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+(*** Bounded ℤ-Like Types *)
+(** This file specifies a ℤ-like type of bounded integers, with
+ operations for Montgomery Reduction and Barrett Reduction. *)
+Require Import Coq.ZArith.ZArith.
+Require Import Crypto.Util.ZUtil.
+Require Import Crypto.Util.Tactics.BreakMatch.
+Require Import Crypto.Util.Notations.
+
+Local Open Scope Z_scope.
+
+Class ZLikeOps (small_bound smaller_bound : Z) (modulus : Z) :=
+ {
+ LargeT : Type;
+ SmallT : Type;
+ modulus_digits : SmallT;
+ decode_large : LargeT -> Z;
+ decode_small : SmallT -> Z;
+ Mod_SmallBound : LargeT -> SmallT;
+ DivBy_SmallBound : LargeT -> SmallT;
+ DivBy_SmallerBound : LargeT -> SmallT;
+ Mul : SmallT -> SmallT -> LargeT;
+ CarryAdd : LargeT -> LargeT -> bool * LargeT;
+ CarrySubSmall : SmallT -> SmallT -> bool * SmallT;
+ ConditionalSubtract : bool -> SmallT -> SmallT;
+ ConditionalSubtractModulus : SmallT -> SmallT
+ }.
+
+Delimit Scope small_zlike_scope with small_zlike.
+Delimit Scope large_zlike_scope with large_zlike.
+Local Open Scope small_zlike_scope.
+Local Open Scope large_zlike_scope.
+Local Open Scope Z_scope.
+Bind Scope small_zlike_scope with SmallT.
+Bind Scope large_zlike_scope with LargeT.
+Arguments decode_large (_ _ _)%Z _ _%large_zlike.
+Arguments decode_small (_ _ _)%Z _ _%small_zlike.
+Arguments Mod_SmallBound (_ _ _)%Z _ _%large_zlike.
+Arguments DivBy_SmallBound (_ _ _)%Z _ _%large_zlike.
+Arguments DivBy_SmallerBound (_ _ _)%Z _ _%large_zlike.
+Arguments Mul (_ _ _)%Z _ (_ _)%small_zlike.
+Arguments CarryAdd (_ _ _)%Z _ (_ _)%large_zlike.
+Arguments CarrySubSmall (_ _ _)%Z _ (_ _)%large_zlike.
+Arguments ConditionalSubtract (_ _ _)%Z _ _%bool _%small_zlike.
+Arguments ConditionalSubtractModulus (_ _ _)%Z _ _%small_zlike.
+
+Infix "*" := Mul : large_zlike_scope.
+Notation "x + y" := (snd (CarryAdd x y)) : large_zlike_scope.
+
+Class ZLikeProperties {small_bound smaller_bound modulus : Z} (Zops : ZLikeOps small_bound smaller_bound modulus) :=
+ {
+ large_valid : LargeT -> Prop;
+ medium_valid : LargeT -> Prop;
+ small_valid : SmallT -> Prop;
+ decode_large_valid : forall v, large_valid v -> 0 <= decode_large v < small_bound * small_bound;
+ decode_medium_valid : forall v, medium_valid v -> 0 <= decode_large v < small_bound * smaller_bound;
+ medium_to_large_valid : forall v, medium_valid v -> large_valid v;
+ decode_small_valid : forall v, small_valid v -> 0 <= decode_small v < small_bound;
+ modulus_digits_valid : small_valid modulus_digits;
+ modulus_digits_correct : decode_small modulus_digits = modulus;
+ Mod_SmallBound_valid : forall v, large_valid v -> small_valid (Mod_SmallBound v);
+ Mod_SmallBound_correct
+ : forall v, large_valid v -> decode_small (Mod_SmallBound v) = decode_large v mod small_bound;
+ DivBy_SmallBound_valid : forall v, large_valid v -> small_valid (DivBy_SmallBound v);
+ DivBy_SmallBound_correct
+ : forall v, large_valid v -> decode_small (DivBy_SmallBound v) = decode_large v / small_bound;
+ DivBy_SmallerBound_valid : forall v, medium_valid v -> small_valid (DivBy_SmallerBound v);
+ DivBy_SmallerBound_correct
+ : forall v, medium_valid v -> decode_small (DivBy_SmallerBound v) = decode_large v / smaller_bound;
+ Mul_valid : forall x y, small_valid x -> small_valid y -> large_valid (Mul x y);
+ Mul_correct
+ : forall x y, small_valid x -> small_valid y -> decode_large (Mul x y) = decode_small x * decode_small y;
+ CarryAdd_valid : forall x y, large_valid x -> large_valid y -> large_valid (snd (CarryAdd x y));
+ CarryAdd_correct_fst
+ : forall x y, large_valid x -> large_valid y -> fst (CarryAdd x y) = (small_bound * small_bound <=? decode_large x + decode_large y);
+ CarryAdd_correct_snd
+ : forall x y, large_valid x -> large_valid y -> decode_large (snd (CarryAdd x y)) = (decode_large x + decode_large y) mod (small_bound * small_bound);
+ CarrySubSmall_valid : forall x y, small_valid x -> small_valid y -> small_valid (snd (CarrySubSmall x y));
+ CarrySubSmall_correct_fst
+ : forall x y, small_valid x -> small_valid y -> fst (CarrySubSmall x y) = (decode_small x - decode_small y <? 0);
+ CarrySubSmall_correct_snd
+ : forall x y, small_valid x -> small_valid y -> decode_small (snd (CarrySubSmall x y)) = (decode_small x - decode_small y) mod small_bound;
+ ConditionalSubtract_valid : forall b x, small_valid x -> small_valid (ConditionalSubtract b x);
+ ConditionalSubtract_correct
+ : forall b x, small_valid x -> decode_small (ConditionalSubtract b x)
+ = if b then (decode_small x - decode_small modulus_digits) mod small_bound else decode_small x;
+ ConditionalSubtractModulus_valid : forall x, small_valid x -> small_valid (ConditionalSubtractModulus x);
+ ConditionalSubtractModulus_correct
+ : forall x, small_valid x -> decode_small (ConditionalSubtractModulus x)
+ = if (decode_small x <? decode_small modulus_digits) then decode_small x else decode_small x - decode_small modulus_digits
+ }.
