phanerozoic/Coq-Prime · Datasets at Hugging Face

fact stringlengths 8 29k	type stringclasses 12 values	library stringclasses 1 value	imports listlengths 0 132	filename stringclasses 381 values	symbolic_name stringlengths 1 2.01k
znz_inj : forall a b, a = b -> val a = val b. intros; subst; auto. Qed.	Theorem	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	znz_inj
Zeq_iok : forall x y, x = y -> Zeq_bool x y = true. intros x y H; subst. apply Zeq_is_eq_bool, eq_refl. Qed.	Theorem	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	Zeq_iok
modz : forall x, (x mod n) = (x mod n) mod n. intros x; rewrite Zmod_mod; auto with zarith. Qed.	Lemma	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	modz
zero := mkznz _ (modz 0).	Definition	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	zero
one := mkznz _ (modz 1).	Definition	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	one
add v1 v2 := mkznz _ (modz (val v1 + val v2)).	Definition	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	add
sub v1 v2 := mkznz _ (modz (val v1 - val v2)).	Definition	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	sub
mul v1 v2 := mkznz _ (modz (val v1 * val v2)).	Definition	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	mul
opp v := mkznz _ (modz (-val v)).	Definition	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	opp
zirr : forall x1 x2 H1 H2, x1 = x2 -> mkznz x1 H1 = mkznz x2 H2. Proof. intros x1 x2 H1 H2 H3. subst x1. rewrite (fun H => eq_proofs_unicity H H1 H2); auto. intros x y; case (Z.eq_dec x y); auto. Qed.	Theorem	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	zirr
znz1 : forall x, x mod 1 = 0. intros x; apply Zdivide_mod; auto with zarith. Qed.	Lemma	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	znz1
RZnZ : ring_theory zero one add mul sub opp (@eq znz). split; auto. intros p; case p; intros x H; refine (zirr _ _ _ _ _); simpl; auto. intros [x Hx] [y Hy]. refine (zirr _ _ _ _ _); simpl. rewrite Zplus_comm; auto. intros [x Hx] [y Hy] [z Hz]. refine (zirr _ _ _ _ _); simpl. rewrite Zplus_mod; auto. rewrite (Zplus_mod((x + y) mod n)); auto. repeat rewrite Zmod_mod; auto. repeat rewrite <- Zplus_mod; auto; rewrite Zplus_assoc; auto. intros p; case p; intros x H. refine (zirr _ _ _ _ _); simpl. case (Zle_lt_or_eq 1 n); auto with zarith; intros Hz. rewrite (Zmod_small 1); auto with zarith. rewrite Zmult_1_l; auto. clear p; subst n; rewrite znz1; rewrite H; rewrite znz1; auto. intros [x Hx] [y Hy]. refine (zirr _ _ _ _ _); simpl. rewrite Zmult_comm; auto. intros [x Hx] [y Hy] [z Hz]. refine (zirr _ _ _ _ _); simpl. rewrite Zmult_mod; auto. rewrite (Zmult_mod ((x * y) mod n)); auto. repeat rewrite Zmod_mod; auto. repeat rewrite <- Zmult_mod; auto; rewrite Zmult_assoc; auto. intros [x Hx] [y Hy] [z Hz]. refine (zirr _ _ _ _ _); simpl. rewrite Zmult_mod; auto. rewrite Zmod_mod; auto. rewrite <- Zmult_mod; auto. rewrite (Zplus_mod ((x*z) mod n)); auto. repeat rewrite Zmod_mod; auto. rewrite <- Zplus_mod; auto. apply f_equal2 with (f := Z.modulo); auto; ring. intros [x Hx] [y Hy]. refine (zirr _ _ _ _ _); simpl. rewrite Zplus_mod; auto. repeat rewrite Zmod_mod; auto. rewrite <- Zplus_mod; auto. intros [x Hx]. refine (zirr _ _ _ _ _); simpl. rewrite Zplus_mod; auto. repeat rewrite Zmod_mod; auto. rewrite <- Zplus_mod; auto. apply f_equal2 with (f := Z.modulo); auto; ring. Defined.	Definition	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	RZnZ
Ring RZnZ : RZnZ. (* It is finite *)	Add	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	Ring
