RecordSubSubtyping with Records

In this chapter, we combine two significant extensions of the pure STLC — records (from chapter Records) and subtyping (from chapter Sub) — and explore their interactions. Most of the concepts have already been discussed in those chapters, so the presentation here is somewhat terse. We just comment where things are nonstandard.
Set Warnings "-notation-overridden,-parsing".
Require Import Maps.
Require Import Smallstep.
Require Import MoreStlc.

Core Definitions


Syntax


Inductive ty : Type :=
  (* proper types *)
  | TTop : ty
  | TBase : idty
  | TArrow : tytyty
  (* record types *)
  | TRNil : ty
  | TRCons : idtytyty.

Inductive tm : Type :=
  (* proper terms *)
  | tvar : idtm
  | tapp : tmtmtm
  | tabs : idtytmtm
  | tproj : tmidtm
  (* record terms *)
  | trnil : tm
  | trcons : idtmtmtm.

Well-Formedness

The syntax of terms and types is a bit too loose, in the sense that it admits things like a record type whose final "tail" is Top or some arrow type rather than Nil. To avoid such cases, it is useful to assume that all the record types and terms that we see will obey some simple well-formedness conditions.
An interesting technical question is whether the basic properties of the system -- progress and preservation -- remain true if we drop these conditions. I believe they do, and I would encourage motivated readers to try to check this by dropping the conditions from the definitions of typing and subtyping and adjusting the proofs in the rest of the chapter accordingly. This is not a trivial exercise (or I'd have done it!), but it should not involve changing the basic structure of the proofs. If someone does do it, please let me know. --BCP 5/16.
Inductive record_ty : tyProp :=
  | RTnil :
        record_ty TRNil
  | RTcons : ∀ i T1 T2,
        record_ty (TRCons i T1 T2).

Inductive record_tm : tmProp :=
  | rtnil :
        record_tm trnil
  | rtcons : ∀ i t1 t2,
        record_tm (trcons i t1 t2).

Inductive well_formed_ty : tyProp :=
  | wfTTop :
        well_formed_ty TTop
  | wfTBase : ∀ i,
        well_formed_ty (TBase i)
  | wfTArrow : ∀ T1 T2,
        well_formed_ty T1
        well_formed_ty T2
        well_formed_ty (TArrow T1 T2)
  | wfTRNil :
        well_formed_ty TRNil
  | wfTRCons : ∀ 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

Substitution and reduction are as before.
Fixpoint subst (x:id) (s:tm) (t:tm) : tm :=
  match t with
  | tvar yif beq_id x y then s else t
  | tabs y T t1tabs y T (if beq_id x y then t1
                             else (subst x s t1))
  | tapp t1 t2tapp (subst x s t1) (subst x s t2)
  | tproj t1 itproj (subst x s t1) i
  | trniltrnil
  | trcons i t1 tr2trcons i (subst x s t1) (subst x s tr2)
  end.

Notation "'[' x ':=' s ']' t" := (subst x s t) (at level 20).

Reduction


Inductive value : tmProp :=
  | v_abs : ∀ x T t,
      value (tabs x T t)
  | v_rnil : value trnil
  | v_rcons : ∀ 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 beq_id 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 beq_id i i' then Some t else tlookup i tr'
  | _None
  end.

Reserved Notation "t1 '==>' t2" (at level 40).

Inductive step : tmtmProp :=
  | ST_AppAbs : ∀ x T t12 v2,
         value v2
         (tapp (tabs x T t12) v2) ==> [x:=v2]t12
  | ST_App1 : ∀ t1 t1' t2,
         t1 ==> t1'
         (tapp t1 t2) ==> (tapp t1' t2)
  | ST_App2 : ∀ v1 t2 t2',
         value v1
         t2 ==> t2'
         (tapp v1 t2) ==> (tapp v1 t2')
  | ST_Proj1 : ∀ tr tr' i,
        tr ==> tr'
        (tproj tr i) ==> (tproj tr' i)
  | ST_ProjRcd : ∀ tr i vi,
        value tr
        tlookup i tr = Some vi
       (tproj tr i) ==> vi
  | ST_Rcd_Head : ∀ i t1 t1' tr2,
        t1 ==> t1'
        (trcons i t1 tr2) ==> (trcons i t1' tr2)
  | ST_Rcd_Tail : ∀ i v1 tr2 tr2',
        value v1
        tr2 ==> tr2'
        (trcons i v1 tr2) ==> (trcons i v1 tr2')

where "t1 '==>' t2" := (step t1 t2).

