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Require Export Bedrock.Word Bedrock.Nomega.
Require Import Coq.NArith.NArith Coq.Numbers.Natural.Peano.NPeano Coq.Lists.List Coq.Bool.Sumbool Coq.Arith.Compare_dec Coq.omega.Omega.
Require Import Crypto.Assembly.QhasmCommon Crypto.Assembly.QhasmEvalCommon Crypto.Assembly.QhasmUtil Crypto.Assembly.Pseudo Crypto.Assembly.State.
Require Export Crypto.Assembly.Wordize Crypto.Assembly.Vectorize.
Require Export Crypto.Util.FixCoqMistakes.
Import Pseudo ListNotations StateCommon EvalUtil ListState.
Section Conversion.
Hint Unfold setList getList getVar setCarry setCarryOpt getCarry
getMem setMem overflows.
Lemma eval_in_length: forall {w s n m} p x M c x' M' c',
@pseudoEval n m w s p (x, M, c) = Some (x', M', c')
-> Datatypes.length x = n.
Proof. Admitted.
Lemma eval_out_length: forall {w s n m} x M c x' M' c' p,
@pseudoEval n m w s p (x, M, c) = Some (x', M', c')
-> Datatypes.length x' = m.
Proof. Admitted.
Lemma pseudo_var: forall {w s n} b k x v m c,
(k < n)%nat
-> nth_error x k = Some v
-> pseudoEval (@PVar w s n b (indexize k)) (x, m, c) = Some ([v], m, c).
Proof.
intros; autounfold; simpl; unfold indexize.
destruct (le_dec n 0); simpl. {
replace k with 0 in * by omega; autounfold; simpl in *.
rewrite H0; simpl; intuition.
}
replace (k mod n) with k by (
assert (n <> 0) as NZ by omega;
pose proof (Nat.div_mod k n NZ);
replace (k mod n) with (k - n * (k / n)) by intuition auto with zarith;
rewrite (Nat.div_small k n); intuition auto with zarith).
autounfold; simpl.
destruct (nth_error x k); simpl; try inversion H0; intuition.
Qed.
Lemma pseudo_mem: forall {w s} n v m c x name len index,
TripleM.find (w, name mod n, index mod len)%nat m = Some (@wordToN w v)
-> pseudoEval (@PMem w s n len (indexize name) (indexize index)) (x, m, c) = Some ([v], m, c).
Proof.
intros; autounfold; simpl.
unfold indexize;
destruct (le_dec n 0), (le_dec len 0);
try replace n with 0 in * by intuition auto with zarith;
try replace len with 0 in * by intuition auto with zarith;
autounfold; simpl in *; rewrite H;
autounfold; simpl; rewrite NToWord_wordToN;
intuition.
Qed.
Lemma pseudo_const: forall {w s n} x v m c,
pseudoEval (@PConst w s n v) (x, m, c) = Some ([v], m, c).
Proof. intros; simpl; intuition. Qed.
Lemma pseudo_plus:
forall {w s n} (p: @Pseudo w s n 2) x out0 out1 m0 m1 c0 c1,
pseudoEval p (x, m0, c0) = Some ([out0; out1], m1, c1)
-> pseudoEval (PBin n IAdd p) (x, m0, c0) =
Some ([out0 ^+ out1], m1,
Some (proj1_sig (bool_of_sumbool
(overflows w (&out0 + &out1)%N)%w))).
Proof.
intros; simpl; rewrite H; simpl.
pose proof (wordize_plus out0 out1).
destruct (overflows w _); autounfold; simpl; try rewrite H0;
try rewrite <- (@Npow2_ignore w (out0 ^+ out1));
try rewrite NToWord_wordToN; intuition.
Qed.
Lemma pseudo_bin:
forall {w s n} (p: @Pseudo w s n 2) x out0 out1 m0 m1 c0 c1 op,
op <> IAdd
-> pseudoEval p (x, m0, c0) = Some ([out0; out1], m1, c1)
-> pseudoEval (PBin n op p) (x, m0, c0) =
Some ([fst (evalIntOp op out0 out1)], m1, c1).
Proof.
intros; simpl; rewrite H0; simpl.
induction op;
try (contradict H; reflexivity);
unfold evalIntOp; autounfold; simpl;
reflexivity.
Qed.
Lemma pseudo_and:
forall {w s n} (p: @Pseudo w s n 2) x out0 out1 m0 m1 c0 c1,
pseudoEval p (x, m0, c0) = Some ([out0; out1], m1, c1)
-> pseudoEval (PBin n IAnd p) (x, m0, c0) =
Some ([out0 ^& out1], m1, c1).
Proof.
intros.
replace (out0 ^& out1) with (fst (evalIntOp IAnd out0 out1)).
- apply pseudo_bin; intuition; inversion H0.
- unfold evalIntOp; simpl; intuition.
Qed.