mklist (n: nat): list nat := match n with O => nil \| (S n) => cons n (mklist n) end.	Fixpoint	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	mklist
mklist_length : forall n1, length (mklist n1) = n1. Proof. intros n1; elim n1; simpl; auto; clear n1. Qed.	Lemma	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	mklist_length
mklist_lt : forall n1 x, (In x (mklist n1)) -> (x < n1)%nat. intros n1; elim n1; simpl; auto; clear n1. intros x H; case H. intros n1 Hrec x [H1 \| H1]; try subst x; auto with arith. Qed.	Theorem	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	mklist_lt
lt_mklist_lt : forall n1 x, (x < n1)%nat -> (In x (mklist n1)). intros n1 x H; elim H; simpl; auto. Qed.	Theorem	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	lt_mklist_lt
uniq_mklist : forall m, ulist (mklist m). intros m; elim m; simpl; auto; clear m. intros m H; constructor; auto. intros H1; absurd (m < m)%nat; auto with arith. apply mklist_lt; auto. Qed.	Theorem	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	uniq_mklist
nat_z_kt : forall x, (x < Z.abs_nat n)%nat -> (Z_of_nat x) = (Z_of_nat x) mod n. Proof. intros x H; rewrite Zmod_small; lia. Qed.	Theorem	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	nat_z_kt
mkzlist : forall (l: list nat), (forall x, In x l -> (x < Z.abs_nat n)%nat) -> list znz. fix mkzlist 1; intros l; case l. intros; exact nil. intros n1 l1 Hn. assert (F1: forall x, In x l1 -> (x < Z.abs_nat n)%nat). intros; apply Hn; simpl; auto. assert (F2: (n1 < Z.abs_nat n)%nat). apply Hn; simpl; auto. exact (cons (mkznz _ (nat_z_kt _ F2)) (mkzlist _ F1)). Defined.	Definition	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	mkzlist
mkzlist_length : forall l H, length (mkzlist l H) = length l. Proof. intros l; elim l; simpl; auto; clear l. Qed.	Lemma	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	mkzlist_length
in_mkzlist : forall l a Ha Hl, In (mkznz (Z_of_nat a) Ha) (mkzlist l Hl) -> In a l. intros l1; elim l1; simpl; auto; clear l1. intros a1 l1 Hrec1 a2 l2 Hl2 [H4 \| H4]. generalize (znz_inj _ _ H4); simpl; clear H4; intros H4; left. rewrite <- (Zabs_nat_Z_of_nat a1); rewrite H4; rewrite Zabs_nat_Z_of_nat; auto. right; apply (Hrec1 _ _ _ H4). Qed.	Theorem	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	in_mkzlist
mkzlist_in : forall l a Ha Hl, In (Z.abs_nat a) l -> In (mkznz a Ha) (mkzlist l Hl). intros l1; elim l1; simpl; auto; clear l1. intros a1 l1 Hrec1 a2 l2 Hl2 [H4 \| H4]; auto. left; apply zirr; auto. rewrite H4; rewrite inj_Zabs_nat; auto. rewrite Z.abs_eq; auto with zarith. case (Z_mod_lt a2 n); auto with zarith. Qed.	Theorem	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	mkzlist_in
mkzlist_uniq : forall l H, ulist l -> ulist (mkzlist l H). intros l H H1; generalize H; elim H1; simpl; auto; clear l H H1. intros a l H1 H2 Hrec H3; constructor; auto. intros HH; case H1; generalize HH; clear HH H1. assert (F1: forall l a Ha Hl, In (mkznz (Z_of_nat a) Ha) (mkzlist l Hl) -> In a l); auto. intros l1; elim l1; simpl; auto; clear l1. intros a1 l1 Hrec1 a2 l2 Hl2 [H4 \| H4]. generalize (znz_inj _ _ H4); simpl; clear H4; intros H4; left. rewrite <- (Zabs_nat_Z_of_nat a1); rewrite H4; rewrite Zabs_nat_Z_of_nat; auto. right; apply (Hrec1 _ _ _ H4). apply in_mkzlist. Qed.	Theorem	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	mkzlist_uniq
all_znz : list znz := (mkzlist (mklist (Z.abs_nat n)) (mklist_lt _)).	Definition	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	all_znz