Hint Constructors step.

Subtyping

Now we come to the interesting part, where the features we've added start to interact. 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 discussion of record subtyping in chapter Sub, but we need to add well-formedness side conditions to some of the rules. Also, we replace the "n-ary" width, depth, and permutation subtyping rules by binary rules that deal with just the first field.
Reserved Notation "T '<:' U" (at level 40).

Inductive subtype : tytyProp :=
  (* Subtyping between proper types *)
  | S_Refl : ∀ T,
    well_formed_ty T
    T <: T
  | S_Trans : ∀ S U T,
    S <: U
    U <: T
    S <: T
  | S_Top : ∀ S,
    well_formed_ty S
    S <: TTop
  | S_Arrow : ∀ S1 S2 T1 T2,
    T1 <: S1
    S2 <: T2
    TArrow S1 S2 <: TArrow T1 T2
  (* Subtyping between record types *)
  | S_RcdWidth : ∀ i T1 T2,
    well_formed_ty (TRCons i T1 T2) →
    TRCons i T1 T2 <: TRNil
  | S_RcdDepth : ∀ i S1 T1 Sr2 Tr2,
    S1 <: T1
    Sr2 <: Tr2
    record_ty Sr2
    record_ty Tr2
    TRCons i S1 Sr2 <: TRCons i T1 Tr2
  | S_RcdPerm : ∀ i1 i2 T1 T2 Tr3,
    well_formed_ty (TRCons i1 T1 (TRCons i2 T2 Tr3)) →
    i1i2
       TRCons i1 T1 (TRCons i2 T2 Tr3)
    <: TRCons i2 T2 (TRCons i1 T1 Tr3)

where "T '<:' U" := (subtype T U).

Hint Constructors subtype.

Examples


Module Examples.

Notation x := (Id "x").
Notation y := (Id "y").
Notation z := (Id "z").
Notation j := (Id "j").
Notation k := (Id "k").
Notation i := (Id "i").
Notation A := (TBase (Id "A")).
Notation B := (TBase (Id "B")).
Notation C := (TBase (Id "C")).

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, make sure you also understand how to prove them on paper!

Exercise: 2 stars (subtyping_example_1)

Example subtyping_example_1 :
  subtype TRcd_kj TRcd_j.
(* {k:A->A,j:B->B} <: {j:B->B} *)
Proof with eauto.
  (* FILL IN HERE *) Admitted.

Exercise: 1 star (subtyping_example_2)

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.
  (* FILL IN HERE *) Admitted.

Exercise: 1 star (subtyping_example_3)

Example subtyping_example_3 :
  subtype (TArrow TRNil (TRCons j A TRNil))
          (TArrow (TRCons k B TRNil) TRNil).
(* {}->{j:A} <: {k:B}->{} *)
Proof with eauto.
  (* FILL IN HERE *) Admitted.

Exercise: 2 stars (subtyping_example_4)

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.
  (* FILL IN HERE *) Admitted.
End Examples.

Properties of Subtyping

Well-Formedness

To get started proving things about subtyping, we need a couple of technical lemmas that intuitively (1) allow us to extract the well-formedness assumptions embedded in subtyping derivations and (2) record the fact that fields of well-formed record types are themselves well-formed types.
Lemma subtype__wf : ∀ S T,
  subtype S T
  well_formed_ty Twell_formed_ty S.
Proof with eauto.
  intros S T Hsub.
  induction Hsub;
    intros; try (destruct IHHsub1; destruct IHHsub2)...
  - (* S_RcdPerm *)
    split... inversion H. subst. inversion H5... Qed.

