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Require Import Crypto.Compilers.Syntax.
Require Import Crypto.Compilers.Wf.
Require Import Crypto.Compilers.ExprInversion.
Require Import Crypto.Util.Sigma.
Require Import Crypto.Util.Option.
Require Import Crypto.Util.Equality.
Require Import Crypto.Util.Tactics.BreakMatch.
Require Import Crypto.Util.Tactics.DestructHead.
Require Import Crypto.Util.Notations.
Section language.
Context {base_type_code : Type}
{op : flat_type base_type_code -> flat_type base_type_code -> Type}.
Local Notation flat_type := (flat_type base_type_code).
Local Notation type := (type base_type_code).
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 Notation Wf := (@Wf base_type_code op).
Section with_var.
Context {var1 var2 : base_type_code -> Type}.
Local Notation eP := (fun t => var1 t * var2 t)%type (only parsing).
Local Notation "x == y" := (existT eP _ (x, y)).
Definition wff_code (G : list (sigT eP)) {t} (e1 : @exprf var1 t) : forall (e2 : @exprf var2 t), Prop
:= match e1 in Syntax.exprf _ _ t return exprf t -> Prop with
| TT
=> fun e2
=> TT = e2
| Var t v1
=> fun e2
=> match invert_Var e2 with
| Some v2 => List.In (v1 == v2) G
| None => False
end
| Op t1 tR opc1 args1
=> fun e2
=> match invert_Op e2 with
| Some (existT t2 (opc2, args2))
=> { pf : t1 = t2
| eq_rect _ (fun t => op t tR) opc1 _ pf = opc2
/\ wff G (eq_rect _ exprf args1 _ pf) args2 }
| None => False
end
| LetIn tx1 ex1 tC1 eC1
=> fun e2
=> match invert_LetIn e2 with
| Some (existT tx2 (ex2, eC2))
=> { pf : tx1 = tx2
| wff G (eq_rect _ exprf ex1 _ pf) ex2
/\ (forall x1 x2,
wff (flatten_binding_list x1 x2 ++ G)%list
(eC1 x1) (eC2 (eq_rect _ _ x2 _ pf))) }
| None => False
end
| Pair tx1 ex1 ty1 ey1
=> fun e2
=> match invert_Pair e2 with
| Some (ex2, ey2) => wff G ex1 ex2 /\ wff G ey1 ey2
| None => False
end
end.
Local Ltac t :=
repeat match goal with
| _ => progress simpl in *
| _ => progress subst
| _ => progress inversion_option
| _ => progress invert_expr_subst
| [ H : Some _ = _ |- _ ] => symmetry in H
| _ => assumption
| _ => reflexivity
| _ => constructor
| _ => progress destruct_head False
| _ => progress destruct_head and
| _ => progress destruct_head sig
| _ => progress break_match_hyps
| _ => progress break_match
| [ |- and _ _ ] => split
| _ => exists eq_refl
| _ => intro
| [ e : expr (Arrow _ _) |- _ ]
=> let H := fresh in
let f := fresh in
remember (invert_Abs e) as f eqn:H;
symmetry in H;
apply invert_Abs_Some in H
end.
Definition wff_encode {G t e1 e2} (v : @wff var1 var2 G t e1 e2) : @wff_code G t e1 e2.
Proof.
destruct v; t.
Defined.
Definition wff_decode {G t e1 e2} (v : @wff_code G t e1 e2) : @wff var1 var2 G t e1 e2.
Proof.
destruct e1; t.
Defined.
Definition wff_endecode {G t e1 e2} v : @wff_decode G t e1 e2 (@wff_encode G t e1 e2 v) = v.
Proof using Type.
destruct v; reflexivity.
Qed.
Definition wff_deencode {G t e1 e2} v : @wff_encode G t e1 e2 (@wff_decode G t e1 e2 v) = v.
Proof using Type.
destruct e1; simpl in *;
move e2 at top;
lazymatch type of e2 with
| exprf Unit
=> subst; reflexivity
| exprf (Tbase ?t)
=> revert dependent t;
intros ? e2
| exprf (Prod ?A ?B)
=> revert dependent A;
intros ? e2;
move e2 at top;
revert dependent B;
intros ? e2
| exprf ?t
=> revert dependent t;
intros ? e2
end;
refine match e2 with
| TT => _
| _ => _
end;
t.
Qed.
Definition wf_code {t} (e1 : @expr var1 t) : forall (e2 : @expr var2 t), Prop
:= match e1 in Syntax.expr _ _ t return expr t -> Prop with
| Abs src dst f1
=> fun e2
=> let f2 := invert_Abs e2 in
forall (x : interp_flat_type var1 src) (x' : interp_flat_type var2 src),
wff (flatten_binding_list x x') (f1 x) (f2 x')
end.
Definition wf_encode {t e1 e2} (v : @wf var1 var2 t e1 e2) : @wf_code t e1 e2.
Proof.
destruct v; t.
Defined.
Definition wf_decode {t e1 e2} (v : @wf_code t e1 e2) : @wf var1 var2 t e1 e2.
Proof.
destruct e1; t.
Defined.
Definition wf_endecode {t e1 e2} v : @wf_decode t e1 e2 (@wf_encode t e1 e2 v) = v.
Proof using Type.
destruct v; reflexivity.
Qed.
Definition wf_deencode {t e1 e2} v : @wf_encode t e1 e2 (@wf_decode t e1 e2 v) = v.
Proof using Type.
destruct e1 as [src dst f1].
revert dependent f1.
refine match e2 with
| Abs _ _ f2 => _
end.
reflexivity.
Qed.
End with_var.
End language.
Ltac is_expr_constructor arg :=
lazymatch arg with
| Op _ _ => idtac
| TT => idtac
| Var _ => idtac
| LetIn _ _ => idtac
| Pair _ _ => idtac
| Abs _ => idtac
end.
Ltac inversion_wf_step_gen guard_tac :=
let postprocess H :=
(cbv [wff_code wf_code] in H;
simpl in H;
try match type of H with
| True => clear H
| False => exfalso; exact H
end) in
match goal with
| [ H : wff _ ?x ?y |- _ ]
=> guard_tac x y;
apply wff_encode in H; postprocess H
| [ H : wf ?x ?y |- _ ]
=> guard_tac x y;
apply wf_encode in H; postprocess H
end.
Ltac inversion_wf_step_constr :=
inversion_wf_step_gen ltac:(fun x y => is_expr_constructor x; is_expr_constructor y).
Ltac inversion_wf_step_one_constr :=
inversion_wf_step_gen ltac:(fun x y => first [ is_expr_constructor x | is_expr_constructor y]).
Ltac inversion_wf_step_var :=
inversion_wf_step_gen ltac:(fun x y => first [ is_var x; is_var y; fail 1 | idtac ]).
Ltac inversion_wf_step := first [ inversion_wf_step_constr | inversion_wf_step_var ].
Ltac inversion_wf_constr := repeat inversion_wf_step_constr.
Ltac inversion_wf_one_constr := repeat inversion_wf_step_one_constr.
Ltac inversion_wf := repeat inversion_wf_step.
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