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Require Import Coq.ZArith.ZArith.
Require Import Coq.Lists.List.
Local Open Scope Z_scope.
Require Import Crypto.Arithmetic.Core.
Require Import Crypto.Arithmetic.Saturated.Core.
Require Import Crypto.Arithmetic.Saturated.UniformWeight.
Require Import Crypto.Util.ZUtil.Definitions.
Require Import Crypto.Util.ZUtil.AddGetCarry.
Require Import Crypto.Util.Tuple Crypto.Util.LetIn.
Local Notation "A ^ n" := (tuple A n) : type_scope.
Module B.
Module Positional.
Section Positional.
Context {s:Z} {s_pos : 0 < s}. (* s is bitwidth *)
Let small {n} := @small s n.
Section GenericOp.
Context {op : Z -> Z -> Z}
{op_get_carry : Z -> Z -> Z * Z} (* no carry in, carry out *)
{op_with_carry : Z -> Z -> Z -> Z * Z}. (* carry in, carry out *)
Section chain_op'_cps.
Context (T : Type).
Fixpoint chain_op'_cps {n} (c:option Z) (p q:Z^n)
: (Z*Z^n->T)->T :=
match n return option Z -> Z^n -> Z^n -> (Z*Z^n -> T) -> T with
| O => fun c p _ f =>
let carry := match c with | None => 0 | Some x => x end in
f (carry,p)
| S n' =>
fun c p q f =>
(* for the first call, use op_get_carry, then op_with_carry *)
let op' := match c with
| None => op_get_carry
| Some x => op_with_carry x end in
dlet carry_result := op' (hd p) (hd q) in
chain_op'_cps (Some (snd carry_result)) (tl p) (tl q)
(fun carry_pq =>
f (fst carry_pq,
append (fst carry_result) (snd carry_pq)))
end c p q.
End chain_op'_cps.
Definition chain_op' {n} c p q := @chain_op'_cps _ n c p q id.
Definition chain_op_cps {n} p q {T} f := @chain_op'_cps T n None p q f.
Definition chain_op {n} p q : Z * Z^n := chain_op_cps p q id.
Lemma chain_op'_id {n} : forall c p q T f,
@chain_op'_cps T n c p q f = f (chain_op' c p q).
Proof.
cbv [chain_op']; induction n; intros; destruct c;
simpl chain_op'_cps; cbv [Let_In]; try reflexivity.
{ etransitivity; rewrite IHn; reflexivity. }
{ etransitivity; rewrite IHn; reflexivity. }
Qed.
Lemma chain_op_id {n} p q T f :
@chain_op_cps n p q T f = f (chain_op p q).
Proof. apply (@chain_op'_id n None). Qed.
End GenericOp.
Hint Opaque chain_op chain_op' : uncps.
Hint Rewrite @chain_op_id @chain_op'_id : uncps.
Section AddSub.
Create HintDb divmod discriminated.
Hint Rewrite
Z.add_get_carry_full_mod
Z.add_get_carry_full_div
Z.add_with_get_carry_full_mod
Z.add_with_get_carry_full_div
Z.sub_get_borrow_full_mod
Z.sub_get_borrow_full_div
Z.sub_with_get_borrow_full_mod
Z.sub_with_get_borrow_full_div
: divmod.
Let eval {n} := B.Positional.eval (n:=n) (uweight s).
Definition sat_add_cps {n} p q T (f:Z*Z^n->T) :=
chain_op_cps (op_get_carry := Z.add_get_carry_full s)
(op_with_carry := Z.add_with_get_carry_full s)
p q f.
Definition sat_add {n} p q := @sat_add_cps n p q _ id.
Lemma sat_add_id n p q T f :
@sat_add_cps n p q T f = f (sat_add p q).
Proof. cbv [sat_add sat_add_cps]. rewrite !chain_op_id. reflexivity. Qed.
