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RecordSub.v
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RecordSub.v
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(** * RecordSub: Subtyping with Records *)
Require Export MoreStlc.
(* ###################################################### *)
(** * Core Definitions *)
(* ################################### *)
(** *** Syntax *)
Inductive ty : Type :=
(* proper types *)
| TTop : ty
| TBase : id -> ty
| TArrow : ty -> ty -> ty
(* record types *)
| TRNil : ty
| TRCons : id -> ty -> ty -> ty.
Tactic Notation "T_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "TTop" | Case_aux c "TBase" | Case_aux c "TArrow"
| Case_aux c "TRNil" | Case_aux c "TRCons" ].
Inductive tm : Type :=
(* proper terms *)
| tvar : id -> tm
| tapp : tm -> tm -> tm
| tabs : id -> ty -> tm -> tm
| tproj : tm -> id -> tm
(* record terms *)
| trnil : tm
| trcons : id -> tm -> tm -> tm.
Tactic Notation "t_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "tvar" | Case_aux c "tapp" | Case_aux c "tabs"
| Case_aux c "tproj" | Case_aux c "trnil" | Case_aux c "trcons" ].
(* ################################### *)
(** *** Well-Formedness *)
Inductive record_ty : ty -> Prop :=
| RTnil :
record_ty TRNil
| RTcons : forall i T1 T2,
record_ty (TRCons i T1 T2).
Inductive record_tm : tm -> Prop :=
| rtnil :
record_tm trnil
| rtcons : forall i t1 t2,
record_tm (trcons i t1 t2).
Inductive well_formed_ty : ty -> Prop :=
| wfTTop :
well_formed_ty TTop
| wfTBase : forall i,
well_formed_ty (TBase i)
| wfTArrow : forall T1 T2,
well_formed_ty T1 ->
well_formed_ty T2 ->
well_formed_ty (TArrow T1 T2)
| wfTRNil :
well_formed_ty TRNil
| wfTRCons : forall i T1 T2,
well_formed_ty T1 ->
well_formed_ty T2 ->
record_ty T2 ->
well_formed_ty (TRCons i T1 T2).
Hint Constructors record_ty record_tm well_formed_ty.
(* ################################### *)
(** *** Substitution *)
Fixpoint subst (x:id) (s:tm) (t:tm) : tm :=
match t with
| tvar y => if eq_id_dec x y then s else t
| tabs y T t1 => tabs y T (if eq_id_dec x y then t1 else (subst x s t1))
| tapp t1 t2 => tapp (subst x s t1) (subst x s t2)
| tproj t1 i => tproj (subst x s t1) i
| trnil => trnil
| trcons i t1 tr2 => trcons i (subst x s t1) (subst x s tr2)
end.
Notation "'[' x ':=' s ']' t" := (subst x s t) (at level 20).
(* ################################### *)
(** *** Reduction *)
Inductive value : tm -> Prop :=
| v_abs : forall x T t,
value (tabs x T t)
| v_rnil : value trnil
| v_rcons : forall i v vr,
value v ->
value vr ->
value (trcons i v vr).
Hint Constructors value.
Fixpoint Tlookup (i:id) (Tr:ty) : option ty :=
match Tr with
| TRCons i' T Tr' => if eq_id_dec i i' then Some T else Tlookup i Tr'
| _ => None
end.
Fixpoint tlookup (i:id) (tr:tm) : option tm :=
match tr with
| trcons i' t tr' => if eq_id_dec i i' then Some t else tlookup i tr'
| _ => None
end.
Reserved Notation "t1 '==>' t2" (at level 40).
Inductive step : tm -> tm -> Prop :=
| ST_AppAbs : forall x T t12 v2,
value v2 ->
(tapp (tabs x T t12) v2) ==> [x:=v2]t12
| ST_App1 : forall t1 t1' t2,
t1 ==> t1' ->
(tapp t1 t2) ==> (tapp t1' t2)
| ST_App2 : forall v1 t2 t2',
value v1 ->
t2 ==> t2' ->
(tapp v1 t2) ==> (tapp v1 t2')
| ST_Proj1 : forall tr tr' i,
tr ==> tr' ->
(tproj tr i) ==> (tproj tr' i)
| ST_ProjRcd : forall tr i vi,
value tr ->
tlookup i tr = Some vi ->
(tproj tr i) ==> vi
| ST_Rcd_Head : forall i t1 t1' tr2,
t1 ==> t1' ->
(trcons i t1 tr2) ==> (trcons i t1' tr2)
| ST_Rcd_Tail : forall i v1 tr2 tr2',
value v1 ->
tr2 ==> tr2' ->
(trcons i v1 tr2) ==> (trcons i v1 tr2')
where "t1 '==>' t2" := (step t1 t2).
