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Require Import Coq.ZArith.ZArith.
Require Import Coq.micromega.Lia.
Require Import Coq.Lists.List. Import ListNotations.
Require Import Crypto.Fancy.Spec.
Require Import Crypto.Fancy.Compiler.
Require Import Crypto.Util.Tactics.BreakMatch.
Import Spec.Registers.
Section Prod.
Definition Mul256 (out src1 src2 tmp : register) (cont : expr) : expr :=
Instr MUL128LL out (src1, src2)
(Instr MUL128UL tmp (src1, src2)
(Instr (ADD 128) out (out, tmp)
(Instr MUL128LU tmp (src1, src2)
(Instr (ADD 128) out (out, tmp) cont)))).
Definition Mul256x256 (out outHigh src1 src2 tmp : register) (cont : expr) : expr :=
Instr MUL128LL out (src1, src2)
(Instr MUL128UU outHigh (src1, src2)
(Instr MUL128UL tmp (src1, src2)
(Instr (ADD 128) out (out, tmp)
(Instr (ADDC (-128)) outHigh (outHigh, tmp)
(Instr MUL128LU tmp (src1, src2)
(Instr (ADD 128) out (out, tmp)
(Instr (ADDC (-128)) outHigh (outHigh, tmp) cont))))))).
Definition MontRed256 lo hi y t1 t2 scratch RegPInv : @expr register :=
Mul256 y lo RegPInv t1
(Mul256x256 t1 t2 y RegMod scratch
(Instr (ADD 0) lo (lo, t1)
(Instr (ADDC 0) hi (hi, t2)
(Instr SELC y (RegMod, RegZero)
(Instr (SUB 0) lo (hi, y)
(Instr ADDM lo (lo, RegZero, RegMod)
(Ret lo))))))).
(* Barrett reduction -- this is only the "reduce" part, excluding the initial multiplication. *)
Definition MulMod x xHigh RegMuLow scratchp1 scratchp2 scratchp3 scratchp4 scratchp5 : @expr register :=
let q1Bottom256 := scratchp1 in
let muSelect := scratchp2 in
let q2 := scratchp3 in
let q2High := scratchp4 in
let q2High2 := scratchp5 in
let q3 := scratchp1 in
let r2 := scratchp2 in
let r2High := scratchp3 in
let maybeM := scratchp1 in
Instr SELM muSelect (RegMuLow, RegZero)
(Instr (RSHI 255) q1Bottom256 (xHigh, x)
(Mul256x256 q2 q2High q1Bottom256 RegMuLow scratchp5
(Instr (RSHI 255) q2High2 (RegZero, xHigh)
(Instr (ADD 0) q2High (q2High, q1Bottom256)
(Instr (ADDC 0) q2High2 (q2High2, RegZero)
(Instr (ADD 0) q2High (q2High, muSelect)
(Instr (ADDC 0) q2High2 (q2High2, RegZero)
(Instr (RSHI 1) q3 (q2High2, q2High)
(Mul256x256 r2 r2High RegMod q3 scratchp4
(Instr (SUB 0) muSelect (x, r2)
(Instr (SUBC 0) xHigh (xHigh, r2High)
(Instr SELL maybeM (RegMod, RegZero)
(Instr (SUB 0) q3 (muSelect, maybeM)
(Instr ADDM x (q3, RegZero, RegMod)
(Ret x))))))))))))))).
End Prod.
Section ProdEquiv.
Context (wordmax : Z).
Let interp256 := interp reg_eqb wordmax cc_spec.
Lemma cc_overwrite_full x1 x2 l1 cc :
CC.update [CC.C; CC.M; CC.L; CC.Z] x2 cc_spec (CC.update l1 x1 cc_spec cc) = CC.update [CC.C; CC.M; CC.L; CC.Z] x2 cc_spec cc.
Proof.
cbv [CC.update]. cbn [CC.cc_c CC.cc_m CC.cc_l CC.cc_z].
break_match; try match goal with H : ~ In _ _ |- _ => cbv [In] in H; tauto end.
reflexivity.
Qed.
Definition value_unused r e : Prop :=
forall x cc ctx, interp256 e cc ctx = interp256 e cc (fun r' => if reg_eqb r' r then x else ctx r').
Lemma value_unused_skip r i rd args cont (Hcont: value_unused r cont) :
r <> rd ->
(~ In r (Tuple.to_list _ args)) ->
value_unused r (Instr i rd args cont).
Proof.
cbv [value_unused interp256] in *; intros.
rewrite !interp_step; cbv zeta.
rewrite Hcont with (x:=x).
match goal with |- ?lhs = ?rhs =>
match lhs with context [Tuple.map ?f ?t] =>
match rhs with context [Tuple.map ?g ?t] =>
rewrite (Tuple.map_ext_In f g) by (intros; cbv [reg_eqb]; break_match; congruence)
end end end.
apply interp_state_equiv; [ congruence | ].
