LamSF_Marks.v

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(* 02110-1301 USA                                                     *)
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(*             Intensional Lambda Calculus                            *)
(*                                                                    *)
(* is implemented in Coq by adapting the implementation of            *) 
(* Lambda Calculus from Project Coq                                   *)
(* 2015                                                               *)
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(*                        LamSF_Marks.v                               *)
(*                                                                    *)
(* adapted from Marks.v for Lambda Calculus                           *)
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(*                          Barry Jay                                 *)
(*                                                                    *)
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Require Import Arith.
Require Import Test.
Require Import General.
Require Import LamSF_Terms.
Require Import LamSF_Redexes.
Require Import LamSF_Residuals.

(* Translation from terms to redexes *)

Fixpoint mark (e : lamSF) : redexes :=
  match e with
  | Ref i => Var i
  | Op o => Opp o
  | App M N => Ap false (mark M) (mark N)
  | Abs M => Fun (mark M)
  end. 


(* Reverse translation : erasing the marks *)

Fixpoint unmark (e : redexes) : lamSF :=
  match e with
  | Var i => Ref i
  | Opp o => Op o
  | Ap b U V => App (unmark U) (unmark V)
  | Fun U => Abs (unmark U)
  end.

Lemma inverse : forall M : lamSF, M = unmark (mark M).
Proof.
simple induction M; simpl in |- *; trivial; simple induction 1; trivial.
simple induction 1; trivial.
Qed.

Lemma comp_unmark_eq : forall U V : redexes, comp U V -> unmark U = unmark V.
Proof.
simple induction 1; simpl in |- *; trivial; split_all. 
Qed.

(* The converse is true, but not needed in the rest of the development *)

Lemma residuals_mark :
forall M, residuals (mark M) (mark M) (mark M).
Proof. induction M; split_all. Qed.

Hint Resolve residuals_mark.