109 lines
2.9 KiB
Coq
109 lines
2.9 KiB
Coq
From Coq Require Import Strings.String.
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Require Import terms.
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Require Import subst.
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Require Import typing.
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Include Terms.
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Include Subst.
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Include Typing.
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Module Smallstep.
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Reserved Notation " s '-->α' t " (at level 40).
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Reserved Notation " s '-->β' t " (at level 40).
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Inductive expr_alpha : expr_term -> expr_term -> Prop :=
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| EAlpha_Rename : forall x x' τ e,
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(expr_abs x τ e) -->α (expr_abs x' τ (expr_subst x (expr_var x') e))
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| EAlpha_TyRename : forall α α' e,
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(expr_ty_abs α e) -->α (expr_ty_abs α' (expr_specialize α (type_var α') e))
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| EAlpha_SubAbs : forall x τ e e',
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(e -->α e') ->
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(expr_abs x τ e) -->α (expr_abs x τ e')
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| EAlpha_SubTyAbs : forall α e e',
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(e -->α e') ->
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(expr_ty_abs α e) -->α (expr_ty_abs α e')
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| EAlpha_SubApp1 : forall e1 e1' e2,
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(e1 -->α e1') ->
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(expr_app e1 e2) -->α (expr_app e1' e2)
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| EAlpha_SubApp2 : forall e1 e2 e2',
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(e2 -->α e2') ->
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(expr_app e1 e2) -->α (expr_app e1 e2')
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where "s '-->α' t" := (expr_alpha s t).
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Example a1 : polymorphic_identity1 -->α polymorphic_identity2.
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Proof.
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unfold polymorphic_identity1.
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unfold polymorphic_identity2.
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apply EAlpha_SubTyAbs.
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apply EAlpha_Rename.
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Qed.
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Inductive beta_step : expr_term -> expr_term -> Prop :=
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| E_App1 : forall e1 e1' e2,
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e1 -->β e1' ->
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(expr_app e1 e2) -->β (expr_app e1' e2)
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| E_App2 : forall e1 e2 e2',
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e2 -->β e2' ->
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(expr_app e1 e2) -->β (expr_app e1 e2')
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| E_TypApp : forall e e' τ,
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e -->β e' ->
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(expr_ty_app e τ) -->β (expr_ty_app e' τ)
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| E_TypAppLam : forall x e a,
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(expr_ty_app (expr_ty_abs x e) a) -->β (expr_specialize x a e)
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| E_AppLam : forall x τ e a,
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(expr_tm_app (expr_tm_abs x τ e) a) -->β (expr_subst x a e)
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| E_AppLet : forall x t e a,
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(expr_let x t a e) -->β (expr_subst x a e)
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where "s '-->β' t" := (beta_step s t).
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Inductive delta_step : expr_term -> expr_term -> Prop :=
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| E_ImplicitCast : forall (Γ:context) (f:expr_term) (h:string) (a:expr_term) (τ:type_term) (s:type_term) (p:type_term),
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(context_contains Γ h (type_morph p s)) ->
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Γ |- f \is (type_fun s τ) ->
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Γ |- a \is p ->
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(expr_tm_app f a) -->δ (expr_tm_app f (expr_tm_app (expr_var h) a))
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where "s '-->δ' t" := (delta_step s t).
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Inductive eval_step : expr_term -> expr_term -> Prop :=
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| E_Beta : forall s t,
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(s -->β t) ->
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(s -->eval t)
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| E_Delta : forall s t,
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(s -->δ t) ->
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(s -->eval t)
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where "s '-->eval' t" := (eval_step s t).
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Inductive multi {X : Type} (R : X -> X -> Prop) : X -> X -> Prop :=
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| Multi_Refl : forall (x : X), multi R x x
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| Multi_Step : forall (x y z : X),
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R x y ->
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multi R y z ->
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multi R x z.
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Notation " s -->α* t " := (multi expr_alpha s t) (at level 40).
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Notation " s -->β* t " := (multi beta_step s t) (at level 40).
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End Smallstep.
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