+
+Arguments ZLikeProperties [small_bound smaller_bound modulus] Zops, small_bound smaller_bound modulus {Zops}.
+
+Existing Class large_valid.
+Existing Class medium_valid.
+Existing Class small_valid.
+Existing Instances Mod_SmallBound_valid DivBy_SmallerBound_valid DivBy_SmallBound_valid Mul_valid CarryAdd_valid CarrySubSmall_valid ConditionalSubtract_valid ConditionalSubtractModulus_valid modulus_digits_valid medium_to_large_valid.
+
+Lemma ConditionalSubtractModulus_correct'
+ {small_bound smaller_bound modulus : Z} {Zops : ZLikeOps small_bound smaller_bound modulus}
+ {Zprops : ZLikeProperties Zops}
+ : forall x, small_valid x -> decode_small (ConditionalSubtractModulus x)
+ = if (decode_small modulus_digits <=? decode_small x) then decode_small x - decode_small modulus_digits else decode_small x.
+Proof.
+ intros; rewrite ConditionalSubtractModulus_correct by assumption.
+ break_match; Z.ltb_to_lt; omega.
+Qed.
+
+Lemma modulus_nonneg {small_bound smaller_bound modulus} (Zops : ZLikeOps small_bound smaller_bound modulus) {_ : ZLikeProperties Zops} : 0 <= modulus.
+Proof.
+ pose proof (decode_small_valid _ modulus_digits_valid) as H.
+ rewrite modulus_digits_correct in H.
+ omega.
+Qed.
+
+Create HintDb push_zlike_decode discriminated.
+Create HintDb pull_zlike_decode discriminated.
+Hint Rewrite @Mod_SmallBound_correct @DivBy_SmallBound_correct @DivBy_SmallerBound_correct @Mul_correct @CarryAdd_correct_fst @CarryAdd_correct_snd @CarrySubSmall_correct_fst @CarrySubSmall_correct_snd @ConditionalSubtract_correct @ConditionalSubtractModulus_correct @ConditionalSubtractModulus_correct' @modulus_digits_correct using solve [ typeclasses eauto ] : push_zlike_decode.
+Hint Rewrite <- @Mod_SmallBound_correct @DivBy_SmallBound_correct @DivBy_SmallerBound_correct @Mul_correct @CarryAdd_correct_fst @CarryAdd_correct_snd @CarrySubSmall_correct_fst @CarrySubSmall_correct_snd @ConditionalSubtract_correct @ConditionalSubtractModulus_correct @modulus_digits_correct using solve [ typeclasses eauto ] : pull_zlike_decode.
+
+Ltac get_modulus :=
+ match goal with
+ | [ _ : ZLikeOps _ _ ?modulus |- _ ] => modulus
+ end.
+
+Ltac push_zlike_decode :=
+ let modulus := get_modulus in
+ repeat first [ erewrite !Mod_SmallBound_correct by typeclasses eauto
+ | erewrite !DivBy_SmallBound_correct by typeclasses eauto
+ | erewrite !DivBy_SmallerBound_correct by typeclasses eauto
+ | erewrite !Mul_correct by typeclasses eauto
+ | erewrite !CarryAdd_correct_fst by typeclasses eauto
+ | erewrite !CarryAdd_correct_snd by typeclasses eauto
+ | erewrite !CarrySubSmall_correct_fst by typeclasses eauto
+ | erewrite !CarrySubSmall_correct_snd by typeclasses eauto
+ | erewrite !ConditionalSubtract_correct by typeclasses eauto
+ | erewrite !ConditionalSubtractModulus_correct by typeclasses eauto
+ | erewrite !ConditionalSubtractModulus_correct' by typeclasses eauto
+ | erewrite !(@modulus_digits_correct _ modulus _ _) by typeclasses eauto ].
+Ltac pull_zlike_decode :=
+ let modulus := get_modulus in
+ repeat first [ match goal with
+ | [ |- context G[modulus] ]
+ => let G' := context G[decode_small modulus_digits] in
+ cut G'; [ rewrite !modulus_digits_correct by typeclasses eauto; exact (fun x => x) | ]
+ end
+ | erewrite <- !Mod_SmallBound_correct by typeclasses eauto
+ | erewrite <- !DivBy_SmallBound_correct by typeclasses eauto
+ | erewrite <- !DivBy_SmallerBound_correct by typeclasses eauto
+ | erewrite <- !Mul_correct by typeclasses eauto
+ | erewrite <- !CarryAdd_correct_fst by typeclasses eauto
+ | erewrite <- !CarryAdd_correct_snd by typeclasses eauto
+ | erewrite <- !ConditionalSubtract_correct by typeclasses eauto
+ | erewrite <- !CarrySubSmall_correct_fst by typeclasses eauto
+ | erewrite <- !CarrySubSmall_correct_snd by typeclasses eauto
+ | erewrite <- !ConditionalSubtractModulus_correct by typeclasses eauto
+ | erewrite <- !ConditionalSubtractModulus_correct' by typeclasses eauto
+ | erewrite <- !(@modulus_digits_correct _ modulus _ _) by typeclasses eauto ].
generated by cgit v1.2.3 (git 2.39.1) at 2024-06-13 01:05:14 +0000