Lemma wf_rcd_lookup : ∀ i T Ti,
  well_formed_ty T
  Tlookup i T = Some Ti
  well_formed_ty Ti.
Proof with eauto.
  intros i T.
  induction T; intros; try solve_by_invert.
  - (* TRCons *)
    inversion H. subst. unfold Tlookup in H0.
    destruct (beq_id i i0)... inversion H0; subst... Qed.

Field Lookup

The 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 the corresponding terms. And second, reasoning by induction on typing derivations becomes harder in general, because typing is no longer syntax directed.
Lemma rcd_types_match : ∀ S T i Ti,
  subtype S T
  Tlookup i T = Some Ti
  ∃ Si, Tlookup i S = Some Sisubtype Si Ti.
Proof with (eauto using wf_rcd_lookup).
  intros S T i Ti Hsub Hget. generalize dependent Ti.
  induction Hsub; intros Ti Hget;
    try solve_by_invert.
  - (* S_Refl *)
    ∃ Ti...
  - (* S_Trans *)
    destruct (IHHsub2 Ti) as [Ui Hui]... destruct Hui.
    destruct (IHHsub1 Ui) as [Si Hsi]... destruct Hsi.
    ∃ Si...
  - (* S_RcdDepth *)
    rename i0 into k.
    unfold Tlookup. unfold Tlookup in Hget.
    destruct (beq_id i k)...
    + (* i = k -- we're looking up the first field *)
      inversion Hget. subst. ∃ S1...
  - (* S_RcdPerm *)
    ∃ Ti. split.
    + (* lookup *)
      unfold Tlookup. unfold Tlookup in Hget.
      destruct (beq_idP i i1)...
      * (* i = i1 -- we're looking up the first field *)
        destruct (beq_idP i i2)...
        (* i = i2 -- contradictory *)
        destruct H0.
        subst...
    + (* 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.
(* FILL IN HERE *)

Inversion Lemmas

Exercise: 3 stars, optional (sub_inversion_arrow)

Lemma sub_inversion_arrow : ∀ U V1 V2,
     subtype U (TArrow V1 V2) →
     ∃ U1, ∃ 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.
  (* FILL IN HERE *) Admitted.

Typing


Definition context := partial_map ty.

Reserved Notation "Gamma '|-' t '∈' T" (at level 40).

Inductive has_type : contexttmtyProp :=
  | T_Var : ∀ Gamma x T,
      Gamma x = Some T
      well_formed_ty T
      Gamma |- tvar xT
  | T_Abs : ∀ Gamma x T11 T12 t12,
      well_formed_ty T11
      update Gamma x T11 |- t12T12
      Gamma |- tabs x T11 t12TArrow T11 T12
  | T_App : ∀ T1 T2 Gamma t1 t2,
      Gamma |- t1TArrow T1 T2
      Gamma |- t2T1
      Gamma |- tapp t1 t2T2
  | T_Proj : ∀ Gamma i t T Ti,
      Gamma |- tT
      Tlookup i T = Some Ti
      Gamma |- tproj t iTi
  (* Subsumption *)
  | T_Sub : ∀ Gamma t S T,
      Gamma |- tS
      subtype S T
      Gamma |- tT
  (* Rules for record terms *)
  | T_RNil : ∀ Gamma,
      Gamma |- trnilTRNil
  | T_RCons : ∀ Gamma i t T tr Tr,
      Gamma |- tT
      Gamma |- trTr
      record_ty Tr
      record_tm tr
      Gamma |- trcons i t trTRCons i T Tr

where "Gamma '|-' t '∈' T" := (has_type Gamma t T).

Hint Constructors has_type.

Typing Examples


Module Examples2.
Import Examples.

Exercise: 1 star (typing_example_0)

Definition trcd_kj :=
  (trcons k (tabs z A (tvar z))
           (trcons j (tabs z B (tvar z))
                      trnil)).