Lemma pseudo_awc:
forall {w s n} (p: @Pseudo w s n 2) x out0 out1 m0 m1 c0 c,
pseudoEval p (x, m0, c0) = Some ([out0; out1], m1, Some c)
-> pseudoEval (PCarry n AddWithCarry p) (x, m0, c0) =
Some ([addWithCarry out0 out1 c], m1,
Some (proj1_sig (bool_of_sumbool (overflows w
(&out0 + &out1 + (if c then 1 else 0))%N)%w))).
Proof.
intros; simpl; rewrite H; simpl.
pose proof (wordize_awc out0 out1); unfold evalCarryOp.
destruct (overflows w ((& out0)%w + (& out1)%w +
(if c then 1%N else 0%N)));
autounfold; simpl; try rewrite H0; intuition.
Qed.
Lemma pseudo_shiftr:
forall {w s n} (p: @Pseudo w s n 1) x out m0 m1 c0 c1 k,
pseudoEval p (x, m0, c0) = Some ([out], m1, c1)
-> pseudoEval (PShift n Shr k p) (x, m0, c0) =
Some ([shiftr out k], m1, c1).
Proof.
intros; simpl; rewrite H; autounfold; simpl.
rewrite wordize_shiftr; rewrite NToWord_wordToN; intuition.
Qed.
Lemma pseudo_comb:
forall {w s n a b} (p0: @Pseudo w s n a) (p1: @Pseudo w s n b)
input out0 out1 m0 m1 m2 c0 c1 c2,
pseudoEval p0 (input, m0, c0) = Some (out0, m1, c1)
-> pseudoEval p1 (input, m1, c1) = Some (out1, m2, c2)
-> pseudoEval (@PComb w s n _ _ p0 p1) (input, m0, c0) =
Some (out0 ++ out1, m2, c2).
Proof.
intros; autounfold; simpl.
rewrite H; autounfold; simpl.
rewrite H0; autounfold; simpl; intuition.
Qed.
Lemma pseudo_cons:
forall {w s n b} (p0: @Pseudo w s n 1) (p1: @Pseudo w s n b)
(p2: @Pseudo w s n (S b)) input x xs m0 m1 m2 c0 c1 c2,
pseudoEval p0 (input, m0, c0) = Some ([x], m1, c1)
-> pseudoEval p1 (input, m1, c1) = Some (xs, m2, c2)
-> p2 = (@PComb w s n _ _ p0 p1)
-> pseudoEval p2 (input, m0, c0) = Some (x :: xs, m2, c2).
Proof.
intros.
replace (x :: xs) with ([x] ++ xs) by (simpl; intuition).
rewrite H1.
apply (pseudo_comb p0 p1 input _ _ m0 m1 m2 c0 c1 c2); intuition.
Qed.
Lemma pseudo_let:
forall {w s n k m} (p0: @Pseudo w s n k) (p1: @Pseudo w s (n + k) m)
input out0 out1 m0 m1 m2 c0 c1 c2,
pseudoEval p0 (input, m0, c0) = Some (out0, m1, c1)
-> pseudoEval p1 (input ++ out0, m1, c1) = Some (out1, m2, c2)
-> pseudoEval (@PLet w s n k m p0 p1) (input, m0, c0) =
Some (out1, m2, c2).
Proof.
intros; autounfold; simpl.
rewrite H; autounfold; simpl.
rewrite H0; autounfold; simpl; intuition.
Qed.
Lemma pseudo_let_var:
forall {w s n k m} (p0: @Pseudo w s n k) (p1: @Pseudo w s (n + k) m)
input a f m0 m1 m2 c0 c1 c2,
pseudoEval p0 (input, m0, c0) = Some ([a], m1, c1)
-> pseudoEval p1 (input ++ [a], m1, c1) = Some (f (nth n (input ++ [a]) (wzero _)), m2, c2)
-> pseudoEval (@PLet w s n k m p0 p1) (input, m0, c0) =
Some (Let_In a f, m2, c2).
Proof.
intros; unfold Let_In; cbv zeta.
eapply pseudo_let; try eassumption.
replace (f a) with (f (nth n (input ++ [a]) (wzero w))); try assumption.
apply f_equal.
assert (Datatypes.length input = n) as L by (
eapply eval_in_length; eassumption).
rewrite app_nth2; try rewrite L; intuition.
replace (n - n) with 0 by omega; simpl; intuition.
Qed.
Lemma pseudo_let_list:
forall {w s n k m} (p0: @Pseudo w s n k) (p1: @Pseudo w s (n + k) m)
input lst f m0 m1 m2 c0 c1 c2,
pseudoEval p0 (input, m0, c0) = Some (lst, m1, c1)
-> pseudoEval p1 (input ++ lst, m1, c1) = Some (f lst, m2, c2)
-> pseudoEval (@PLet w s n k m p0 p1) (input, m0, c0) =
Some (Let_In lst f, m2, c2).
Proof.
intros; unfold Let_In; cbv zeta.
eapply pseudo_let; try eassumption.
Qed.
