ladder-calculus/coq/soundness.v

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From Coq Require Import Strings.String.
Require Import terms.
Require Import subst.
Require Import equiv.
Require Import subtype.
Require Import smallstep.
Require Import typing.
Include Terms.
Include Subst.
Include Equiv.
Include Subtype.
Include Smallstep.
Include Typing.
Module Soundness.
(* e is stuck when it is neither a value, nor can it be reduced *)
Definition is_stuck (e:expr_term) : Prop :=
~(is_value e) ->
~(exists e', e -->β e')
.
(* every exactly typed term is not stuck *)
Lemma exact_progress :
forall (e:expr_term),
(is_exactly_typed e) -> ~(is_stuck e)
.
Proof.
Admitted.
(* every well typed term is not stuck *)
Lemma progress :
forall (e:expr_term),
(is_well_typed e) -> ~(is_stuck e)
.
Proof.
Admitted.
(* reduction step preserves the type *)
Lemma exact_preservation :
forall Γ e e' τ,
(Γ |- e \is τ) ->
(e -->β e') ->
(Γ |- e' \is τ)
.
Proof.
(*
intros.
generalize dependent e'.
induction H.
intros e' I.
inversion I.
*)
Admitted.
(* reduction step preserves well-typedness *)
Lemma preservation :
forall Γ e e' τ,
(Γ |- e \compatible τ) ->
(e -->β e') ->
(Γ |- e' \compatible τ)
.
Proof.
Admitted.
(* every well-typed expression can be reduced to a value *)
Theorem soundness :
forall (e:expr_term),
(is_well_typed e) ->
(exists e', e -->β* e' /\ (is_value e'))
.
Proof.
intros.
Admitted.
End Soundness.