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Require Import Coq.Classes.Morphisms.
Require Import Coq.ZArith.ZArith.
Require Import Coq.micromega.Psatz.
Require Import Crypto.Compilers.Z.Syntax.
Require Import Crypto.Compilers.Z.Syntax.Util.
Require Import Crypto.Compilers.Syntax.
Require Import Crypto.Compilers.Z.Bounds.Interpretation.
Require Import Crypto.Compilers.Z.Bounds.InterpretationLemmas.Tactics.
Require Import Crypto.Compilers.SmartMap.
Require Import Crypto.Util.ZUtil.
Require Import Crypto.Util.ZUtil.Stabilization.
Require Import Crypto.Util.Bool.
Require Import Crypto.Util.FixedWordSizesEquality.
Require Import Crypto.Util.Tactics.DestructHead.
Require Import Crypto.Util.Tactics.BreakMatch.
Require Import Crypto.Util.Tactics.UniquePose.
Require Import Crypto.Util.Tactics.SpecializeBy.
Local Open Scope Z_scope.
Local Arguments Bounds.is_bounded_by' !_ _ _ / .
Lemma is_bounded_by_truncation_bounds Tout bs v
(H : Bounds.is_bounded_by (T:=Tbase TZ) bs v)
: Bounds.is_bounded_by (T:=Tbase Tout)
(Bounds.truncation_bounds (Bounds.bit_width_of_base_type Tout) bs)
(ZToInterp v).
Proof.
destruct bs as [l u]; cbv [Bounds.truncation_bounds Bounds.is_bounded_by Bounds.is_bounded_by' Bounds.bit_width_of_base_type Bounds.log_bit_width_of_base_type option_map ZRange.is_bounded_by'] in *; simpl in *.
repeat first [ break_t_step
| fin_t
| progress simpl in *
| Zarith_t_step
| rewriter_t
| word_arith_t ].
Qed.
Lemma monotone_four_corners_genb
(f : Z -> Z -> Z)
(R := fun b : bool => if b then Z.le else Basics.flip Z.le)
(Hmonotone1 : forall x, exists b, Proper (R b ==> Z.le) (f x))
(Hmonotone2 : forall y, exists b, Proper (R b ==> Z.le) (fun x => f x y))
x_bs y_bs x y
(Hboundedx : ZRange.is_bounded_by' None x_bs x)
(Hboundedy : ZRange.is_bounded_by' None y_bs y)
: ZRange.is_bounded_by' None (Bounds.four_corners f x_bs y_bs) (f x y).
Proof.
unfold ZRange.is_bounded_by' in *; split; trivial.
destruct x_bs as [lx ux], y_bs as [ly uy]; simpl in *.
destruct Hboundedx as [Hboundedx _], Hboundedy as [Hboundedy _].
pose proof (Hmonotone1 lx); pose proof (Hmonotone1 x); pose proof (Hmonotone1 ux).
pose proof (Hmonotone2 ly); pose proof (Hmonotone2 y); pose proof (Hmonotone2 uy).
destruct_head'_ex.
repeat match goal with
| [ H : Proper (R ?b ==> Z.le) (f _) |- _ ]
=> unique assert (R b (if b then ly else y) (if b then y else ly)
/\ R b (if b then y else uy) (if b then uy else y))
by (unfold R, Basics.flip; destruct b; omega)
| [ H : Proper (R ?b ==> Z.le) (fun x => f x _) |- _ ]
=> unique assert (R b (if b then lx else x) (if b then x else lx)
/\ R b (if b then x else ux) (if b then ux else x))
by (unfold R, Basics.flip; destruct b; omega)
end.
destruct_head' and.
repeat match goal with
| [ H : Proper (R ?b ==> Z.le) _, H' : R ?b _ _ |- _ ]
=> unique pose proof (H _ _ H')
end.
destruct_head bool; split_min_max; omega.
Qed.