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.
  (* FILL IN HERE *) Admitted.

Exercise: 2 stars (typing_example_1)

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.
  (* FILL IN HERE *) Admitted.

Exercise: 2 stars, optional (typing_example_2)

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.
  (* FILL IN HERE *) Admitted.

End Examples2.

Properties of Typing

Well-Formedness


Lemma has_type__wf : ∀ Gamma t T,
  has_type Gamma t Twell_formed_ty T.
Proof with eauto.
  intros Gamma t T Htyp.
  induction Htyp...
  - (* T_App *)
    inversion IHHtyp1...
  - (* T_Proj *)
    eapply wf_rcd_lookup...
  - (* T_Sub *)
    apply subtype__wf in H.
    destruct H...
Qed.

Lemma step_preserves_record_tm : ∀ 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 : ∀ v T i Ti,
  value v
  has_type empty v T
  Tlookup i T = Some Ti
  ∃ vi, tlookup i v = Some vihas_type empty vi Ti.
Proof with eauto.
  remember empty as Gamma.
  intros t T i Ti Hval Htyp. revert Ti HeqGamma Hval.
  induction Htyp; intros; subst; try solve_by_invert.
  - (* T_Sub *)
    apply (rcd_types_match S) in H0...
    destruct H0 as [Si [HgetSi Hsub]].
    destruct (IHHtyp Si) as [vi [Hget Htyvi]]...
  - (* T_RCons *)
    simpl in H0. simpl. simpl in H1.
    destruct (beq_id i i0).
    + (* i is first *)
      inversion H1. subst. ∃ t...
    + (* 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 : ∀ Gamma s T1 T2,
     has_type Gamma s (TArrow T1 T2) →
     value s
     ∃ x, ∃ S1, ∃ s2,
        s = tabs x S1 s2.
Proof with eauto.
  (* FILL IN HERE *) Admitted.

Theorem progress : ∀ t T,
     has_type empty t T
     value t ∨ ∃ t', t ==> t'.
Proof with eauto.
  intros t T Ht.
  remember empty as Gamma.
  revert HeqGamma.
  induction Ht;
    intros HeqGamma; subst...
  - (* T_Var *)
    inversion H.
  - (* T_App *)
    right.
    destruct IHHt1; subst...
    + (* t1 is a value *)
      destruct IHHt2; subst...
      * (* t2 is a value *)
        destruct (canonical_forms_of_arrow_types empty t1 T1 T2)
          as [x [S1 [t12 Heqt1]]]...
        subst. ∃ ([x:=t2]t12)...
      * (* t2 steps *)
        destruct H0 as [t2' Hstp]. ∃ (tapp t1 t2')...
    + (* t1 steps *)
      destruct H as [t1' Hstp]. ∃ (tapp t1' t2)...
  - (* T_Proj *)
    right. destruct IHHt...
    + (* rcd is value *)
      destruct (lookup_field_in_value t T i Ti)
        as [t' [Hget Ht']]...
    + (* rcd_steps *)
      destruct H0 as [t' Hstp]. ∃ (tproj t' i)...
  - (* T_RCons *)
    destruct IHHt1...
    + (* head is a value *)
      destruct IHHt2...
      * (* tail steps *)
        right. destruct H2 as [tr' Hstp].
        ∃ (trcons i t tr')...
    + (* head steps *)
      right. destruct H1 as [t' Hstp].
      ∃ (trcons i t' tr)... Qed.
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 proceed by induction on the given typing derivation.
  • The cases where the last step in the typing derivation is T_Abs or T_RNil are immediate because abstractions and {} are always values. The case for 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.
    • 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. But then (\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 are a term tr, a type Tr, and a label i such that t = tr.i, empty |- tr : Tr, and Tlookup i Tr = Some T.
    By the IH, either tr is a value or it steps. If tr ==> tr' for some term tr', then tr.i ==> tr'.i by rule ST_Proj1.
    If tr is a value, then 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_ty 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 : ∀ Gamma x T,
  has_type Gamma (tvar x) T
  ∃ S,
    Gamma x = Some Ssubtype S T.
Proof with eauto.
  intros Gamma x T Hty.
  remember (tvar x) as t.
  induction Hty; intros;
    inversion Heqt; subst; try solve_by_invert.
  - (* T_Var *)
    ∃ T...
  - (* T_Sub *)
    destruct IHHty as [U [Hctx HsubU]]... Qed.

