diff options
author | jadep <jade.philipoom@gmail.com> | 2016-04-14 21:26:17 -0400 |
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committer | jadep <jade.philipoom@gmail.com> | 2016-04-14 21:26:17 -0400 |
commit | addb070c83e0cd33e096f80781103db3ac883e5f (patch) | |
tree | 3ac52be6ce25f0626421c169e720daa06e7b4312 /src/Util/IterAssocOp.v | |
parent | e8f3cdec7613d26d4cd15bf2fb80e8576d4f2d62 (diff) |
Retrieved updated version of Util/IterAssocOp and modified ExtendedCoordinates and Ed25519 to use it.
Diffstat (limited to 'src/Util/IterAssocOp.v')
-rw-r--r-- | src/Util/IterAssocOp.v | 74 |
1 files changed, 44 insertions, 30 deletions
diff --git a/src/Util/IterAssocOp.v b/src/Util/IterAssocOp.v index 016a4f7bd..5e23bb987 100644 --- a/src/Util/IterAssocOp.v +++ b/src/Util/IterAssocOp.v @@ -9,7 +9,11 @@ Section IterAssocOp. (assoc: forall a b c, op a (op b c) === op (op a b) c) (neutral:T) (neutral_l : forall a, op neutral a === a) - (neutral_r : forall a, op a neutral === a). + (neutral_r : forall a, op a neutral === a) + {scalar : Type} + (testbit : scalar -> nat -> bool) + (scToN : scalar -> N) + (testbit_spec : forall x i, testbit x i = N.testbit_nat (scToN x) i). Existing Instance op_proper. Fixpoint nat_iter_op n a := @@ -51,19 +55,19 @@ Section IterAssocOp. | S exp' => f (funexp f a exp') end. - Definition test_and_op n a (state : nat * T) := + Definition test_and_op sc a (state : nat * T) := let '(i, acc) := state in let acc2 := op acc acc in match i with | O => (0, acc) - | S i' => (i', if N.testbit_nat n i' then op a acc2 else acc2) + | S i' => (i', if testbit sc i' then op a acc2 else acc2) end. - Definition iter_op n a : T := - snd (funexp (test_and_op n a) (N.size_nat n, neutral) (N.size_nat n)). + Definition iter_op sc a bound : T := + snd (funexp (test_and_op sc a) (bound, neutral) bound). - Definition test_and_op_inv n a (s : nat * T) := - snd s === nat_iter_op (N.to_nat (N.shiftr_nat n (fst s))) a. + Definition test_and_op_inv sc a (s : nat * T) := + snd s === nat_iter_op (N.to_nat (N.shiftr_nat (scToN sc) (fst s))) a. Hint Rewrite N.succ_double_spec @@ -91,7 +95,7 @@ Section IterAssocOp. reflexivity. Qed. - Lemma shiftr_succ : forall n i, + Lemma Nshiftr_succ : forall n i, N.to_nat (N.shiftr_nat n i) = if N.testbit_nat n i then S (2 * N.to_nat (N.shiftr_nat n (S i))) @@ -115,21 +119,23 @@ Section IterAssocOp. apply N2Nat.id. Qed. - Lemma test_and_op_inv_step : forall n a s, - test_and_op_inv n a s -> - test_and_op_inv n a (test_and_op n a s). + Lemma test_and_op_inv_step : forall sc a s, + test_and_op_inv sc a s -> + test_and_op_inv sc a (test_and_op sc a s). Proof. destruct s as [i acc]. unfold test_and_op_inv, test_and_op; simpl; intro Hpre. destruct i; [ apply Hpre | ]. simpl. - rewrite shiftr_succ. - case_eq (N.testbit_nat n i); intro; simpl; rewrite Hpre, <- plus_n_O, nat_iter_op_plus; reflexivity. + rewrite Nshiftr_succ. + case_eq (testbit sc i); intro testbit_eq; simpl; + rewrite testbit_spec in testbit_eq; rewrite testbit_eq; + rewrite Hpre, <- plus_n_O, nat_iter_op_plus; reflexivity. Qed. - Lemma test_and_op_inv_holds : forall n a i s, - test_and_op_inv n a s -> - test_and_op_inv n a (funexp (test_and_op n a) s i). + Lemma test_and_op_inv_holds : forall sc a i s, + test_and_op_inv sc a s -> + test_and_op_inv sc a (funexp (test_and_op sc a) s i). Proof. induction i; intros; auto; simpl; apply test_and_op_inv_step; auto. Qed. @@ -144,14 +150,14 @@ Section IterAssocOp. destruct i; rewrite NPeano.Nat.sub_succ_r; subst; rewrite <- IHy; simpl; reflexivity. Qed. - Lemma iter_op_termination : forall n a, - test_and_op_inv n a - (funexp (test_and_op n a) (N.size_nat n, neutral) (N.size_nat n)) -> - iter_op n a === nat_iter_op (N.to_nat n) a. + Lemma iter_op_termination : forall sc a bound, + N.size_nat (scToN sc) <= bound -> + test_and_op_inv sc a + (funexp (test_and_op sc a) (bound, neutral) bound) -> + iter_op sc a bound === nat_iter_op (N.to_nat (scToN sc)) a. Proof. - unfold test_and_op_inv, iter_op; simpl; intros ? ? Hinv. + unfold test_and_op_inv, iter_op; simpl; intros ? ? ? ? Hinv. rewrite Hinv, funexp_test_and_op_index, Minus.minus_diag. - replace (N.shiftr_nat n 0) with n by auto. reflexivity. Qed. @@ -160,25 +166,33 @@ Section IterAssocOp. destruct n; auto; simpl; induction p; simpl; auto; rewrite IHp, Pnat.Pos2Nat.inj_succ; reflexivity. Qed. - Lemma Nshiftr_size : forall n, N.shiftr_nat n (N.size_nat n) = 0%N. + Lemma Nshiftr_size : forall n bound, N.size_nat n <= bound -> + N.shiftr_nat n bound = 0%N. Proof. intros. - rewrite Nsize_nat_equiv. + rewrite <- (Nat2N.id bound). rewrite Nshiftr_nat_equiv. - destruct (N.eq_dec n 0); subst; auto. + destruct (N.eq_dec n 0); subst; [apply N.shiftr_0_l|]. apply N.shiftr_eq_0. - rewrite N.size_log2 by auto. - apply N.lt_succ_diag_r. + rewrite Nsize_nat_equiv in *. + rewrite N.size_log2 in * by auto. + apply N.le_succ_l. + rewrite <- N.compare_le_iff. + rewrite N2Nat.inj_compare. + rewrite <- Compare_dec.nat_compare_le. + rewrite Nat2N.id. + auto. Qed. - Lemma iter_op_spec : forall n a, iter_op n a === nat_iter_op (N.to_nat n) a. + Lemma iter_op_spec : forall sc a bound, N.size_nat (scToN sc) <= bound -> + iter_op sc a bound === nat_iter_op (N.to_nat (scToN sc)) a. Proof. intros. - apply iter_op_termination. + apply iter_op_termination; auto. apply test_and_op_inv_holds. unfold test_and_op_inv. simpl. - rewrite Nshiftr_size. + rewrite Nshiftr_size by auto. reflexivity. Qed. |