88 lines
2.4 KiB
Coq
88 lines
2.4 KiB
Coq
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).
|
||
Reserved Notation " s '-->δ' t " (at level 40).
|
||
Reserved Notation " s '-->eval' t " (at level 40).
|
||
|
||
Inductive alpha_step : expr_term -> expr_term -> Prop :=
|
||
| E_Rename : forall x x' e,
|
||
(expr_tm_abs x e) -->α (expr_tm_abs x' (expr_subst x (type_var x')))
|
||
where "s '-->α' t" := (alpha_step s t).
|
||
|
||
|
||
Example a1 : polymorphic_identity1 -->α polymorphic_identity2.
|
||
Proof.
|
||
Qed.
|
||
|
||
|
||
Inductive beta_step : expr_term -> expr_term -> Prop :=
|
||
| E_App1 : forall e1 e1' e2,
|
||
e1 -->β e1' ->
|
||
(expr_tm_app e1 e2) -->β (expr_tm_app e1' e2)
|
||
|
||
| E_App2 : forall e1 e2 e2',
|
||
e2 -->β e2' ->
|
||
(expr_tm_app e1 e2) -->β (expr_tm_app e1 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_tm_app (expr_tm_abs 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 delta_step : expr_term -> expr_term -> Prop :=
|
||
|
||
| E_ImplicitCast : forall (Γ:context) (f:expr_term) (h:string) (a:expr_term) (τ:type_term) (s:type_term) (p:type_term),
|
||
|
||
(context_contains Γ h (type_morph p s)) ->
|
||
Γ |- f \is (type_fun s τ) ->
|
||
Γ |- a \is p ->
|
||
(expr_tm_app f a) -->δ (expr_tm_app f (expr_tm_app (expr_var h) a))
|
||
|
||
where "s '-->δ' t" := (delta_step s t).
|
||
|
||
|
||
Inductive eval_step : expr_term -> expr_term -> Prop :=
|
||
| E_Beta : forall s t,
|
||
(s -->β t) ->
|
||
(s -->eval t)
|
||
|
||
| E_Delta : forall s t,
|
||
(s -->δ t) ->
|
||
(s -->eval t)
|
||
|
||
where "s '-->eval' t" := (eval_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 -->β* t " := (multi beta_step s t) (at level 40).
|
||
Notation " s -->δ* t " := (multi delta_step s t) (at level 40).
|
||
Notation " s -->eval* t " := (multi eval_step s t) (at level 40).
|
||
|
||
End Smallstep.
|