Tactic Notation "step_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "ST_AppAbs" | Case_aux c "ST_App1" | Case_aux c "ST_App2"
| Case_aux c "ST_Proj1" | Case_aux c "ST_ProjRcd" | Case_aux c "ST_Rcd"
| Case_aux c "ST_Rcd_Head" | Case_aux c "ST_Rcd_Tail" ].
Hint Constructors step.
(* ###################################################################### *)
(** * Subtyping *)
(** Now we come to the interesting part. We begin by defining
the subtyping relation and developing some of its important
technical properties. *)
(* ################################### *)
(** ** Definition *)
(** The definition of subtyping is essentially just what we
sketched in the motivating discussion, but we need to add
well-formedness side conditions to some of the rules. *)
Inductive subtype : ty -> ty -> Prop :=
(* Subtyping between proper types *)
| S_Refl : forall T,
well_formed_ty T ->
subtype T T
| S_Trans : forall S U T,
subtype S U ->
subtype U T ->
subtype S T
| S_Top : forall S,
well_formed_ty S ->
subtype S TTop
| S_Arrow : forall S1 S2 T1 T2,
subtype T1 S1 ->
subtype S2 T2 ->
subtype (TArrow S1 S2) (TArrow T1 T2)
(* Subtyping between record types *)
| S_RcdWidth : forall i T1 T2,
well_formed_ty (TRCons i T1 T2) ->
subtype (TRCons i T1 T2) TRNil
| S_RcdDepth : forall i S1 T1 Sr2 Tr2,
subtype S1 T1 ->
subtype Sr2 Tr2 ->
record_ty Sr2 ->
record_ty Tr2 ->
subtype (TRCons i S1 Sr2) (TRCons i T1 Tr2)
| S_RcdPerm : forall i1 i2 T1 T2 Tr3,
well_formed_ty (TRCons i1 T1 (TRCons i2 T2 Tr3)) ->
i1 <> i2 ->
subtype (TRCons i1 T1 (TRCons i2 T2 Tr3))
(TRCons i2 T2 (TRCons i1 T1 Tr3)).
Hint Constructors subtype.
Tactic Notation "subtype_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "S_Refl" | Case_aux c "S_Trans" | Case_aux c "S_Top"
| Case_aux c "S_Arrow" | Case_aux c "S_RcdWidth"
| Case_aux c "S_RcdDepth" | Case_aux c "S_RcdPerm" ].
(* ############################################### *)
(** ** Subtyping Examples and Exercises *)
Module Examples.
Notation x := (Id 0).
Notation y := (Id 1).
Notation z := (Id 2).
Notation j := (Id 3).
Notation k := (Id 4).
Notation i := (Id 5).
Notation A := (TBase (Id 6)).
Notation B := (TBase (Id 7)).
Notation C := (TBase (Id 8)).
Definition TRcd_j :=
(TRCons j (TArrow B B) TRNil). (* {j:B->B} *)
Definition TRcd_kj :=
TRCons k (TArrow A A) TRcd_j. (* {k:C->C,j:B->B} *)
Example subtyping_example_0 :
subtype (TArrow C TRcd_kj)
(TArrow C TRNil).
(* C->{k:A->A,j:B->B} <: C->{} *)
Proof.
apply S_Arrow.
apply S_Refl. auto.
unfold TRcd_kj, TRcd_j. apply S_RcdWidth; auto.
Qed.
(** The following facts are mostly easy to prove in Coq. To get
full benefit from the exercises, make sure you also
understand how to prove them on paper! *)
(** **** Exercise: 2 stars *)
Hint Extern 2 (_ <> _) => intros contra; inversion contra.
Example subtyping_example_1 :
subtype TRcd_kj TRcd_j.
(* {k:A->A,j:B->B} <: {j:B->B} *)
Proof with eauto.
unfold TRcd_kj, TRcd_j.
eapply S_Trans. apply S_RcdPerm...
apply S_RcdDepth...
Qed.
(** [] *)
(** **** Exercise: 1 star *)
Example subtyping_example_2 :
subtype (TArrow TTop TRcd_kj)
(TArrow (TArrow C C) TRcd_j).
(* Top->{k:A->A,j:B->B} <: (C->C)->{j:B->B} *)
Proof with eauto using subtyping_example_1.
apply S_Arrow...
Qed.
(** [] *)
(** **** Exercise: 1 star *)
Example subtyping_example_3 :
subtype (TArrow TRNil (TRCons j A TRNil))
(TArrow (TRCons k B TRNil) TRNil).
(* {}->{j:A} <: {k:B}->{} *)
Proof with eauto.
apply S_Arrow...
Qed.