all_znz_length : length all_znz = (Z.abs_nat n). Proof. unfold all_znz; rewrite mkzlist_length. rewrite mklist_length; auto. Qed.	Lemma	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	all_znz_length
uniq_all_znz : ulist all_znz. unfold all_znz; apply mkzlist_uniq. apply uniq_mklist. Qed.	Theorem	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	uniq_all_znz
in_all_znz : forall z, In z all_znz. intros (z1, Hz1). unfold all_znz; apply mkzlist_in. apply lt_mklist_lt. case (Z_mod_lt z1 n). auto with zarith. rewrite <- Hz1; intros H1 H2. case (Nat.le_gt_cases (Z.abs_nat n) (Z.abs_nat z1)); auto; intros H3. absurd (z1 < n); auto; apply Zle_not_lt. rewrite <- Z.abs_eq; auto. rewrite <- inj_Zabs_nat; auto. rewrite <- (Z.abs_eq n) by auto with zarith. rewrite <- (inj_Zabs_nat n); auto. apply inj_le; auto. Qed.	Theorem	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	in_all_znz
p_pos : 0 < p. generalize (prime_ge_2 _ p_prime); auto with zarith. Qed.	Theorem	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	p_pos
inv v := mkznz _ _ (modz p (fst (fst (Zegcd (val p v) p)))).	Definition	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	inv
div v1 v2 := mul _ v1 (inv v2).	Definition	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	div
FZpZ : field_theory (zero _) (one _) (add _) (mul _) (sub _) (opp _) div inv (@eq (znz p)). assert (Hmp := p_pos). split; auto. exact (RZnZ _ p_pos). intros H; injection H; repeat rewrite Zmod_small; auto with zarith. generalize (prime_ge_2 _ p_prime); auto with zarith. intros (n, Hn); unfold zero, one, inv, mul; simpl. intros H; apply zirr. generalize (Zegcd_is_egcd n p); case Zegcd; intros (u,v) w (Hu, (Hv, Hw)); simpl. assert (F1: rel_prime n p). apply rel_prime_le_prime; auto. rewrite Hn; auto. case (Z_mod_lt n p); try (intros H1 H2; split); auto with zarith. case (Zle_lt_or_eq _ _ H1); auto with zarith. rewrite <- Hn; intros H3; case H; apply zirr; rewrite <- H3; auto. red in F1. case (Zis_gcd_unique _ _ _ _ Hv F1); auto with zarith; intros; subst w. rewrite <- H0. rewrite Zmult_mod; repeat rewrite Zmod_mod; try rewrite <- Zmult_mod; auto. rewrite Z_mod_plus; auto with zarith. Defined.	Definition	src	[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]	src/Coqprime/elliptic/GZnZ.v	FZpZ
ell_theory : Prop := mk_ell_theory { (* field properties ) Kfth : field_theory kO kI kplus kmul ksub kopp kdiv kinv (@eq K); NonSingular: 4 A * A * A + 27 * B * B <> 0; (* Characteristic greater than 2 *) one_not_zero: 1 <> 0; two_not_zero: 2 <> 0; is_zero_correct: forall k, is_zero k = true <-> k = 0 }. Variable Eth: ell_theory.	Record	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	ell_theory
pow (k: K) (n: nat) := match n with O => 1 \| 1%nat => k \| S n1 => k * pow k n1 end.	Fixpoint	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	pow
pow_S : forall k n, k ^ (S n) = k * k ^ n. intros k n; simpl; auto; case n; auto. simpl; rewrite Eth.(Kfth).(F_R).(Rmul_comm). rewrite Eth.(Kfth).(F_R).(Rmul_1_l); auto. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	pow_S
Mkmul := rmul_ext3_Proper (Eq_ext kplus kmul kopp).	Let	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Mkmul
Kpower_theory : power_theory 1 kmul (eq (A:=K)) BinNat.nat_of_N pow. constructor. intros r n; case n; simpl; auto. intros p; elim p using BinPos.Pind; auto. intros p1 H. rewrite Pnat.nat_of_P_succ_morphism; rewrite pow_S. rewrite (pow_pos_succ (Eqsth K) Mkmul); auto. rewrite H; auto. exact Eth.(Kfth).(F_R).(Rmul_assoc). Qed.	Lemma	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Kpower_theory
