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Diffstat (limited to 'coqprime/PrimalityTest/Proth.v')
| -rw-r--r-- | coqprime/PrimalityTest/Proth.v | 120 |
1 files changed, 0 insertions, 120 deletions
diff --git a/coqprime/PrimalityTest/Proth.v b/coqprime/PrimalityTest/Proth.v deleted file mode 100644 index b087f1854..000000000 --- a/coqprime/PrimalityTest/Proth.v +++ /dev/null @@ -1,120 +0,0 @@ - -(*************************************************************) -(* This file is distributed under the terms of the *) -(* GNU Lesser General Public License Version 2.1 *) -(*************************************************************) -(* Benjamin.Gregoire@inria.fr Laurent.Thery@inria.fr *) -(*************************************************************) - -(********************************************************************** - Proth.v - - Proth's Test - - Definition: ProthTest - **********************************************************************) -Require Import ZArith. -Require Import ZCAux. -Require Import Pocklington. - -Open Scope Z_scope. - -Theorem ProthTest: forall h k a, let n := h * 2 ^ k + 1 in 1 < a -> 0 < h < 2 ^k -> (a ^ ((n - 1) / 2) + 1) mod n = 0 -> prime n. -intros h k a n; unfold n; intros H H1 H2. -assert (Hu: 0 < h * 2 ^ k). -apply Zmult_lt_O_compat; auto with zarith. -assert (Hu1: 0 < k). -case (Zle_or_lt k 0); intros Hv; auto. -generalize H1 Hv; case k; simpl. -intros (Hv1, Hv2); contradict Hv2; auto with zarith. -intros p1 _ Hv1; contradict Hv1; auto with zarith. -intros p (Hv1, Hv2); contradict Hv2; auto with zarith. -apply PocklingtonCorollary1 with (F1 := 2 ^ k) (R1 := h); auto with zarith. -ring. -apply Zlt_le_trans with ((h + 1) * 2 ^ k); auto with zarith. -rewrite Zmult_plus_distr_l; apply Zplus_lt_compat_l. -rewrite Zmult_1_l; apply Zlt_le_trans with 2; auto with zarith. -intros p H3 H4. -generalize H2; replace (h * 2 ^ k + 1 - 1) with (h * 2 ^k); auto with zarith; clear H2; intros H2. -exists a; split; auto; split. -pattern (h * 2 ^k) at 1; rewrite (Zdivide_Zdiv_eq 2 (h * 2 ^ k)); auto with zarith. -rewrite (Zmult_comm 2); rewrite Zpower_mult; auto with zarith. -rewrite Zpower_mod; auto with zarith. -assert (tmp: forall p, p = (p + 1) -1); auto with zarith; rewrite (fun x => (tmp (a ^ x))). -rewrite Zminus_mod; auto with zarith. -rewrite H2. -rewrite (Zmod_small 1); auto with zarith. -rewrite <- Zpower_mod; auto with zarith. -rewrite Zmod_small; auto with zarith. -simpl; unfold Zpower_pos; simpl; auto with zarith. -apply Z_div_pos; auto with zarith. -apply Zdivide_trans with (2 ^ k). -apply Zpower_divide; auto with zarith. -apply Zdivide_factor_l; auto with zarith. -apply Zis_gcd_gcd; auto with zarith. -apply Zis_gcd_intro; auto with zarith. -intros x HD1 HD2. -assert (Hd1: p = 2). -apply prime_div_Zpower_prime with (4 := H4); auto with zarith. -apply prime_2. -assert (Hd2: (x | 2)). -replace 2 with ((a ^ (h * 2 ^ k / 2) + 1) - (a ^ (h * 2 ^ k/ 2) - 1)); auto with zarith. -apply Zdivide_minus_l; auto. -apply Zdivide_trans with (1 := HD2). -apply Zmod_divide; auto with zarith. -pattern 2 at 2; rewrite <- Hd1; auto. -replace 1 with ((h * 2 ^k + 1) - (h * 2 ^ k)); auto with zarith. -apply Zdivide_minus_l; auto. -apply Zdivide_trans with (1 := Hd2); auto. -apply Zdivide_trans with (2 ^ k). -apply Zpower_divide; auto with zarith. -apply Zdivide_factor_l; auto with zarith. -Qed. - - -Definition proth_test h k a := - let n := h * 2 ^ k + 1 in - if (Z_lt_dec 1 a) then - if (Z_lt_dec 0 h) then - if (Z_lt_dec h (2 ^k)) then - if Z_eq_dec (Zpow_mod a ((n - 1) / 2) n) (n - 1) then true - else false else false else false else false. - - -Theorem ProthTestOp: forall h k a, proth_test h k a = true -> prime (h * 2 ^ k + 1). -intros h k a; unfold proth_test. -repeat match goal with |- context[if ?X then _ else _] => case X end; try (intros; discriminate). -intros H1 H2 H3 H4 _. -assert (Hu: 0 < h * 2 ^ k). -apply Zmult_lt_O_compat; auto with zarith. -apply ProthTest with (a := a); auto. -rewrite Zplus_mod; auto with zarith. -rewrite <- Zpow_mod_Zpower_correct; auto with zarith. -rewrite H1. -rewrite (Zmod_small 1); auto with zarith. -replace (h * 2 ^ k + 1 - 1 + 1) with (h * 2 ^ k + 1); auto with zarith. -apply Zdivide_mod; auto with zarith. -apply Z_div_pos; auto with zarith. -Qed. - -Theorem prime5: prime 5. -exact (ProthTestOp 1 2 2 (refl_equal _)). -Qed. - -Theorem prime17: prime 17. -exact (ProthTestOp 1 4 3 (refl_equal _)). -Qed. - -Theorem prime257: prime 257. -exact (ProthTestOp 1 8 3 (refl_equal _)). -Qed. - -Theorem prime65537: prime 65537. -exact (ProthTestOp 1 16 3 (refl_equal _)). -Qed. - -(* Too tough !! -Theorem prime4294967297: prime 4294967297. -exact (ProthTestOp 1 32 3 (refl_equal _)). -Qed. -*) |