(** [] *)
(** **** Exercise: 2 stars *)
Example subtyping_example_4 :
subtype (TRCons x A (TRCons y B (TRCons z C TRNil)))
(TRCons z C (TRCons y B (TRCons x A TRNil))).
(* {x:A,y:B,z:C} <: {z:C,y:B,x:A} *)
Proof with eauto.
apply S_Trans with (TRCons y B (TRCons x A (TRCons z C TRNil))).
apply S_RcdPerm...
apply S_Trans with (TRCons y B (TRCons z C (TRCons x A TRNil))).
apply S_RcdDepth...
apply S_RcdPerm...
Qed.
(** [] *)
Definition trcd_kj :=
(trcons k (tabs z A (tvar z))
(trcons j (tabs z B (tvar z))
trnil)).
End Examples.
(* ###################################################################### *)
(** ** Properties of Subtyping *)
(** *** Well-Formedness *)
Lemma subtype__wf : forall S T,
subtype S T ->
well_formed_ty T /\ well_formed_ty S.
Proof with eauto.
intros S T Hsub.
subtype_cases (induction Hsub) Case;
intros; try (destruct IHHsub1; destruct IHHsub2)...
Case "S_RcdPerm".
split... inversion H. subst. inversion H5... Qed.
Lemma wf_rcd_lookup : forall i T Ti,
well_formed_ty T ->
Tlookup i T = Some Ti ->
well_formed_ty Ti.
Proof with eauto.
intros i T.
T_cases (induction T) Case; intros; try solve by inversion.
Case "TRCons".
inversion H. subst. unfold Tlookup in H0.
destruct (eq_id_dec i i0)... inversion H0; subst... Qed.
(** *** Field Lookup *)
(** Our record matching lemmas get a little more complicated in
the presence of subtyping for two reasons: First, record
types no longer necessarily describe the exact structure of
corresponding terms. Second, reasoning by induction on
[has_type] derivations becomes harder in general, because
[has_type] is no longer syntax directed. *)
Lemma rcd_types_match : forall S T i Ti,
subtype S T ->
Tlookup i T = Some Ti ->
exists Si, Tlookup i S = Some Si /\ subtype Si Ti.
Proof with (eauto using wf_rcd_lookup).
intros S T i Ti Hsub Hget. generalize dependent Ti.
subtype_cases (induction Hsub) Case; intros Ti Hget;
try solve by inversion.
Case "S_Refl".
exists Ti...
Case "S_Trans".
destruct (IHHsub2 Ti) as [Ui Hui]... destruct Hui.
destruct (IHHsub1 Ui) as [Si Hsi]... destruct Hsi.
exists Si...
Case "S_RcdDepth".
rename i0 into k.
unfold Tlookup. unfold Tlookup in Hget.
destruct (eq_id_dec i k)...
SCase "i = k -- we're looking up the first field".
inversion Hget. subst. exists S1...
Case "S_RcdPerm".
exists Ti. split.
SCase "lookup".
unfold Tlookup. unfold Tlookup in Hget.
destruct (eq_id_dec i i1)...
SSCase "i = i1 -- we're looking up the first field".
destruct (eq_id_dec i i2)...
SSSCase "i = i2 - -contradictory".
destruct H0.
subst...
SCase "subtype".
inversion H. subst. inversion H5. subst... Qed.
(** **** Exercise: 3 stars (rcd_types_match_informal) *)
(** Write a careful informal proof of the [rcd_types_match]
lemma. *)
(* ... *)
(** [] *)
(** *** Inversion Lemmas *)
(** **** Exercise: 3 stars, optional (sub_inversion_arrow) *)
Lemma sub_inversion_arrow : forall U V1 V2,
subtype U (TArrow V1 V2) ->
exists U1, exists U2,
(U=(TArrow U1 U2)) /\ (subtype V1 U1) /\ (subtype U2 V2).
Proof with eauto.
intros U V1 V2 Hs.
remember (TArrow V1 V2) as V.
generalize dependent V2. generalize dependent V1.
induction Hs; intros; try solve by inversion.
subst. inversion H. exists V1. exists V2...
apply IHHs2 in HeqV. destruct HeqV as [W1 [W2 [H1 [H2 H3]]]].
apply IHHs1 in H1. destruct H1 as [X1 [X2 [H4 [H5 H6]]]].
exists X1. exists X2...
inversion HeqV; subst. exists S1. exists S2...
Qed.
(** [] *)
(* ###################################################################### *)
(** * Typing *)
Definition context := id -> (option ty).
Definition empty : context := (fun _ => None).
Definition extend (Gamma : context) (x:id) (T : ty) :=
fun x' => if eq_id_dec x x' then Some T else Gamma x'.
Reserved Notation "Gamma '|-' t '\in' T" (at level 40).
