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(** * Linearize: Place all and only operations in let binders *)
Require Import Crypto.Reflection.Syntax.
Require Import Crypto.Reflection.Relations.
Require Import Crypto.Reflection.LinearizeWf.
Require Import Crypto.Reflection.InterpProofs.
Require Import Crypto.Reflection.Linearize.
Require Import Crypto.Util.Tactics Crypto.Util.Sigma Crypto.Util.Prod.
Local Open Scope ctype_scope.
Section language.
Context (base_type_code : Type).
Context (interp_base_type : base_type_code -> Type).
Context (op : flat_type base_type_code -> flat_type base_type_code -> Type).
Context (interp_op : forall src dst, op src dst -> interp_flat_type interp_base_type src -> interp_flat_type interp_base_type dst).
Local Notation flat_type := (flat_type base_type_code).
Local Notation type := (type base_type_code).
Local Notation interp_type := (interp_type interp_base_type).
Local Notation interp_flat_type := (interp_flat_type interp_base_type).
Local Notation exprf := (@exprf base_type_code op).
Local Notation expr := (@expr base_type_code op).
Local Notation Expr := (@Expr base_type_code op).
Local Notation wff := (@wff base_type_code op).
Local Notation wf := (@wf base_type_code op).
Local Hint Extern 1 => eapply interpf_SmartVarVarf.
Local Ltac t_fin :=
repeat match goal with
| _ => reflexivity
| _ => progress unfold LetIn.Let_In
| _ => progress simpl in *
| _ => progress intros
| _ => progress inversion_sigma
| _ => progress inversion_prod
| _ => solve [ intuition eauto ]
| _ => apply (f_equal (interp_op _ _ _))
| _ => apply (f_equal2 (@pair _ _))
| _ => progress specialize_by assumption
| _ => progress subst
| [ H : context[List.In _ (_ ++ _)] |- _ ] => setoid_rewrite List.in_app_iff in H
| [ H : or _ _ |- _ ] => destruct H
| _ => progress break_match
| _ => rewrite <- !surjective_pairing
| [ H : ?x = _, H' : context[?x] |- _ ] => rewrite H in H'
| [ H : _ |- _ ] => apply H
| [ H : _ |- _ ] => rewrite H
end.
Lemma interpf_under_letsf {t tC} (ex : exprf t) (eC : _ -> exprf tC)
: interpf interp_op (under_letsf ex eC) = let x := interpf interp_op ex in interpf interp_op (eC x).
Proof.
clear.
induction ex; t_fin.
Qed.
Lemma interpf_linearizef {t} e
: interpf interp_op (linearizef (t:=t) e) = interpf interp_op e.
Proof.
clear.
induction e;
repeat first [ progress rewrite ?interpf_under_letsf, ?interpf_SmartVarf
| progress simpl
| t_fin ].
Qed.
Local Hint Resolve interpf_linearizef.
Lemma interp_linearize {t} e
: interp_type_gen_rel_pointwise (fun _ => @eq _)
(interp interp_op (linearize (t:=t) e))
(interp interp_op e).
Proof.
induction e; eauto.
eapply interpf_linearizef.
Qed.
Lemma Interp_Linearize {t} (e : Expr t)
: interp_type_gen_rel_pointwise (fun _ => @eq _)
(Interp interp_op (Linearize e))
(Interp interp_op e).
Proof.
unfold Interp, Linearize.
eapply interp_linearize.
Qed.
End language.
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