morph with (L:=L).
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1-2:assumption.
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apply typing_inv_morph with (L:=L).
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1-2:assumption.
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(* app *)
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- apply T_App with (σ':=σ) (σ:=σ); auto.
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apply T_App with (σ':=σ') (σ:=σ'); auto.
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2-3: apply id_morphism_path.
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apply T_MorphFun.
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apply morphism_path_correct with (τ:=σ') (τ':=σ).
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3: assumption.
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2:admit. (* env wf *)
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apply typing_regular_type_lc with (Γ:=Γ) (e:=a).
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assumption.
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apply typing_regular_type_lc with (Γ:=Γ) (e:=a).
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assumption.
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apply morph_regular_lc with (Γ:=Γ) (τ:=σ') (τ':=σ).
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admit. (* env wf *)
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apply typing_regular_type_lc with (Γ:=Γ) (e:=a).
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assumption.
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assumption.
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- auto with typing_hints.
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- auto with typing_hints.
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- eauto with typing_hints.
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Admitted.
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