Inductive has_type : context -> tm -> ty -> Prop :=
| T_Var : forall Gamma x T,
Gamma x = Some T ->
well_formed_ty T ->
has_type Gamma (tvar x) T
| T_Abs : forall Gamma x T11 T12 t12,
well_formed_ty T11 ->
has_type (extend Gamma x T11) t12 T12 ->
has_type Gamma (tabs x T11 t12) (TArrow T11 T12)
| T_App : forall T1 T2 Gamma t1 t2,
has_type Gamma t1 (TArrow T1 T2) ->
has_type Gamma t2 T1 ->
has_type Gamma (tapp t1 t2) T2
| T_Proj : forall Gamma i t T Ti,
has_type Gamma t T ->
Tlookup i T = Some Ti ->
has_type Gamma (tproj t i) Ti
(* Subsumption *)
| T_Sub : forall Gamma t S T,
has_type Gamma t S ->
subtype S T ->
has_type Gamma t T
(* Rules for record terms *)
| T_RNil : forall Gamma,
has_type Gamma trnil TRNil
| T_RCons : forall Gamma i t T tr Tr,
has_type Gamma t T ->
has_type Gamma tr Tr ->
record_ty Tr ->
record_tm tr ->
has_type Gamma (trcons i t tr) (TRCons i T Tr)
where "Gamma '|-' t '\in' T" := (has_type Gamma t T).
Hint Constructors has_type.
Tactic Notation "has_type_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "T_Var" | Case_aux c "T_Abs" | Case_aux c "T_App"
| Case_aux c "T_Proj" | Case_aux c "T_Sub"
| Case_aux c "T_RNil" | Case_aux c "T_RCons" ].
(* ############################################### *)
(** ** Typing Examples *)
Module Examples2.
Import Examples.
Hint Extern 2 (has_type _ (trcons _ _ _) _) =>
eapply T_RCons; auto.
Hint Extern 2 (has_type _ (tapp _ _) _) =>
eapply T_App; auto.
Hint Extern 2 (has_type _ (tabs _ _ _) _) =>
eapply T_Abs; auto.
Hint Extern 2 (has_type _ (tproj _ _) _) =>
eapply T_Proj; auto.
Hint Extern 2 (_ = _) => compute; reflexivity.
(** **** Exercise: 1 star *)
Example typing_example_0 :
has_type empty
(trcons k (tabs z A (tvar z))
(trcons j (tabs z B (tvar z))
trnil))
TRcd_kj.
(* empty |- {k=(\z:A.z), j=(\z:B.z)} : {k:A->A,j:B->B} *)
Proof with eauto.
unfold TRcd_kj, TRcd_j...
Qed.
(** [] *)
(** **** Exercise: 2 stars *)
Example typing_example_1 :
has_type empty
(tapp (tabs x TRcd_j (tproj (tvar x) j))
(trcd_kj))
(TArrow B B).
(* empty |- (\x:{k:A->A,j:B->B}. x.j) {k=(\z:A.z), j=(\z:B.z)} : B->B *)
Proof with eauto using subtyping_example_1.
unfold trcd_kj, TRcd_j...
Qed.
(** [] *)
(** **** Exercise: 2 stars, optional *)
Example typing_example_2 :
has_type empty
(tapp (tabs z (TArrow (TArrow C C) TRcd_j)
(tproj (tapp (tvar z)
(tabs x C (tvar x)))
j))
(tabs z (TArrow C C) trcd_kj))
(TArrow B B).
(* empty |- (\z:(C->C)->{j:B->B}. (z (\x:C.x)).j)
(\z:C->C. {k=(\z:A.z), j=(\z:B.z)})
: B->B *)
Proof with eauto using subtyping_example_1.
unfold trcd_kj, TRcd_j...
Qed.
(** [] *)
End Examples2.
(* ###################################################################### *)
(** ** Properties of Typing *)
(** *** Well-Formedness *)
Lemma has_type__wf : forall Gamma t T,
has_type Gamma t T -> well_formed_ty T.
Proof with eauto.
intros Gamma t T Htyp.
has_type_cases (induction Htyp) Case...
Case "T_App".
inversion IHHtyp1...
Case "T_Proj".
eapply wf_rcd_lookup...
Case "T_Sub".
apply subtype__wf in H.
destruct H...
Qed.
Lemma step_preserves_record_tm : forall tr tr',
record_tm tr ->
tr ==> tr' ->
record_tm tr'.
Proof.
intros tr tr' Hrt Hstp.
inversion Hrt; subst; inversion Hstp; subst; eauto.
Qed.
(** *** Field Lookup *)
Lemma lookup_field_in_value : forall v T i Ti,
value v ->
has_type empty v T ->
Tlookup i T = Some Ti ->
exists vi, tlookup i v = Some vi /\ has_type empty vi Ti.
