initial definition of soundness theorems

coq/_CoqProject

@@ -6,5 +6,5 @@ subtype.v
 typing.v
 morph.v
 smallstep.v
+soundness.v
 bbencode.v
-

coq/soundness.v

@@ -0,0 +1,83 @@
+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.
+