ladder-calculus/coq/smallstep.v

88 lines
2.3 KiB
Coq
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

From Coq Require Import Strings.String.
Require Import terms.
Require Import subst.
Require Import typing.
Include Terms.
Include Subst.
Include Typing.
Module Smallstep.
Reserved Notation " s '-->α' t " (at level 40).
Reserved Notation " s '-->β' t " (at level 40).
Inductive expr_alpha : expr_term -> expr_term -> Prop :=
| EAlpha_Rename : forall x x' τ e,
(expr_abs x τ e) -->α (expr_abs x' τ (expr_subst x (expr_var x') e))
| EAlpha_TyRename : forall α α' e,
(expr_ty_abs α e) -->α (expr_ty_abs α' (expr_specialize α (type_var α') e))
| EAlpha_SubAbs : forall x τ e e',
(e -->α e') ->
(expr_abs x τ e) -->α (expr_abs x τ e')
| EAlpha_SubTyAbs : forall α e e',
(e -->α e') ->
(expr_ty_abs α e) -->α (expr_ty_abs α e')
| EAlpha_SubApp1 : forall e1 e1' e2,
(e1 -->α e1') ->
(expr_app e1 e2) -->α (expr_app e1' e2)
| EAlpha_SubApp2 : forall e1 e2 e2',
(e2 -->α e2') ->
(expr_app e1 e2) -->α (expr_app e1 e2')
where "s '-->α' t" := (expr_alpha s t).
Example a1 : polymorphic_identity1 -->α polymorphic_identity2.
Proof.
unfold polymorphic_identity1.
unfold polymorphic_identity2.
apply EAlpha_SubTyAbs.
apply EAlpha_Rename.
Qed.
Inductive beta_step : expr_term -> expr_term -> Prop :=
| E_App1 : forall e1 e1' e2,
e1 -->β e1' ->
(expr_app e1 e2) -->β (expr_app e1' e2)
| E_App2 : forall v1 e2 e2',
(is_value v1) ->
e2 -->β e2' ->
(expr_app v1 e2) -->β (expr_app v1 e2')
| E_TypApp : forall e e' τ,
e -->β e' ->
(expr_ty_app e τ) -->β (expr_ty_app e' τ)
| E_TypAppLam : forall x e a,
(expr_ty_app (expr_ty_abs x e) a) -->β (expr_specialize x a e)
| E_AppLam : forall x τ e a,
(expr_app (expr_abs x τ e) a) -->β (expr_subst x a e)
| E_AppMorph : forall x τ e a,
(expr_app (expr_morph x τ e) a) -->β (expr_subst x a e)
| E_AppLet : forall x t e a,
(expr_let x t a e) -->β (expr_subst x a e)
where "s '-->β' t" := (beta_step s t).
Inductive multi {X : Type} (R : X -> X -> Prop) : X -> X -> Prop :=
| Multi_Refl : forall (x : X), multi R x x
| Multi_Step : forall (x y z : X),
R x y ->
multi R y z ->
multi R x z.
Notation " s -->α* t " := (multi expr_alpha s t) (at level 40).
Notation " s -->β* t " := (multi beta_step s t) (at level 40).
End Smallstep.