Proof with eauto.
remember empty as Gamma.
intros t T i Ti Hval Htyp. revert Ti HeqGamma Hval.
has_type_cases (induction Htyp) Case; intros; subst; try solve by inversion.
Case "T_Sub".
apply (rcd_types_match S) in H0... destruct H0 as [Si [HgetSi Hsub]].
destruct (IHHtyp Si) as [vi [Hget Htyvi]]...
Case "T_RCons".
simpl in H0. simpl. simpl in H1.
destruct (eq_id_dec i i0).
SCase "i is first".
inversion H1. subst. exists t...
SCase "i in tail".
destruct (IHHtyp2 Ti) as [vi [get Htyvi]]...
inversion Hval... Qed.
(* ########################################## *)
(** *** Progress *)
(** **** Exercise: 3 stars (canonical_forms_of_arrow_types) *)
Lemma canonical_forms_of_arrow_types : forall Gamma s T1 T2,
has_type Gamma s (TArrow T1 T2) ->
value s ->
exists x, exists S1, exists s2,
s = tabs x S1 s2.
Proof with eauto.
intros.
remember (TArrow T1 T2) as ta.
generalize dependent T1.
generalize dependent T2.
induction H; intros; subst; try solve by inversion.
inversion Heqta; subst...
apply sub_inversion_arrow in H1.
destruct H1 as [U1 [U2 [H2 [H3 H4]]]]...
Qed.
(** [] *)
Theorem progress : forall t T,
has_type empty t T ->
value t \/ exists t', t ==> t'.
Proof with eauto.
intros t T Ht.
remember empty as Gamma.
revert HeqGamma.
has_type_cases (induction Ht) Case;
intros HeqGamma; subst...
Case "T_Var".
inversion H.
Case "T_App".
right.
destruct IHHt1; subst...
SCase "t1 is a value".
destruct IHHt2; subst...
SSCase "t2 is a value".
destruct (canonical_forms_of_arrow_types empty t1 T1 T2)
as [x [S1 [t12 Heqt1]]]...
subst. exists ([x:=t2]t12)...
SSCase "t2 steps".
destruct H0 as [t2' Hstp]. exists (tapp t1 t2')...
SCase "t1 steps".
destruct H as [t1' Hstp]. exists (tapp t1' t2)...
Case "T_Proj".
right. destruct IHHt...
SCase "rcd is value".
destruct (lookup_field_in_value t T i Ti) as [t' [Hget Ht']]...
SCase "rcd_steps".
destruct H0 as [t' Hstp]. exists (tproj t' i)...
Case "T_RCons".
destruct IHHt1...
SCase "head is a value".
destruct IHHt2...
SSCase "tail steps".
right. destruct H2 as [tr' Hstp].
exists (trcons i t tr')...
SCase "head steps".
right. destruct H1 as [t' Hstp].
exists (trcons i t' tr)... Qed.
(** Informal proof of progress:
Theorem : For any term [t] and type [T], if [empty |- t : T]
then [t] is a value or [t ==> t'] for some term [t'].
Proof : Let [t] and [T] be given such that [empty |- t : T]. We go
by induction on the typing derivation. Cases [T_Abs] and
[T_RNil] are immediate because abstractions and [{}] are always
values. Case [T_Var] is vacuous because variables cannot be
typed in the empty context.
- If the last step in the typing derivation is by [T_App], then
there are terms [t1] [t2] and types [T1] [T2] such that
[t = t1 t2], [T = T2], [empty |- t1 : T1 -> T2] and
[empty |- t2 : T1].
The induction hypotheses for these typing derivations yield
that [t1] is a value or steps, and that [t2] is a value or
steps. We consider each case:
- Suppose [t1 ==> t1'] for some term [t1']. Then
[t1 t2 ==> t1' t2] by [ST_App1].
- Otherwise [t1] is a value.
- Suppose [t2 ==> t2'] for some term [t2']. Then
[t1 t2 ==> t1 t2'] by rule [ST_App2] because [t1] is a value.
- Otherwise, [t2] is a value. By lemma
[canonical_forms_for_arrow_types], [t1 = \x:S1.s2] for some
[x], [S1], and [s2]. And [(\x:S1.s2) t2 ==> [x:=t2]s2] by
[ST_AppAbs], since [t2] is a value.
- If the last step of the derivation is by [T_Proj], then there
is a term [tr], type [Tr] and label [i] such that [t = tr.i],
[empty |- tr : Tr], and [Tlookup i Tr = Some T].
The IH for the typing subderivation gives us that either [tr]
is a value or it steps. If [tr ==> tr'] for some term [tr'],
then [tr.i ==> tr'.i] by rule [ST_Proj1].