Lemma sat_add_mod_step n c d :
c mod s + s * ((d + c / s) mod (uweight s n))
= (s * d + c) mod (s * uweight s n).
Proof.
assert (0 < uweight s n) as wt_pos
by auto using Z.lt_gt, Z.gt_lt, uweight_positive.
rewrite <-(Columns.compact_mod_step s (uweight s n) c d s_pos wt_pos).
repeat (ring_simplify; f_equal; ring_simplify; try omega).
Qed.
Lemma sat_add_div_step n c d :
(d + c / s) / uweight s n = (s * d + c) / (s * uweight s n).
Proof.
assert (0 < uweight s n) as wt_pos
by auto using Z.lt_gt, Z.gt_lt, uweight_positive.
rewrite <-(Columns.compact_div_step s (uweight s n) c d s_pos wt_pos).
repeat (ring_simplify; f_equal; ring_simplify; try omega).
Qed.
Lemma sat_add_divmod n p q :
eval (snd (@sat_add n p q)) = (eval p + eval q) mod (uweight s n)
/\ fst (@sat_add n p q) = (eval p + eval q) / (uweight s n).
Proof.
cbv [sat_add sat_add_cps chain_op_cps].
remember None as c.
replace (eval p + eval q) with
(eval p + eval q + match c with | None => 0 | Some x => x end)
by (subst; ring).
destruct Heqc. revert c.
induction n; [|destruct c]; intros; simpl chain_op'_cps;
repeat match goal with
| _ => progress cbv [eval Let_In] in *
| _ => progress autorewrite with uncps divmod push_id cancel_pair push_basesystem_eval
| _ => rewrite uweight_0, ?Z.mod_1_r, ?Z.div_1_r
| _ => rewrite uweight_succ
| _ => rewrite Z.sub_opp_r
| _ => rewrite sat_add_mod_step
| _ => rewrite sat_add_div_step
| p : Z ^ 0 |- _ => destruct p
| _ => rewrite uweight_eval_step, ?hd_append, ?tl_append
| |- context[B.Positional.eval _ (snd (chain_op' ?c ?p ?q))]
=> specialize (IHn p q c); autorewrite with push_id uncps in IHn;
rewrite (proj1 IHn); rewrite (proj2 IHn)
| _ => split; ring
| _ => solve [split; repeat (f_equal; ring_simplify; try omega)]
end.
Qed.
Lemma sat_add_mod n p q :
eval (snd (@sat_add n p q)) = (eval p + eval q) mod (uweight s n).
Proof. exact (proj1 (sat_add_divmod n p q)). Qed.
Lemma sat_add_div n p q :
fst (@sat_add n p q) = (eval p + eval q) / (uweight s n).
Proof. exact (proj2 (sat_add_divmod n p q)). Qed.
Lemma small_sat_add n p q : small (snd (@sat_add n p q)).
Proof.
cbv [small UniformWeight.small sat_add sat_add_cps chain_op_cps].
remember None as c. destruct Heqc. revert c.
induction n; intros;
repeat match goal with
| p: Z^0 |- _ => destruct p
| _ => progress (cbv [Let_In] in * )
| _ => progress (simpl chain_op'_cps in * )
| _ => progress autorewrite with uncps push_id cancel_pair in H
| H : _ |- _ => rewrite to_list_append in H;
simpl In in H
| H : _ \/ _ |- _ => destruct H
| _ => contradiction
end.
{ subst x.
destruct c; rewrite ?Z.add_with_get_carry_full_mod,
?Z.add_get_carry_full_mod;
apply Z.mod_pos_bound; omega. }
{ apply IHn in H. assumption. }
Qed.
Definition sat_sub_cps {n} p q T (f:Z*Z^n->T) :=
chain_op_cps (op_get_carry := Z.sub_get_borrow_full s)
(op_with_carry := Z.sub_with_get_borrow_full s)
p q f.
Definition sat_sub {n} p q := @sat_sub_cps n p q _ id.