{ intros; cbv [reg_eqb] in *; break_match; congruence. }
Qed.
Lemma value_unused_overwrite r i args cont :
(~ In r (Tuple.to_list _ args)) ->
value_unused r (Instr i r args cont).
Proof.
cbv [value_unused interp256]; intros; rewrite !interp_step; cbv zeta.
match goal with |- ?lhs = ?rhs =>
match lhs with context [Tuple.map ?f ?t] =>
match rhs with context [Tuple.map ?g ?t] =>
rewrite (Tuple.map_ext_In f g) by (intros; cbv [reg_eqb]; break_match; congruence)
end end end.
apply interp_state_equiv; [ congruence | ].
{ intros; cbv [reg_eqb] in *; break_match; congruence. }
Qed.
Lemma value_unused_ret r r' :
r <> r' ->
value_unused r (Ret r').
Proof.
cbv - [reg_dec]; intros.
break_match; congruence.
Qed.
Lemma interp_Mul256 out src1 src2 tmp tmp2 cont cc ctx:
out <> src1 ->
out <> src2 ->
out <> tmp ->
out <> tmp2 ->
src1 <> src2 ->
src1 <> tmp ->
src1 <> tmp2 ->
src2 <> tmp ->
src2 <> tmp2 ->
tmp <> tmp2 ->
value_unused tmp cont ->
value_unused tmp2 cont ->
interp256 (Mul256 out src1 src2 tmp cont) cc ctx =
interp256 (
Instr MUL128LU tmp (src1, src2)
(Instr MUL128UL tmp2 (src1, src2)
(Instr MUL128LL out (src1, src2)
(Instr (ADD 128) out (out, tmp2)
(Instr (ADD 128) out (out, tmp) cont))))) cc ctx.
Proof.
intros; cbv [Mul256 interp256].
repeat
(rewrite interp_step; cbn - [interp spec cc_spec];
rewrite ?reg_eqb_refl, ?reg_eqb_neq by congruence;
remember_single_result;
cbn [spec MUL128LL MUL128LU MUL128UL ADD] in *).
match goal with H : value_unused tmp _ |- _ => erewrite H end.
match goal with H : value_unused tmp2 _ |- _ => erewrite H end.
apply interp_state_equiv.
{ rewrite !cc_overwrite_full.
f_equal; subst; lia. }
{ intros; cbv [reg_eqb].
break_innermost_match; try congruence; reflexivity. }
Qed.
Lemma interp_Mul256x256 out outHigh src1 src2 tmp tmp2 cont cc ctx:
out <> src1 ->
out <> outHigh ->
out <> src2 ->
out <> tmp ->
out <> tmp2 ->
outHigh <> src1 ->
outHigh <> src2 ->
outHigh <> tmp ->
outHigh <> tmp2 ->
src1 <> src2 ->
src1 <> tmp ->
src1 <> tmp2 ->
src2 <> tmp ->
src2 <> tmp2 ->
tmp <> tmp2 ->
value_unused tmp cont ->
value_unused tmp2 cont ->
interp256 (Mul256x256 out outHigh src1 src2 tmp cont) cc ctx =
interp256 (
Instr MUL128LL out (src1, src2)
(Instr MUL128LU tmp (src1, src2)
(Instr MUL128UL tmp2 (src1, src2)
(Instr MUL128UU outHigh (src1, src2)
(Instr (ADD 128) out (out, tmp2)
(Instr (ADDC (-128)) outHigh (outHigh, tmp2)
(Instr (ADD 128) out (out, tmp)
(Instr (ADDC (-128)) outHigh (outHigh, tmp) cont)))))))) cc ctx.
Proof.
intros; cbv [Mul256x256 interp256].
repeat
(rewrite interp_step; cbn - [interp spec cc_spec];
rewrite ?reg_eqb_refl, ?reg_eqb_neq by congruence;
remember_single_result;
cbn [spec MUL128LL MUL128LU MUL128UL MUL128UU ADD ADDC] in *).
match goal with H : value_unused tmp _ |- _ => erewrite H end.
match goal with H : value_unused tmp2 _ |- _ => erewrite H end.
apply interp_state_equiv.
{ rewrite !cc_overwrite_full.
f_equal.
subst. cbn - [Z.add Z.modulo Z.testbit Z.mul Z.shiftl lower128 upper128].
lia. }
{ intros; cbv [reg_eqb].
break_innermost_match; try congruence; try reflexivity; [ ].
subst. cbn - [Z.add Z.modulo Z.testbit Z.mul Z.shiftl lower128 upper128].
lia. }
Qed.
End ProdEquiv.
Ltac push_value_unused :=
repeat match goal with
| |- ~ In _ _ => cbn; intuition; congruence
| _ => apply value_unused_overwrite
| _ => apply value_unused_skip; [ | congruence | ]
| _ => apply value_unused_ret; congruence
end.
|