Otherwise, [tr] is a value. In this case, lemma
[lookup_field_in_value] yields that there is a term [ti] such
that [tlookup i tr = Some ti]. It follows that [tr.i ==> ti]
by rule [ST_ProjRcd].
- If the final step of the derivation is by [T_Sub], then there
is a type [S] such that [S <: T] and [empty |- t : S]. The
desired result is exactly the induction hypothesis for the
typing subderivation.
- If the final step of the derivation is by [T_RCons], then there
exist some terms [t1] [tr], types [T1 Tr] and a label [t] such
that [t = {i=t1, tr}], [T = {i:T1, Tr}], [record_tm tr],
[record_tm Tr], [empty |- t1 : T1] and [empty |- tr : Tr].
The induction hypotheses for these typing derivations yield
that [t1] is a value or steps, and that [tr] is a value or
steps. We consider each case:
- Suppose [t1 ==> t1'] for some term [t1']. Then
[{i=t1, tr} ==> {i=t1', tr}] by rule [ST_Rcd_Head].
- Otherwise [t1] is a value.
- Suppose [tr ==> tr'] for some term [tr']. Then
[{i=t1, tr} ==> {i=t1, tr'}] by rule [ST_Rcd_Tail],
since [t1] is a value.
- Otherwise, [tr] is also a value. So, [{i=t1, tr}] is a
value by [v_rcons]. *)
(* ########################################## *)
(** *** Inversion Lemmas *)
Lemma typing_inversion_var : forall Gamma x T,
has_type Gamma (tvar x) T ->
exists S,
Gamma x = Some S /\ subtype S T.
Proof with eauto.
intros Gamma x T Hty.
remember (tvar x) as t.
has_type_cases (induction Hty) Case; intros;
inversion Heqt; subst; try solve by inversion.
Case "T_Var".
exists T...
Case "T_Sub".
destruct IHHty as [U [Hctx HsubU]]... Qed.
Lemma typing_inversion_app : forall Gamma t1 t2 T2,
has_type Gamma (tapp t1 t2) T2 ->
exists T1,
has_type Gamma t1 (TArrow T1 T2) /\
has_type Gamma t2 T1.
Proof with eauto.
intros Gamma t1 t2 T2 Hty.
remember (tapp t1 t2) as t.
has_type_cases (induction Hty) Case; intros;
inversion Heqt; subst; try solve by inversion.
Case "T_App".
exists T1...
Case "T_Sub".
destruct IHHty as [U1 [Hty1 Hty2]]...
assert (Hwf := has_type__wf _ _ _ Hty2).
exists U1... Qed.
Lemma typing_inversion_abs : forall Gamma x S1 t2 T,
has_type Gamma (tabs x S1 t2) T ->
(exists S2, subtype (TArrow S1 S2) T
/\ has_type (extend Gamma x S1) t2 S2).
Proof with eauto.
intros Gamma x S1 t2 T H.
remember (tabs x S1 t2) as t.
has_type_cases (induction H) Case;
inversion Heqt; subst; intros; try solve by inversion.
Case "T_Abs".
assert (Hwf := has_type__wf _ _ _ H0).
exists T12...
Case "T_Sub".
destruct IHhas_type as [S2 [Hsub Hty]]...
Qed.
Lemma typing_inversion_proj : forall Gamma i t1 Ti,
has_type Gamma (tproj t1 i) Ti ->
exists T, exists Si,
Tlookup i T = Some Si /\ subtype Si Ti /\ has_type Gamma t1 T.
Proof with eauto.
intros Gamma i t1 Ti H.
remember (tproj t1 i) as t.
has_type_cases (induction H) Case;
inversion Heqt; subst; intros; try solve by inversion.
Case "T_Proj".
assert (well_formed_ty Ti) as Hwf.
SCase "pf of assertion".
apply (wf_rcd_lookup i T Ti)...
apply has_type__wf in H...
exists T. exists Ti...
Case "T_Sub".
destruct IHhas_type as [U [Ui [Hget [Hsub Hty]]]]...
exists U. exists Ui... Qed.
Lemma typing_inversion_rcons : forall Gamma i ti tr T,
has_type Gamma (trcons i ti tr) T ->
exists Si, exists Sr,
subtype (TRCons i Si Sr) T /\ has_type Gamma ti Si /\
record_tm tr /\ has_type Gamma tr Sr.
Proof with eauto.
intros Gamma i ti tr T Hty.
remember (trcons i ti tr) as t.
has_type_cases (induction Hty) Case;
inversion Heqt; subst...
Case "T_Sub".
apply IHHty in H0.
destruct H0 as [Ri [Rr [HsubRS [HtypRi HtypRr]]]].
exists Ri. exists Rr...
Case "T_RCons".
assert (well_formed_ty (TRCons i T Tr)) as Hwf.