Lemma pseudo_mult_single:
forall {w s n m} (p0: @Pseudo w s n 2)
(p1: @Pseudo w s (n + 2) m)
out0 out1 f x m0 m1 m2 c0 c1 c2,
pseudoEval p0 (x, m0, c0) = Some ([out0; out1], m1, c1)
-> pseudoEval p1 (x ++ [out0 ^* out1; multHigh out0 out1], m1, c1) =
Some (f (nth n (x ++ [out0 ^* out1; multHigh out0 out1]) (wzero _)), m2, c2)
-> pseudoEval (@PLet w s n 2 m (PDual n Mult p0) p1) (x, m0, c0) =
Some (Let_In (out0 ^* out1) f, m2, c2).
Proof.
intros; simpl; rewrite H; autounfold; simpl; rewrite H0; simpl; intuition.
replace (nth n (x ++ _) _) with (out0 ^* out1); simpl; intuition.
assert (Datatypes.length x = n) as L by (
eapply eval_in_length; eassumption).
rewrite app_nth2; try rewrite L; intuition.
replace (n - n) with 0 by omega.
simpl; intuition.
Qed.
Lemma pseudo_mult_dual:
forall {w s n m} (p0: @Pseudo w s n 2)
(p1: @Pseudo w s (n + 2) m)
out0 out1 f x m0 m1 m2 c0 c1 c2,
pseudoEval p0 (x, m0, c0) = Some ([out0; out1], m1, c1)
-> pseudoEval p1 (x ++ [out0 ^* out1; multHigh out0 out1], m1, c1) =
Some (f (nth n (x ++ [out0 ^* out1; multHigh out0 out1]) (wzero _))
(nth (S n) (x ++ [out0 ^* out1; multHigh out0 out1]) (wzero _)),
m2, c2)
-> pseudoEval (@PLet w s n 2 m (PDual n Mult p0) p1) (x, m0, c0) =
Some (Let_In (multHigh out0 out1) (fun x =>
Let_In (out0 ^* out1) (fun y =>
f y x)), m2, c2).
Proof.
intros; simpl; rewrite H; autounfold; simpl; rewrite H0; simpl; intuition.
assert (Datatypes.length x = n) as L by (eapply eval_in_length; eassumption).
replace (nth n (x ++ _) _) with (out0 ^* out1); simpl; intuition.
replace (nth (S n) (x ++ _) _) with (multHigh out0 out1); simpl; intuition.
- rewrite app_nth2; try rewrite L; intuition.
replace (S n - n) with 1 by omega.
simpl; intuition.
- rewrite app_nth2; try rewrite L; intuition.
replace (n - n) with 0 by omega.
simpl; intuition.
Qed.
Definition pseudeq {w s} (n m: nat) (f: list (word w) -> list (word w)) : Type :=
{p: @Pseudo w s n m | forall x: (list (word w)),
List.length x = n -> exists m' c',
pseudoEval p (x, TripleM.empty N, None) = Some (f x, m', c')}.
End Conversion.
Ltac autodestruct :=
repeat match goal with
| [H: context[Datatypes.length (cons _ _)] |- _] => simpl in H
| [H: context[Datatypes.length nil] |- _] => simpl in H
| [H: S ?a = S ?b |- _] => inversion H; clear H
| [H: (S ?a) = 0 |- _] => contradict H; intuition
| [H: 0 = (S ?a) |- _] => contradict H; intuition
| [H: 0 = 0 |- _] => clear H
| [x: list ?T |- _] =>
match goal with
| [H: context[Datatypes.length x] |- _] => destruct x
end
end.
Ltac pseudo_step :=
match goal with
| [ |- pseudoEval ?p _ = Some ((
Let_In (multHigh ?a ?b) (fun x =>
Let_In (?a ^* ?b) (fun y => _))), _, _) ] =>
is_evar p; eapply pseudo_mult_dual
| [ |- pseudoEval ?p _ = Some (Let_In (?a ^* ?b) _, _, _) ] =>
is_evar p; eapply pseudo_mult_single
| [ |- pseudoEval ?p _ = Some ([?x ^& ?y], _, _) ] =>
is_evar p; eapply pseudo_and
| [ |- pseudoEval ?p _ = Some ([?x ^+ ?y], _, _) ] =>
is_evar p; eapply pseudo_plus
| [ |- pseudoEval ?p _ = Some (cons ?x (cons _ _), _, _) ] =>
is_evar p; eapply pseudo_cons; try reflexivity
| [ |- pseudoEval ?p _ = Some ([natToWord _ ?x], _, _)%p ] =>
is_evar p; eapply pseudo_const
| [ |- pseudoEval ?p _ = Some ((Let_In ?a ?f), _, _) ] =>
is_evar p;
match (type of a) with
| list _ => eapply pseudo_let_list
| word _ => eapply pseudo_let_var
| (_ * _)%type => rewrite detuple_let
end
| [ |- @pseudoEval ?n _ _ _ ?P _ =
Some ([nth ?i ?lst _], _, _)%p ] =>
eapply (pseudo_var None i); simpl; intuition
end.
Ltac pseudo_solve :=
repeat eexists;
autounfold;
autodestruct;
repeat pseudo_step.
|