Lemma sat_sub_id n p q T f :
@sat_sub_cps n p q T f = f (sat_sub p q).
Proof. cbv [sat_sub sat_sub_cps]. rewrite !chain_op_id. reflexivity. Qed.
Lemma sat_sub_divmod n p q :
eval (snd (@sat_sub n p q)) = (eval p - eval q) mod (uweight s n)
/\ fst (@sat_sub n p q) = - ((eval p - eval q) / (uweight s n)).
Proof.
cbv [sat_sub sat_sub_cps chain_op_cps].
remember None as c.
replace (eval p - eval q) with
(eval p - eval q - match c with | None => 0 | Some x => x end)
by (subst; ring).
destruct Heqc. revert c.
induction n; [|destruct c]; intros; simpl chain_op'_cps;
repeat match goal with
| _ => progress cbv [eval Let_In] in *
| _ => progress autorewrite with uncps divmod push_id cancel_pair push_basesystem_eval
| _ => rewrite uweight_0, ?Z.mod_1_r, ?Z.div_1_r
| _ => rewrite uweight_succ
| _ => rewrite Z.sub_opp_r
| _ => rewrite sat_add_mod_step
| _ => rewrite sat_add_div_step
| p : Z ^ 0 |- _ => destruct p
| _ => rewrite uweight_eval_step, ?hd_append, ?tl_append
| |- context[B.Positional.eval _ (snd (chain_op' ?c ?p ?q))]
=> specialize (IHn p q c); autorewrite with push_id uncps in IHn;
rewrite (proj1 IHn); rewrite (proj2 IHn)
| _ => split; ring
| _ => solve [split; repeat (f_equal; ring_simplify; try omega)]
end.
Qed.
Lemma sat_sub_mod n p q :
eval (snd (@sat_sub n p q)) = (eval p - eval q) mod (uweight s n).
Proof. exact (proj1 (sat_sub_divmod n p q)). Qed.
Lemma sat_sub_div n p q :
fst (@sat_sub n p q) = - ((eval p - eval q) / uweight s n).
Proof. exact (proj2 (sat_sub_divmod n p q)). Qed.
Lemma small_sat_sub n p q : small (snd (@sat_sub n p q)).
Proof.
cbv [small UniformWeight.small sat_sub sat_sub_cps chain_op_cps].
remember None as c. destruct Heqc. revert c.
induction n; intros;
repeat match goal with
| p: Z^0 |- _ => destruct p
| _ => progress (cbv [Let_In] in * )
| _ => progress (simpl chain_op'_cps in * )
| _ => progress autorewrite with uncps push_id cancel_pair in H
| H : _ |- _ => rewrite to_list_append in H;
simpl In in H
| H : _ \/ _ |- _ => destruct H
| _ => contradiction
end.
{ subst x.
destruct c; rewrite ?Z.sub_with_get_borrow_full_mod,
?Z.sub_get_borrow_full_mod;
apply Z.mod_pos_bound; omega. }
{ apply IHn in H. assumption. }
Qed.
End AddSub.
End Positional.
End Positional.
End B.
Hint Opaque B.Positional.sat_sub B.Positional.sat_add B.Positional.chain_op B.Positional.chain_op' : uncps.
Hint Rewrite @B.Positional.sat_sub_id @B.Positional.sat_add_id @B.Positional.chain_op_id @B.Positional.chain_op' : uncps.
Hint Rewrite @B.Positional.sat_sub_mod @B.Positional.sat_sub_div @B.Positional.sat_add_mod @B.Positional.sat_add_div using (omega || assumption) : push_basesystem_eval.
Hint Unfold
B.Positional.chain_op'_cps
B.Positional.chain_op'
B.Positional.chain_op_cps
B.Positional.chain_op
B.Positional.sat_add_cps
B.Positional.sat_add
B.Positional.sat_sub_cps
B.Positional.sat_sub
: basesystem_partial_evaluation_unfolder.
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