# Library Tutorial_STLC_Iso

Set Implicit Arguments.

Require Export LNameless_Isomorphism.
Require Export Equality.

# Step 1. Defining syntax

atm stands for the set of atomic types.

Notation atm := nat.

Grammar of pseudo-type

Inductive typ : Type :=
| typ_var : atm -> typ
| typ_arrow : typ -> typ -> typ.

var stands for the set of parameters (free variables).

Notation var := nat.

Grammar of pseudo-term

Inductive trm : Set :=
| trm_bvar : nat -> trm
| trm_fvar : var -> trm
| trm_abs : trm -> trm
| trm_app : trm -> trm -> trm.

# Step 2. Defining isomorphisms

## Isomorphism module for natural number

Module Iso_nat.

Definition TT := nat.

Definition RR := PLUS UNIT REC.

Fixpoint From (T: TT) : Interpret RR :=
match T with
| O => Term (InL _ Unit)
| S n => Term (InR _ (Rec (From n)))
end.

Fixpoint To_ (T : InterpretRep RR RR) : TT :=
match T return _ with
| InL _ _ Unit => 0
| InR _ _ (Rec U) => S (To U)
| _ => 0
end

with To (T:Interpret RR) : TT :=
match T with
| Var _ => 0
| Term U => To_ U
end.

Lemma To_From : forall (T:TT), To (From T) = T.
Proof.
intro T; induction T; simpl; f_equal; auto.
Qed.

End Iso_nat.

For automation of some proofs:
• This should be mentioned every time a iso module is used below in another definition using Repr.
• Example: Iso_typ.RR occurs in the defintion of Iso_trm.RR below.

Hint Resolve Iso_nat.To_From.

## Isomorphism module for type

Note that atm stands for atomic types, not for type variables.

Module Iso_typ <: Iso_partial.

Definition TT := typ.

Definition RR := PLUS (CONST Iso_nat.RR) (PROD REC REC).

Fixpoint From (x : TT) : Interpret RR :=
match x with
| typ_var k => Term (InL _ (Const _ (Iso_nat.From k)))
| typ_arrow e1 e2 =>
Term (InR _ (Pair (Rec (From e1)) (Rec (From e2))))
end.

Fixpoint To_ (T : InterpretRep RR RR) : typ :=
match T return _ with
| InL _ _ (Const (PLUS UNIT REC) k) =>
typ_var (Iso_nat.To k)
| InR _ _ (Pair _ _ (Rec k1) (Rec k2)) =>
typ_arrow (To k1) (To k2)
| _ => typ_var O
end

with To (T : Interpret RR) : typ :=
match T with
| Term T0 => To_ T0
| _ => typ_var O
end.

Lemma To_From : forall (T:TT), To (From T) = T.

Lemma From_inj : forall T U : TT, From T = From U -> T = U.
Proof.
intros T U H.
rewrite <- (To_From T), <- (To_From U).
rewrite H; auto.
Qed.

Definition ISO : Iso TT (Interpret RR) := (From, To).

End Iso_typ.

Hint Resolve Iso_typ.To_From.

## Isomorphism module for term

For the generic representation of trm, the case of abs case needs special care:
• PROD should come before BIND,
• otherwise, the definition of substitution would be impossible.

Module Iso_trm <: Iso_full.

Definition TT := trm.

Definition RR := PLUS (BIND REC REC) (PROD REC REC).

Fixpoint From (x : TT) : Interpret RR :=
match x with
| trm_bvar j => Var _ (inr j)
| trm_fvar n => Var _ (inl n)
| trm_abs e => Term (InL _ (Bind REC tt (Rec (From e))))
| trm_app e1 e2 =>
Term (InR _ (Pair (Rec (From e1)) (Rec (From e2))))
end.

Fixpoint To_ (T : InterpretRep RR RR) : trm :=
match T return _ with
| InL _ _ (Bind REC _ _ (Rec T1)) =>
trm_abs (To T1)
| InR _ _ (Pair _ _ (Rec T0) (Rec T1)) =>
trm_app (To T0) (To T1)
| _ =>
trm_fvar 0
end

with To (T : Interpret RR) : trm :=
match T with
| Var (inl k) => trm_fvar k
| Var (inr j) => trm_bvar j
| Term T0 => To_ T0
end.

Lemma To_From : forall (T:TT), To (From T) = T.

Lemma From_inj : forall T U : TT, From T = From U -> T = U.

Lemma From_To_Rep : forall (T : InterpretRep RR RR), From (To_ T) = Term T

with From_To : forall (T : Interpret RR), From (To T) = T.
Proof.
from_to_Rep T From_To.
from_to T From_To_Rep.
Qed.

Lemma To_inj : forall T U : Interpret RR, To T = To U -> T = U.
Proof.
intros.
rewrite <- (From_To T), <- (From_To U).
rewrite H; auto.
Qed.

Definition ISO : Iso TT (Interpret RR) := (From, To).

End Iso_trm.

Making arguments implicit

Implicit Arguments Iso_typ.From_inj [T U].

Implicit Arguments Iso_trm.From_inj [T U].
Implicit Arguments Iso_trm.To_inj [T U].