Lemma monotone_four_corners_gen
(f : Z -> Z -> Z)
(Hmonotone1 : forall x, Proper (Z.le ==> Z.le) (f x) \/ Proper (Basics.flip Z.le ==> Z.le) (f x))
(Hmonotone2 : forall y, Proper (Z.le ==> Z.le) (fun x => f x y) \/ Proper (Basics.flip Z.le ==> Z.le) (fun x => f x y))
x_bs y_bs x y
(Hboundedx : ZRange.is_bounded_by' None x_bs x)
(Hboundedy : ZRange.is_bounded_by' None y_bs y)
: ZRange.is_bounded_by' None (Bounds.four_corners f x_bs y_bs) (f x y).
Proof.
eapply monotone_four_corners_genb; auto.
{ intro x'; destruct (Hmonotone1 x'); [ exists true | exists false ]; assumption. }
{ intro x'; destruct (Hmonotone2 x'); [ exists true | exists false ]; assumption. }
Qed.
Lemma monotone_four_corners
(b1 b2 : bool)
(f : Z -> Z -> Z)
(R1 := if b1 then Z.le else Basics.flip Z.le) (R2 := if b2 then Z.le else Basics.flip Z.le)
(Hmonotone : Proper (R1 ==> R2 ==> Z.le) f)
x_bs y_bs x y
(Hboundedx : ZRange.is_bounded_by' None x_bs x)
(Hboundedy : ZRange.is_bounded_by' None y_bs y)
: ZRange.is_bounded_by' None (Bounds.four_corners f x_bs y_bs) (f x y).
Proof.
apply monotone_four_corners_genb; auto; intro x'; subst R1 R2;
[ exists b2 | exists b1 ];
[ eapply (Hmonotone x' x'); destruct b1; reflexivity
| intros ???; apply Hmonotone; auto; destruct b2; reflexivity ].
Qed.
Lemma monotonify2 (f : Z -> Z -> Z) (upper : Z -> Z -> Z)
(Hbounded : forall a b, Z.abs (f a b) <= upper (Z.abs a) (Z.abs b))
(Hupper_monotone : Proper (Z.le ==> Z.le ==> Z.le) upper)
{xb yb x y}
(Hboundedx : ZRange.is_bounded_by' None xb x)
(Hboundedy : ZRange.is_bounded_by' None yb y)
: ZRange.is_bounded_by'
None
{| ZRange.lower := -upper (Bounds.max_abs_bound xb) (Bounds.max_abs_bound yb);
ZRange.upper := upper (Bounds.max_abs_bound xb) (Bounds.max_abs_bound yb) |}
(f x y).
Proof.
split; [ | exact I ]; simpl.
apply Z.abs_le.
destruct Hboundedx as [Hx _], Hboundedy as [Hy _].
etransitivity; [ apply Hbounded | ].
apply Hupper_monotone;
unfold Bounds.max_abs_bound;
repeat (apply Z.max_case_strong || apply Zabs_ind); omega.
Qed.
Local Existing Instances Z.log2_up_le_Proper Z.add_le_Proper.
Lemma land_upper_lor_land_bounds a b
: Z.abs (Z.land a b) <= Bounds.upper_lor_and_bounds (Z.abs a) (Z.abs b).
Proof.
unfold Bounds.upper_lor_and_bounds.
apply stabilizes_bounded; auto with zarith.
rewrite <- !Z.max_mono by exact _.
apply land_stabilizes; apply stabilization_time_weaker.
Qed.
Lemma lor_upper_lor_land_bounds a b
: Z.abs (Z.lor a b) <= Bounds.upper_lor_and_bounds (Z.abs a) (Z.abs b).
Proof.
unfold Bounds.upper_lor_and_bounds.
apply stabilizes_bounded; auto with zarith.
rewrite <- !Z.max_mono by exact _.
apply lor_stabilizes; apply stabilization_time_weaker.
Qed.
Lemma upper_lor_and_bounds_Proper
: Proper (Z.le ==> Z.le ==> Z.le) Bounds.upper_lor_and_bounds.
