diff options
author | 2016-09-13 16:01:28 -0400 | |
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committer | 2016-09-13 22:00:07 -0400 | |
commit | e24e7ecbb3f01e0f58f5a40283a4dc7d0cd86246 (patch) | |
tree | 88e302a055190b4493382f96d13f129d3dc9c7fe /src/ModularArithmetic/ModularBaseSystemProofs.v | |
parent | 271cac4fe54d377d95d503971267be8d783b5d2a (diff) |
Moved lemmas to ZUtil
Diffstat (limited to 'src/ModularArithmetic/ModularBaseSystemProofs.v')
-rw-r--r-- | src/ModularArithmetic/ModularBaseSystemProofs.v | 59 |
1 files changed, 4 insertions, 55 deletions
diff --git a/src/ModularArithmetic/ModularBaseSystemProofs.v b/src/ModularArithmetic/ModularBaseSystemProofs.v index 5361fa858..50b181f91 100644 --- a/src/ModularArithmetic/ModularBaseSystemProofs.v +++ b/src/ModularArithmetic/ModularBaseSystemProofs.v @@ -676,48 +676,6 @@ Section CanonicalizationProofs. repeat (break_if; try omega). Qed. - Lemma lt_pow_2_shiftr : forall a n, 0 <= a < 2 ^ n -> a >> n = 0. - Proof. - intros. - destruct (Z_le_dec 0 n). - + rewrite Z.shiftr_div_pow2 by assumption. - auto using Z.div_small. - + assert (2 ^ n = 0) by (apply Z.pow_neg_r; omega). - omega. - Qed. - - Hint Rewrite Z.pow2_bits_eqb using omega : Ztestbit. - Lemma pow_2_shiftr : forall n, 0 <= n -> (2 ^ n) >> n = 1. - Proof. - intros; apply Z.bits_inj'; intros. - replace 1 with (2 ^ 0) by ring. - repeat match goal with - | |- _ => progress intros - | |- _ => progress rewrite ?Z.eqb_eq, ?Z.eqb_neq in * - | |- _ => progress autorewrite with Ztestbit - | |- appcontext[Z.eqb ?a ?b] => case_eq (Z.eqb a b) - | |- _ => reflexivity || omega - end. - Qed. - - Lemma lt_mul_2_pow_2_shiftr : forall a n, 0 <= a < 2 * 2 ^ n -> - a >> n = if Z_lt_dec a (2 ^ n) then 0 else 1. - Proof. - intros; break_if; [ apply lt_pow_2_shiftr; omega | ]. - destruct (Z_le_dec 0 n). - + replace (2 * 2 ^ n) with (2 ^ (n + 1)) in * - by (rewrite Z.pow_add_r; try omega; ring). - pose proof (Z.shiftr_ones a (n + 1) n H). - pose proof (Z.shiftr_le (2 ^ n) a n). - specialize_by omega. - replace (n + 1 - n) with 1 in * by ring. - replace (Z.ones 1) with 1 in * by reflexivity. - rewrite pow_2_shiftr in * by omega. - omega. - + assert (2 ^ n = 0) by (apply Z.pow_neg_r; omega). - omega. - Qed. - Lemma bound_during_second_loop : forall us, length us = length limb_widths -> (forall n, 0 <= nth_default 0 us n < if eq_nat_dec n 0 then 2 * 2 ^ limb_widths [n] else 2 ^ limb_widths [n]) -> @@ -751,7 +709,7 @@ Section CanonicalizationProofs. |- appcontext [(?u {{ ?i }} [?n]) >> _] => pose proof (IH 0%nat); pose proof (IH (S n)); specialize (IH n); specialize_by omega | H : 0 <= ?a < 2 * 2 ^ ?n |- appcontext [?a >> ?n] => pose proof c_pos; - apply lt_mul_2_pow_2_shiftr in H; break_if; rewrite H; omega + apply Z.lt_mul_2_pow_2_shiftr in H; break_if; rewrite H; omega | |- _ => apply Z.pow2_mod_pos_bound, limb_widths_pos, nth_default_preserves_properties_length_dep; [tauto | omega] | |- 0 <= 0 < _ => solve[split; zero_bounds] | |- _ => omega @@ -785,15 +743,6 @@ Section CanonicalizationProofs. auto using limb_widths_pos. Qed. - Lemma lt_mul_2_mod_sub : forall a b, b <> 0 -> b <= a < 2 * b -> a mod b = a - b. - Proof. - intros. - replace a with (1 * b + (a - b)) at 1 by ring. - rewrite Z.mod_add_l by auto. - apply Z.mod_small. - omega. - Qed. - Lemma bound_during_third_loop : forall us, length us = length limb_widths -> (forall n, 0 <= nth_default 0 us n < if eq_nat_dec n 0 then 2 ^ limb_widths [n] + c else 2 ^ limb_widths [n]) -> @@ -835,16 +784,16 @@ Section CanonicalizationProofs. unique assert (0 <= a < 2 * 2 ^ n) by omega | H : 0 <= ?a < 2 ^ ?n |- appcontext [?a >> ?n] => pose proof c_pos; - apply lt_pow_2_shiftr in H; rewrite H; omega + apply Z.lt_pow_2_shiftr in H; rewrite H; omega | H : 0 <= ?a < 2 * 2 ^ ?n |- appcontext [?a >> ?n] => pose proof c_pos; - apply lt_mul_2_pow_2_shiftr in H; break_if; rewrite H; omega + apply Z.lt_mul_2_pow_2_shiftr in H; break_if; rewrite H; omega | |- _ => apply Z.pow2_mod_pos_bound, limb_widths_pos, nth_default_preserves_properties_length_dep; [tauto | omega] | |- 0 <= 0 < _ => solve[split; zero_bounds] | |- _ => omega end. rewrite Z.pow2_mod_spec by auto. - rewrite lt_mul_2_mod_sub; omega. + rewrite Z.lt_mul_2_mod_sub; omega. Qed. Lemma bound_after_third_loop : forall n us, |