# Normalization for Call-by-Value Simply-Typed Lambda Calculus.

Authors: Martin Elsman and Danil Annenkov, University of Copenhagen
In the present formalization, we consider simply-typed lambda calculus (STLC) with primitive type of integers. We show how to prove the normalization property of STLC using the Tait's method. The key observation is that we can define logical relation required for the proof of normalization using Coq's fixpoint construct.

Require Import String.

## Basic definitions

Inductive Ty : Set :=
| tInt : Ty
| tArr : Ty -> Ty -> Ty.

Notation "A :-> B" := (tArr A B) (at level 70).

The syntax for STLC. We use Coq's string type to represent variable names
Inductive Exp : Set :=
| Int : nat -> Exp
| Var : string -> Exp
| Lam : string -> Exp -> Exp
| App : Exp -> Exp -> Exp.

Environments are defined inductively
Inductive Env {A:Set} : Set :=
| empty : Env
| cons : Env -> string -> A -> Env.

We have two kinds of notation for environment extension.
A typing context extension:
Notation "Gamma , a @ A" := (cons Gamma a A) (at level 201).
A value environment extension:
Notation "E # [ a ~> v ]" := (cons E a v) (at level 99).

Fixpoint lookEnv {T : Set} (E : Env) (x : string) : option T :=
match E with
| empty => None
| cons E y A =>
if string_dec y x then Some A else lookEnv E x
end.

Definition TEnv : Set := Env (A:=Ty).
Reserved Notation "[ Gamma |- a @ A ]".

The usual typing rules for the STLC
Inductive Typing : TEnv -> Exp -> Ty -> Prop :=
| tyInt : forall (Gamma : TEnv) (n : nat),
[ Gamma |- (Int n) @ tInt]
| tyVar : forall (Gamma : TEnv) (x : string) (A : Ty),
lookEnv Gamma x = Some(A) ->
[ Gamma |- (Var x) @ A ]
| tyLam : forall (Gamma : TEnv) (x : string) (b : Exp) (A B : Ty),
[ Gamma, x @ A |- b @ B ] ->
[ Gamma |- (Lam x b) @ (A :-> B)]
| tyApp : forall (Gamma : TEnv) (f a : Exp) (A B : Ty),
[ Gamma |- f @ (A :-> B) ]->
[ Gamma |- a @ A ] ->
[ Gamma |- (App f a) @ B ]
where "[ Gamma |- a @ A ]" := (Typing Gamma a A).

The values are either an integer or a closure, corresponding to a lambda abstraction
Inductive Val : Set :=
| vInt : nat -> Val
| vClos : Env (A:=Val) -> string -> Exp -> Val.

Definition DEnv := Env (A:=Val).

Reserved Notation "[ E |- a ==> v ]".

We define big-step evaluation relation in a call-by-value style. We do not use substitution, instead we are explicitly passing a value environment
Inductive Eval : DEnv -> Exp -> Val -> Prop :=
| eInt : forall (E : DEnv) (n : nat),
[ E |- (Int n) ==> (vInt n) ]
| eVar : forall (E : DEnv) (x : string) (v : Val),
lookEnv E x = Some v ->
[ E |- (Var x) ==> v ]
| eLam : forall (E : DEnv) (x : string) (a : Exp),
[ E |- (Lam x a) ==> (vClos E x a) ]
| eApp : forall (E E0 : DEnv) (f a e0 : Exp) (v va : Val) (x : string),
[ E |- f ==> (vClos E0 x e0) ] ->
[ E |- a ==> va ] ->
[ E0 # [x ~> va] |- e0 ==> v ] ->
[ E |- (App f a) ==> v ]
where "[ E |- a ==> v ]" := (Eval E a v).

## The logical relation

Reserved Notation "[ |= v @ t ]".

The very core of our proof of normalization is a logical relation, defined recursively on a structure of types in our STLC
Fixpoint Equiv (val:Val) (ty:Ty) : Prop :=
match ty with
tInt => exists n : nat, val = (vInt n)
| tArr A B => exists (x:string) (a:Exp) (E:DEnv),
(val = vClos E x a) /\
(forall v1:Val, [ |= v1 @ A ] ->
exists v2:Val, [ E # [x ~> v1] |- a ==> v2] /\ [ |= v2 @ B ])
end
where "[ |= v @ t ]" := (Equiv v t).
It is crucial to use a fixpoint for the definition of Equiv, because naive inductive definition will not pass strict positivity check

Notation "[ |= v @ t ]" := (Equiv v t).

Definition EquivEnv (E : DEnv) (Gamma : TEnv) : Prop :=
(forall (x:string) (val:Val),
lookEnv E x = Some val ->
exists ty:Ty, lookEnv Gamma x = Some ty /\ [ |= val @ ty ])
/\
(forall (x:string) (ty:Ty),
lookEnv Gamma x = Some ty ->
exists val:Val, lookEnv E x = Some val /\ [ |= val @ ty ]).