Proof.
intros ??? ???.
unfold Bounds.upper_lor_and_bounds.
auto with zarith.
Qed.
Local Arguments Z.pow !_ !_.
Local Arguments Z.log2_up !_.
Local Arguments Z.add !_ !_.
Lemma land_bounds_extreme xb yb x y
(Hx : ZRange.is_bounded_by' None xb x)
(Hy : ZRange.is_bounded_by' None yb y)
: ZRange.is_bounded_by' None (Bounds.extreme_lor_land_bounds xb yb) (Z.land x y).
Proof.
apply monotonify2; auto;
unfold Bounds.extreme_lor_land_bounds;
[ apply land_upper_lor_land_bounds
| apply upper_lor_and_bounds_Proper ].
Qed.
Lemma lor_bounds_extreme xb yb x y
(Hx : ZRange.is_bounded_by' None xb x)
(Hy : ZRange.is_bounded_by' None yb y)
: ZRange.is_bounded_by' None (Bounds.extreme_lor_land_bounds xb yb) (Z.lor x y).
Proof.
apply monotonify2; auto;
unfold Bounds.extreme_lor_land_bounds;
[ apply lor_upper_lor_land_bounds
| apply upper_lor_and_bounds_Proper ].
Qed.
Local Arguments N.ldiff : simpl never.
Local Arguments Z.pow : simpl never.
Local Arguments Z.add !_ !_.
Local Existing Instances Z.add_le_Proper Z.sub_le_flip_le_Proper Z.log2_up_le_Proper Z.pow_Zpos_le_Proper Z.sub_le_eq_Proper.
Local Hint Extern 1 => progress cbv beta iota : typeclass_instances.
Lemma is_bounded_by_interp_op t tR (opc : op t tR)
(bs : interp_flat_type Bounds.interp_base_type _)
(v : interp_flat_type interp_base_type _)
(H : Bounds.is_bounded_by bs v)
: Bounds.is_bounded_by (Bounds.interp_op opc bs) (Syntax.interp_op _ _ opc v).
Proof.
destruct opc; apply is_bounded_by_truncation_bounds;
repeat first [ progress simpl in *
| progress cbv [interp_op lift_op cast_const Bounds.interp_base_type Bounds.is_bounded_by' ZRange.is_bounded_by'] in *
| progress destruct_head'_prod
| progress destruct_head'_and
| omega
| match goal with
| [ |- context[interpToZ ?x] ]
=> generalize dependent (interpToZ x); clear x; intros
| [ |- _ /\ True ] => split; [ | tauto ]
end ].
{ apply (@monotone_four_corners true true _ _); split; auto. }
{ apply (@monotone_four_corners true false _ _); split; auto. }
{ apply monotone_four_corners_genb; try (split; auto);
unfold Basics.flip;
let x := fresh "x" in
intro x;
exists (0 <=? x);
break_match; Z.ltb_to_lt;
intros ???; nia. }
{ apply monotone_four_corners_genb; try (split; auto);
[ eexists; apply Z.shiftl_le_Proper1
| exists true; apply Z.shiftl_le_Proper2 ]. }
{ apply monotone_four_corners_genb; try (split; auto);
[ eexists; apply Z.shiftr_le_Proper1
| exists true; apply Z.shiftr_le_Proper2 ]. }
{ break_innermost_match;
try (apply land_bounds_extreme; split; auto);
repeat first [ progress simpl in *
| Zarith_t_step
| rewriter_t
| progress saturate_land_lor_facts
| split_min_max; omega ]. }
{ break_innermost_match;
try (apply lor_bounds_extreme; split; auto);
repeat first [ progress simpl in *
| Zarith_t_step
| rewriter_t
| progress Zarith_land_lor_t_step
| solve [ split_min_max; try omega; try Zarith_land_lor_t_step ] ]. }
{ repeat first [ progress destruct_head Bounds.t
| progress simpl in *
| progress split_min_max
| omega ]. }
Qed.
|