iskpow_coef t := match t with \| (S ?x) => iskpow_coef x \| O => true \| _ => false end.	Ltac	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	iskpow_coef
kpow_tac t := match iskpow_coef t with \| true => constr:(BinNat.N_of_nat t) \| _ => constr:(NotConstant) end.	Ltac	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	kpow_tac
Ring Rfth : (F_R (Eth.(Kfth))) (power_tac Kpower_theory [kpow_tac]).	Add	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Ring
Field Kfth : Eth.(Kfth) (power_tac Kpower_theory [kpow_tac]).	Add	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Field
Kdiv_def := (Fdiv_def Eth.(Kfth)).	Let	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Kdiv_def
Kinv_ext : forall p q, p = q -> / p = / q. Proof. intros p q H; rewrite H; auto. Qed.	Lemma	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Kinv_ext
Ksth := (Eqsth K).	Let	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Ksth
Keqe := (Eq_ext kplus kmul kopp).	Let	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Keqe
AFth := Field_theory.F2AF Ksth Keqe Eth.(Kfth).	Let	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	AFth
Kmorph := InitialRing.gen_phiZ_morph Ksth Keqe (F_R Eth.(Kfth)). Hint Resolve one_not_zero two_not_zero : core.	Let	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Kmorph
Kdiv1 : forall r, r /1 = r. Proof. intros r; field; auto. Qed. (* Some stuff for K *)	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Kdiv1
n2k (n: nat) : K := match n with O => kO \| (S O) => kI \| (S n1) => (1 + n2k n1) end.	Fixpoint	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	n2k
N2k := n2k.	Coercion	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	N2k
Kdiff_2_0 : (2:K) <> 0. Proof. simpl; auto. Qed. Hint Resolve Kdiff_2_0 : core.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Kdiff_2_0
Keq_minus_eq : forall x y, x - y = 0 -> x = y. Proof. intros x y H. apply trans_equal with (y + (x - y)); try ring. rewrite H; ring. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Keq_minus_eq
Keq_minus_eq_inv : forall x y, x = y -> x - y = 0. Proof. intros x y HH; rewrite HH; ring. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Keq_minus_eq_inv
Kdiff_diff_minus_eq : forall x y, x <> y -> x - y <> 0. Proof. intros x y H H1; case H; apply Keq_minus_eq; auto. Qed. Hint Resolve Kdiff_diff_minus_eq : core.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Kdiff_diff_minus_eq
Kmult_integral : forall x y, x * y = 0 -> x = 0 \/ y = 0. Proof. intros x y H. generalize (Eth.(is_zero_correct) x); case (is_zero x); intros (H1, H2); auto; right. apply trans_equal with ((/x) * (x * y)); try field. intros H3; assert (H4 := H2 H3); discriminate. rewrite H; ring. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Kmult_integral
Kmult_integral_contrapositive : forall x y, x <> 0 -> y <> 0 -> x * y <> 0. Proof. intros x y H H1 H2. case (Kmult_integral H2); auto. Qed. Hint Resolve Kmult_integral_contrapositive : core.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Kmult_integral_contrapositive
Kmult_eq_compat_l : forall x y z, y = z -> x * y = x * z. intros x y z H; rewrite H; auto. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Kmult_eq_compat_l
Keq_opp_is_zero : forall x, x = - x -> x = 0. Proof. intros x H. case (@Kmult_integral (1+1:K) x); simpl; auto. apply trans_equal with (x + x); simpl; try ring. pattern x at 1; rewrite H; ring. intros H1; case two_not_zero; auto. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Keq_opp_is_zero