SCase "pf of assertion".
apply has_type__wf in Hty1.
apply has_type__wf in Hty2...
exists T. exists Tr... Qed.
Lemma abs_arrow : forall x S1 s2 T1 T2,
has_type empty (tabs x S1 s2) (TArrow T1 T2) ->
subtype T1 S1
/\ has_type (extend empty x S1) s2 T2.
Proof with eauto.
intros x S1 s2 T1 T2 Hty.
apply typing_inversion_abs in Hty.
destruct Hty as [S2 [Hsub Hty]].
apply sub_inversion_arrow in Hsub.
destruct Hsub as [U1 [U2 [Heq [Hsub1 Hsub2]]]].
inversion Heq; subst... Qed.
(* ########################################## *)
(** *** Context Invariance *)
Inductive appears_free_in : id -> tm -> Prop :=
| afi_var : forall x,
appears_free_in x (tvar x)
| afi_app1 : forall x t1 t2,
appears_free_in x t1 -> appears_free_in x (tapp t1 t2)
| afi_app2 : forall x t1 t2,
appears_free_in x t2 -> appears_free_in x (tapp t1 t2)
| afi_abs : forall x y T11 t12,
y <> x ->
appears_free_in x t12 ->
appears_free_in x (tabs y T11 t12)
| afi_proj : forall x t i,
appears_free_in x t ->
appears_free_in x (tproj t i)
| afi_rhead : forall x i t tr,
appears_free_in x t ->
appears_free_in x (trcons i t tr)
| afi_rtail : forall x i t tr,
appears_free_in x tr ->
appears_free_in x (trcons i t tr).
Hint Constructors appears_free_in.
Lemma context_invariance : forall Gamma Gamma' t S,
has_type Gamma t S ->
(forall x, appears_free_in x t -> Gamma x = Gamma' x) ->
has_type Gamma' t S.
Proof with eauto.
intros. generalize dependent Gamma'.
has_type_cases (induction H) Case;
intros Gamma' Heqv...
Case "T_Var".
apply T_Var... rewrite <- Heqv...
Case "T_Abs".
apply T_Abs... apply IHhas_type. intros x0 Hafi.
unfold extend. destruct (eq_id_dec x x0)...
Qed.
Lemma free_in_context : forall x t T Gamma,
appears_free_in x t ->
has_type Gamma t T ->
exists T', Gamma x = Some T'.
Proof with eauto.
intros x t T Gamma Hafi Htyp.
has_type_cases (induction Htyp) Case; subst; inversion Hafi; subst...
Case "T_Abs".
destruct (IHHtyp H5) as [T Hctx]. exists T.
unfold extend in Hctx. rewrite neq_id in Hctx... Qed.
(* ########################################## *)
(** *** Preservation *)
Lemma substitution_preserves_typing : forall Gamma x U v t S,
has_type (extend Gamma x U) t S ->
has_type empty v U ->
has_type Gamma ([x:=v]t) S.
Proof with eauto.
intros Gamma x U v t S Htypt Htypv.
generalize dependent S. generalize dependent Gamma.
t_cases (induction t) Case; intros; simpl.
Case "tvar".
rename i into y.
destruct (typing_inversion_var _ _ _ Htypt) as [T [Hctx Hsub]].
unfold extend in Hctx.
destruct (eq_id_dec x y)...
SCase "x=y".
subst.
inversion Hctx; subst. clear Hctx.
apply context_invariance with empty...
intros x Hcontra.
destruct (free_in_context _ _ S empty Hcontra) as [T' HT']...
inversion HT'.
SCase "x<>y".
destruct (subtype__wf _ _ Hsub)...
Case "tapp".
destruct (typing_inversion_app _ _ _ _ Htypt) as [T1 [Htypt1 Htypt2]].
eapply T_App...
Case "tabs".
rename i into y. rename t into T1.
destruct (typing_inversion_abs _ _ _ _ _ Htypt)
as [T2 [Hsub Htypt2]].
destruct (subtype__wf _ _ Hsub) as [Hwf1 Hwf2].
inversion Hwf2. subst.
apply T_Sub with (TArrow T1 T2)... apply T_Abs...
destruct (eq_id_dec x y).
SCase "x=y".
eapply context_invariance...
subst.
intros x Hafi. unfold extend.
destruct (eq_id_dec y x)...
SCase "x<>y".
apply IHt. eapply context_invariance...
intros z Hafi. unfold extend.
destruct (eq_id_dec y z)...
subst. rewrite neq_id...
Case "tproj".
destruct (typing_inversion_proj _ _ _ _ Htypt)
as [T [Ti [Hget [Hsub Htypt1]]]]...
Case "trnil".
eapply context_invariance...
intros y Hcontra. inversion Hcontra.
