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Require Import Crypto.Compilers.Syntax.
Require Import Crypto.Util.FixCoqMistakes.
Section language.
Context {base_type_code : Type}.
Section flat.
Context (P : flat_type base_type_code -> Type).
Local Ltac t :=
let H := fresh in
intro H; intros;
match goal with
| [ p : _ |- _ ] => specialize (H _ p)
end;
cbv beta iota in *;
try specialize (H eq_refl); simpl in *;
try assumption.
Definition preinvert_Tbase (Q : forall t, P (Tbase t) -> Type)
: (forall t (p : P t), match t return P t -> Type with Tbase _ => Q _ | _ => fun _ => True end p)
-> forall t p, Q t p.
Proof. t. Defined.
Definition preinvert_Unit (Q : P Unit -> Type)
: (forall t (p : P t), match t return P t -> Type with Unit => Q | _ => fun _ => True end p)
-> forall p, Q p.
Proof. t. Defined.
Definition preinvert_Prod (Q : forall A B, P (Prod A B) -> Type)
: (forall t (p : P t), match t return P t -> Type with Prod _ _ => Q _ _ | _ => fun _ => True end p)
-> forall A B p, Q A B p.
Proof. t. Defined.
Definition preinvert_Prod2 (Q : forall A B, P (Prod (Tbase A) (Tbase B)) -> Type)
: (forall t (p : P t), match t return P t -> Type with Prod (Tbase _) (Tbase _) => Q _ _ | _ => fun _ => True end p)
-> forall A B p, Q A B p.
Proof. t. Defined.
Definition preinvert_Prod2_same (Q : forall A, P (Prod (Tbase A) (Tbase A)) -> Type)
: (forall t (p : P t), match t return P t -> Type with
| Prod (Tbase A) (Tbase B)
=> fun p => forall pf : A = B, Q B (eq_rect _ (fun a => P (Prod (Tbase a) (Tbase B))) p _ pf)
| _ => fun _ => True
end p)
-> forall A p, Q A p.
Proof. t. Defined.
Definition preinvert_Prod3 (Q : forall A B C, P (Tbase A * Tbase B * Tbase C)%ctype -> Type)
: (forall t (p : P t), match t return P t -> Type with Prod (Prod (Tbase _) (Tbase _)) (Tbase _) => Q _ _ _ | _ => fun _ => True end p)
-> forall A B C p, Q A B C p.
Proof. t. Defined.
Definition preinvert_Prod4 (Q : forall A B C D, P (Tbase A * Tbase B * Tbase C * Tbase D)%ctype -> Type)
: (forall t (p : P t), match t return P t -> Type with Prod (Prod (Prod (Tbase _) (Tbase _)) (Tbase _)) (Tbase _) => Q _ _ _ _ | _ => fun _ => True end p)
-> forall A B C D p, Q A B C D p.
Proof. t. Defined.
End flat.
Definition preinvert_Arrow (P : type base_type_code -> Type) (Q : forall A B, P (Arrow A B) -> Type)
: (forall t (p : P t), match t return P t -> Type with
| Arrow A B => Q A B
end p)
-> forall A B p, Q A B p.
Proof.
intros H A B p; specialize (H _ p); assumption.
Defined.
Section encode_decode.
Definition flat_type_code (t1 t2 : flat_type base_type_code) : Prop
:= match t1, t2 with
| Unit, Unit => True
| Tbase t1, Tbase t2 => t1 = t2
| Prod A B, Prod A' B' => A = A' /\ B = B'
| Unit, _
| Tbase _, _
| Prod _ _, _
=> False
end.
Definition type_code (t1 t2 : type base_type_code) : Prop
:= domain t1 = domain t2 /\ codomain t1 = codomain t2.
Definition flat_type_encode (x y : flat_type base_type_code) : x = y -> flat_type_code x y.
Proof. intro p; destruct p, x; repeat constructor. Defined.
Definition type_encode (x y : type base_type_code) : x = y -> type_code x y.
Proof. intro p; destruct p, x; repeat constructor. Defined.
Definition flat_type_decode (x y : flat_type base_type_code) : flat_type_code x y -> x = y.
Proof.
destruct x, y; simpl in *; intro H;
try first [ apply f_equal; assumption
| exfalso; assumption
| reflexivity
| apply f_equal2; destruct H; assumption ].