Lemma typing_inversion_app : ∀ Gamma t1 t2 T2,
  has_type Gamma (tapp t1 t2) T2
  ∃ 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.
  induction Hty; intros;
    inversion Heqt; subst; try solve_by_invert.
  - (* T_App *)
    ∃ T1...
  - (* T_Sub *)
    destruct IHHty as [U1 [Hty1 Hty2]]...
    assert (Hwf := has_type__wf _ _ _ Hty2).
    ∃ U1... Qed.

Lemma typing_inversion_abs : ∀ Gamma x S1 t2 T,
     has_type Gamma (tabs x S1 t2) T
     (∃ S2, subtype (TArrow S1 S2) T
              ∧ has_type (update Gamma x S1) t2 S2).
Proof with eauto.
  intros Gamma x S1 t2 T H.
  remember (tabs x S1 t2) as t.
  induction H;
    inversion Heqt; subst; intros; try solve_by_invert.
  - (* T_Abs *)
    assert (Hwf := has_type__wf _ _ _ H0).
    ∃ T12...
  - (* T_Sub *)
    destruct IHhas_type as [S2 [Hsub Hty]]...
    Qed.

Lemma typing_inversion_proj : ∀ Gamma i t1 Ti,
  has_type Gamma (tproj t1 i) Ti
  ∃ T, ∃ Si,
    Tlookup i T = Some Sisubtype Si Tihas_type Gamma t1 T.
Proof with eauto.
  intros Gamma i t1 Ti H.
  remember (tproj t1 i) as t.
  induction H;
    inversion Heqt; subst; intros; try solve_by_invert.
  - (* T_Proj *)
    assert (well_formed_ty Ti) as Hwf.
    { (* pf of assertion *)
      apply (wf_rcd_lookup i T Ti)...
      apply has_type__wf in H... }
    ∃ T. ∃ Ti...
  - (* T_Sub *)
    destruct IHhas_type as [U [Ui [Hget [Hsub Hty]]]]...
    ∃ U. ∃ Ui... Qed.

Lemma typing_inversion_rcons : ∀ Gamma i ti tr T,
  has_type Gamma (trcons i ti tr) T
  ∃ Si, ∃ Sr,
    subtype (TRCons i Si Sr) Thas_type Gamma ti Si
    record_tm trhas_type Gamma tr Sr.
Proof with eauto.
  intros Gamma i ti tr T Hty.
  remember (trcons i ti tr) as t.
  induction Hty;
    inversion Heqt; subst...
  - (* T_Sub *)
    apply IHHty in H0.
    destruct H0 as [Ri [Rr [HsubRS [HtypRi HtypRr]]]].
    ∃ Ri. ∃ Rr...
  - (* T_RCons *)
    assert (well_formed_ty (TRCons i T Tr)) as Hwf.
    { (* pf of assertion *)
      apply has_type__wf in Hty1.
      apply has_type__wf in Hty2... }
    ∃ T. ∃ Tr... Qed.