Notation "[ |== E @ Gamma ]" := (EquivEnv E Gamma).

Lemma Look : forall (Gamma : TEnv) (ty : Ty) (E : DEnv) (s : string),
[ |== E @ Gamma ] -> lookEnv Gamma s = Some ty
-> exists v:Val, lookEnv E s = Some v /\ [ |= v @ ty ].
Proof.
intros. unfold EquivEnv in H. intuition;auto.
Qed.

Lemma EquivExtend : forall (Gamma : TEnv) (E : DEnv) (s : string) (val : Val) (ty : Ty),
[ |= val @ ty ] -> [ |== E @ Gamma ] -> [ |== (E # [s ~> val]) @ Gamma, s @ ty].
Proof.
intros Gamma E s v ty Hty Heqv. constructor; intros s' v' E'; simpl in *.
- remember (string_dec s s') as b.
destruct b.
+ inversion E';subst. eexists. split;auto.
+ inversion Heqv as [H1 H2]. destruct (H1 s' v' E'). destruct H.
eexists. split;eauto.
- remember (string_dec s s') as b.
destruct b.
+ inversion E';subst. eexists. split;auto.
+ inversion Heqv as [H1 H2]. destruct (H2 s' v' E'). destruct H.
eexists. split;eauto.
Qed.

## Normalization

Hint Constructors Typing Eval.

A tactic for repeatedly destructing all existentials in hypothesis H, creating new variables with the n preffix
Ltac dest_exs H n :=
match goal with
| [_ : ex _ |- _ ] => let i := fresh n in
let Hi := fresh "H" i in (destruct H as [i Hi]; dest_exs Hi n)
| [_ : _ |- _] => idtac
end.

A proof of normalization by induction on typing derivation. We are being very explicit in this proof and use proof automation only in obvious and tedious cases.
Lemma Normalisation : forall (exp : Exp) (Gamma : TEnv) (ty : Ty) (E : DEnv),
[ Gamma |- exp @ ty ] -> [ |== E @ Gamma ] ->
exists val:Val, [ E |- exp ==> val ] /\ [ |= val @ ty ].
Proof.
intros until E. intros Ty He.
generalize dependent E.
induction Ty; intros E He.
- exists (vInt n). split; auto. econstructor;eauto.
- inversion He as [H1 H2]. destruct (H2 x A H) as [v H3]. destruct H3.
exists v. split;auto.
- exists (vClos E x b). split;auto.
simpl. exists x. exists b. exists E. split;auto.
intros v1 Hv1. specialize IHTy with (E:= E # [x ~> v1]). apply IHTy.
apply EquivExtend; auto.
- destruct (IHTy1 E He) as [v H]. clear IHTy1.
destruct (IHTy2 E He) as [v' H']. clear IHTy2.
destruct H as [? Heqv]. destruct H' as [? Heqv']. simpl in *. dest_exs Heqv x.
rename x0 into e0. rename x1 into E0.
destruct Hx1 as [Hveq H3].
destruct (H3 v' Heqv') as [v'' H''].
destruct H''. subst.
exists v''. split;eauto.
Qed.
Check Typing_ind.
An alternative proof of normalization by induction on syntax
Lemma Normalisation_alt : forall (exp : Exp) (Gamma : TEnv) (ty : Ty) (E : DEnv),
[ Gamma |- exp @ ty ] -> [ |== E @ Gamma ] ->
exists val:Val, [ E |- exp ==> val ] /\ [ |= val @ ty ].
Proof.
induction exp; intros Gamma ty E Ty Heqv.
- exists (vInt n). split; auto.
inversion Ty. exists n. reflexivity.
- inversion_clear Ty. inversion Heqv as [H1 H2].
specialize H2 with (x:=s)(ty:=ty).
destruct (H2 H). exists x. intuition.
- exists (vClos E s exp). split. constructor.
inversion_clear Ty. exists s. exists exp. exists E. split;auto.
intros v1 Hv1. specialize IHexp with (Gamma:=cons Gamma s A)(ty:=B)(E:=cons E s v1).
destruct (IHexp H).
apply EquivExtend;auto. exists x. intuition.
- inversion_clear Ty.
specialize IHexp1 with (Gamma:=Gamma)(ty:= A :-> ty) (E:=E).
specialize IHexp2 with (Gamma:=Gamma)(ty:=A)(E:=E).
destruct (IHexp1 H Heqv) as [v1 Hv1]. clear IHexp1.
destruct (IHexp2 H0 Heqv) as [v2 Hv2]. clear IHexp2.
destruct Hv1 as [Ev1 Heqv1]. destruct Hv2 as [Ev2 Tv2].
simpl in *. dest_exs Heqv1 x. destruct Hx1 as [Hv1 Heqv2]. subst.
assert (H' := Heqv2 v2 Tv2). destruct H' as [v' Hv']. destruct Hv'.
exists v'. split.
* econstructor; eauto.
* eauto.
Qed.