Defined.
Definition type_decode (x y : type base_type_code) : type_code x y -> x = y.
Proof.
destruct x, y; simpl; intro H;
try first [ exfalso; assumption
| apply f_equal; assumption
| apply f_equal2; destruct H; assumption ].
Defined.
Definition path_flat_type_rect {x y : flat_type base_type_code} (Q : x = y -> Type)
(f : forall p, Q (flat_type_decode x y p))
: forall p, Q p.
Proof. intro p; specialize (f (flat_type_encode x y p)); destruct x, p; exact f. Defined.
Definition path_type_rect {x y : type base_type_code} (Q : x = y -> Type)
(f : forall p, Q (type_decode x y p))
: forall p, Q p.
Proof. intro p; specialize (f (type_encode x y p)); destruct x, p; exact f. Defined.
End encode_decode.
End language.
Ltac preinvert_one_type e :=
lazymatch type of e with
| ?P (Tbase ?T)
=> is_var T;
move e at top;
revert dependent T;
refine (preinvert_Tbase P _ _)
| ?P (Prod (Tbase ?A) (Tbase ?A))
=> is_var A;
move e at top; revert dependent A;
refine (preinvert_Prod2_same P _ _)
| ?P (Prod (Tbase ?A) (Tbase ?B))
=> is_var A; is_var B;
move e at top; revert dependent A; intros A e;
move e at top; revert dependent B; revert A;
refine (preinvert_Prod2 P _ _)
| ?P (Prod (Prod (Tbase ?A) (Tbase ?B)) (Tbase ?C))
=> is_var A; is_var B; is_var C;
move e at top; revert dependent A; intros A e;
move e at top; revert dependent B; intros B e;
move e at top; revert dependent C; revert A B;
refine (preinvert_Prod3 P _ _)
| ?P (Prod (Prod (Prod (Tbase ?A) (Tbase ?B)) (Tbase ?C)) (Tbase ?D))
=> is_var A; is_var B; is_var C; is_var D;
move e at top; revert dependent A; intros A e;
move e at top; revert dependent B; intros B e;
move e at top; revert dependent C; intros C e;
move e at top; revert dependent D; revert A B C;
refine (preinvert_Prod4 P _ _)
| ?P (Prod ?A ?B)
=> is_var A; is_var B;
move e at top; revert dependent A; intros A e;
move e at top; revert dependent B; revert A;
refine (preinvert_Prod P _ _)
| ?P Unit
=> revert dependent e;
refine (preinvert_Unit P _ _)
| ?P (Arrow ?A ?B)
=> is_var A; is_var B;
move e at top; revert dependent A; intros A e;
move e at top; revert dependent B; revert A;
refine (preinvert_Arrow P _ _)
end.
Ltac induction_type_in_using H rect :=
induction H as [H] using (rect _ _ _);
cbv [flat_type_code type_code] in H;
let H1 := fresh H in
let H2 := fresh H in
try lazymatch type of H with
| False => exfalso; exact H
| True => destruct H
| _ /\ _ => destruct H as [H1 H2]
end.
Ltac inversion_flat_type_step :=
lazymatch goal with
| [ H : _ = Tbase _ |- _ ]
=> induction_type_in_using H @path_flat_type_rect
| [ H : Tbase _ = _ |- _ ]
=> induction_type_in_using H @path_flat_type_rect
| [ H : _ = Prod _ _ |- _ ]
=> induction_type_in_using H @path_flat_type_rect
| [ H : Prod _ _ = _ |- _ ]
=> induction_type_in_using H @path_flat_type_rect
| [ H : _ = Unit |- _ ]
=> induction_type_in_using H @path_flat_type_rect
| [ H : Unit = _ |- _ ]
=> induction_type_in_using H @path_flat_type_rect
end.
Ltac inversion_flat_type := repeat inversion_flat_type_step.
Ltac inversion_type_step :=
lazymatch goal with
| [ H : _ = Arrow _ _ |- _ ]
=> induction_type_in_using H @path_type_rect
| [ H : Arrow _ _ = _ |- _ ]
=> induction_type_in_using H @path_type_rect
end.
Ltac inversion_type := repeat inversion_type_step.
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