Lemma abs_arrow : ∀ x S1 s2 T1 T2,
  has_type empty (tabs x S1 s2) (TArrow T1 T2) →
     subtype T1 S1
  ∧ has_type (update 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 : idtmProp :=
  | afi_var : ∀ x,
      appears_free_in x (tvar x)
  | afi_app1 : ∀ x t1 t2,
      appears_free_in x t1appears_free_in x (tapp t1 t2)
  | afi_app2 : ∀ x t1 t2,
      appears_free_in x t2appears_free_in x (tapp t1 t2)
  | afi_abs : ∀ x y T11 t12,
        yx
        appears_free_in x t12
        appears_free_in x (tabs y T11 t12)
  | afi_proj : ∀ x t i,
      appears_free_in x t
      appears_free_in x (tproj t i)
  | afi_rhead : ∀ x i t tr,
      appears_free_in x t
      appears_free_in x (trcons i t tr)
  | afi_rtail : ∀ 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 : ∀ Gamma Gamma' t S,
     has_type Gamma t S
     (∀ x, appears_free_in x tGamma x = Gamma' x) →
     has_type Gamma' t S.
Proof with eauto.
  intros. generalize dependent Gamma'.
  induction H;
    intros Gamma' Heqv...
  - (* T_Var *)
    apply T_Var... rewrite <- Heqv...
  - (* T_Abs *)
    apply T_Abs... apply IHhas_type. intros x0 Hafi.
    unfold update, t_update. destruct (beq_idP x x0)...
  - (* T_App *)
    apply T_App with T1...
  - (* T_RCons *)
    apply T_RCons... Qed.

Lemma free_in_context : ∀ x t T Gamma,
   appears_free_in x t
   has_type Gamma t T
   ∃ T', Gamma x = Some T'.
Proof with eauto.
  intros x t T Gamma Hafi Htyp.
  induction Htyp; subst; inversion Hafi; subst...
  - (* T_Abs *)
    destruct (IHHtyp H5) as [T Hctx]. ∃ T.
    unfold update, t_update in Hctx.
    rewrite false_beq_id in Hctx... Qed.

Preservation


Lemma substitution_preserves_typing : ∀ Gamma x U v t S,
     has_type (update 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.
  induction t; intros; simpl.
  - (* tvar *)
    rename i into y.
    destruct (typing_inversion_var _ _ _ Htypt) as [T [Hctx Hsub]].
    unfold update, t_update in Hctx.
    destruct (beq_idP x y)...
    + (* 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'.
    + (* x<>y *)
      destruct (subtype__wf _ _ Hsub)...
  - (* tapp *)
    destruct (typing_inversion_app _ _ _ _ Htypt)
      as [T1 [Htypt1 Htypt2]].
    eapply T_App...
  - (* 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 (beq_idP x y).
    + (* x=y *)
      eapply context_invariance...
      subst.
      intros x Hafi. unfold update, t_update.
      destruct (beq_id y x)...
    + (* x<>y *)
      apply IHt. eapply context_invariance...
      intros z Hafi. unfold update, t_update.
      destruct (beq_idP y z)...
      subst. rewrite false_beq_id...
  - (* tproj *)
    destruct (typing_inversion_proj _ _ _ _ Htypt)
      as [T [Ti [Hget [Hsub Htypt1]]]]...
  - (* trnil *)
    eapply context_invariance...
    intros y Hcontra. inversion Hcontra.
  - (* trcons *)
    destruct (typing_inversion_rcons _ _ _ _ _ Htypt) as
      [Ti [Tr [Hsub [HtypTi [Hrcdt2 HtypTr]]]]].
    apply T_Sub with (TRCons i Ti Tr)...
    apply T_RCons...
    + (* record_ty Tr *)
      apply subtype__wf in Hsub. destruct Hsub. inversion H0...
    + (* record_tm (x:=vt2) *)
      inversion Hrcdt2; subst; simpl... Qed.

Theorem preservation : ∀ 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'.
  induction HT;
    intros t' HeqGamma HE; subst; inversion HE; subst...
  - (* T_App *)
    inversion HE; subst...
    + (* ST_AppAbs *)
      destruct (abs_arrow _ _ _ _ _ HT1) as [HA1 HA2].
      apply substitution_preserves_typing with T...
  - (* T_Proj *)
    destruct (lookup_field_in_value _ _ _ _ H2 HT H)
      as [vi [Hget Hty]].
    rewrite H4 in Hget. inversion Hget. subst...
  - (* T_RCons *)
    eauto using step_preserves_record_tm. Qed.
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_ty 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.