Case "trcons".
destruct (typing_inversion_rcons _ _ _ _ _ Htypt) as
[Ti [Tr [Hsub [HtypTi [Hrcdt2 HtypTr]]]]].
apply T_Sub with (TRCons i Ti Tr)...
apply T_RCons...
SCase "record_ty Tr".
apply subtype__wf in Hsub. destruct Hsub. inversion H0...
SCase "record_tm ([x:=v]t2)".
inversion Hrcdt2; subst; simpl... Qed.
Theorem preservation : forall t t' T,
has_type empty t T ->
t ==> t' ->
has_type empty t' T.
Proof with eauto.
intros t t' T HT.
remember empty as Gamma. generalize dependent HeqGamma.
generalize dependent t'.
has_type_cases (induction HT) Case;
intros t' HeqGamma HE; subst; inversion HE; subst...
Case "T_App".
inversion HE; subst...
SCase "ST_AppAbs".
destruct (abs_arrow _ _ _ _ _ HT1) as [HA1 HA2].
apply substitution_preserves_typing with T...
Case "T_Proj".
destruct (lookup_field_in_value _ _ _ _ H2 HT H)
as [vi [Hget Hty]].
rewrite H4 in Hget. inversion Hget. subst...
Case "T_RCons".
eauto using step_preserves_record_tm. Qed.
(** Informal proof of [preservation]:
Theorem: If [t], [t'] are terms and [T] is a type such that
[empty |- t : T] and [t ==> t'], then [empty |- t' : T].
Proof: Let [t] and [T] be given such that [empty |- t : T]. We go
by induction on the structure of this typing derivation, leaving
[t'] general. Cases [T_Abs] and [T_RNil] are vacuous because
abstractions and {} don't step. Case [T_Var] is vacuous as well,
since the context is empty.
- If the final step of the derivation is by [T_App], then there
are terms [t1] [t2] and types [T1] [T2] such that [t = t1 t2],
[T = T2], [empty |- t1 : T1 -> T2] and [empty |- t2 : T1].
By inspection of the definition of the step relation, there are
three ways [t1 t2] can step. Cases [ST_App1] and [ST_App2]
follow immediately by the induction hypotheses for the typing
subderivations and a use of [T_App].
Suppose instead [t1 t2] steps by [ST_AppAbs]. Then
[t1 = \x:S.t12] for some type [S] and term [t12], and
[t' = [x:=t2]t12].
By Lemma [abs_arrow], we have [T1 <: S] and [x:S1 |- s2 : T2].
It then follows by lemma [substitution_preserves_typing] that
[empty |- [x:=t2] t12 : T2] as desired.
- If the final step of the derivation is by [T_Proj], then there
is a term [tr], type [Tr] and label [i] such that [t = tr.i],
[empty |- tr : Tr], and [Tlookup i Tr = Some T].
The IH for the typing derivation gives us that, for any term
[tr'], if [tr ==> tr'] then [empty |- tr' Tr]. Inspection of
the definition of the step relation reveals that there are two
ways a projection can step. Case [ST_Proj1] follows
immediately by the IH.
Instead suppose [tr.i] steps by [ST_ProjRcd]. Then [tr] is a
value and there is some term [vi] such that
[tlookup i tr = Some vi] and [t' = vi]. But by lemma
[lookup_field_in_value], [empty |- vi : Ti] as desired.
- If the final step of the derivation is by [T_Sub], then there
is a type [S] such that [S <: T] and [empty |- t : S]. The
result is immediate by the induction hypothesis for the typing
subderivation and an application of [T_Sub].
- If the final step of the derivation is by [T_RCons], then there
exist some terms [t1] [tr], types [T1 Tr] and a label [t] such
that [t = {i=t1, tr}], [T = {i:T1, Tr}], [record_tm tr],
[record_tm Tr], [empty |- t1 : T1] and [empty |- tr : Tr].
By the definition of the step relation, [t] must have stepped
by [ST_Rcd_Head] or [ST_Rcd_Tail]. In the first case, the
result follows by the IH for [t1]'s typing derivation and
[T_RCons]. In the second case, the result follows by the IH
for [tr]'s typing derivation, [T_RCons], and a use of the
[step_preserves_record_tm] lemma. *)
(* ###################################################### *)
(** ** Exercises on Typing *)
(** **** Exercise: 2 stars, optional (variations) *)
(** Each part of this problem suggests a different way of
changing the definition of the STLC with records and
subtyping. (These changes are not cumulative: each part
starts from the original language.) In each part, list which
properties (Progress, Preservation, both, or neither) become
false. If a property becomes false, give a counterexample.
- Suppose we add the following typing rule:
Gamma |- t : S1->S2
S1 <: T1 T1 <: S1 S2 <: T2
----------------------------------- (T_Funny1)