Kdiv_inv_eq_0 : forall x y, x/y = 0 -> y<>0 -> x = 0. Proof. intros x y H1 H2. apply trans_equal with (y * (x/y)); try field; auto. rewrite H1; ring. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Kdiv_inv_eq_0
is_zero_diff : forall x y, (x - y) ?0 = true -> x = y. Proof. intros x y H. apply trans_equal with (y + (x - y)); try ring. case (Eth.(is_zero_correct) (x - y)); intros H1 H2; rewrite H1; auto; ring. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	is_zero_diff
is_zero_diff_inv : forall x y, x = y -> (x - y) ?0 = true. Proof. intros x y H; rewrite H. case (Eth.(is_zero_correct) (y - y)); intros H1 H2; apply H2; ring. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	is_zero_diff_inv
Ksqr_eq : forall x y, x^2 = y^2 -> x = y \/ x = - y. Proof. intros x y H. case (@Kmult_integral (x - y) (x + y)); auto. ring [H]. intros H1; left; apply trans_equal with (y + (x - y)); try ring. rewrite H1; ring. intros H1; right; apply trans_equal with (-y + (x + y)); try ring. rewrite H1; ring. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Ksqr_eq
diff_rm_quo : forall x y, x/y <> 0 -> y<>0 -> x <> 0. intros x y H H0 H1; case H; field [H1]; auto. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	diff_rm_quo
dtac H := match type of H with ?X <> 0 => field_simplify X in H end; [ match type of H with ?X/?Y <> 0 => cut (X <> 0); [clear H; intros H \| apply diff_rm_quo with Y; auto] end \| auto]. (**********************************************************) ( ) ( Definition of the elements of the curve ) ( ) (**********************************************************)	Ltac	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	dtac
elt : Set := (* The infinity point ) inf_elt: elt ( A point of the curve ) \| curve_elt: forall x y, y^2 = x^3 + A x + B -> elt.	Inductive	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	elt
Kdec : forall a b: K, {a = b} + {a <> b}. intros a b; case_eq ((a - b) ?0); intros H. left; apply is_zero_diff; auto. right; intros H1. rewrite (is_zero_diff_inv H1) in H; discriminate. Defined.	Definition	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	Kdec
curve_elt_irr : forall x1 x2 y1 y2 H1 H2, x1 = x2 -> y1 = y2 -> @curve_elt x1 y1 H1 = @curve_elt x2 y2 H2. Proof. intros x1 x2 y1 y2 H1 H2 H3 H4. subst. rewrite (fun H => eq_proofs_unicity H H1 H2); auto. intros x y; case (Kdec x y); auto. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	curve_elt_irr
curve_elt_irr1 : forall x1 x2 y1 y2 H1 H2, x1 = x2 -> (x1 = x2 -> y1 = y2) -> @curve_elt x1 y1 H1 = @curve_elt x2 y2 H2. Proof. intros x1 x2 y1 y2 H1 H2 H3 H4. apply curve_elt_irr; auto. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	curve_elt_irr1
ceqb : forall a b: elt, {a = b} + {a<>b}. Proof. intros a b; case a; case b; auto; try (intros; right; intros; discriminate). intros x1 y1 H1 x2 y2 H2; case (Kdec x1 x2); intros H3. case (Kdec y1 y2); intros H4. left; apply curve_elt_irr1; auto. right; intros H; injection H; intros H5 H6; case H4; auto. right; intros H; injection H; intros H4 H5; case H3; auto. Defined.	Definition	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	ceqb
is_zero_true : forall e, is_zero e = true -> e = 0. intro e; generalize (Eth.(is_zero_correct) e); case is_zero; auto; intros (H,_); auto. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	is_zero_true
is_zero_false : forall e, is_zero e = false -> e <> 0. intro e; generalize (Eth.(is_zero_correct) e); case is_zero; auto; intros (_,H); auto. intros; discriminate. intros _ H1; generalize (H H1); discriminate. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	is_zero_false
opp_lem : forall x y, y ^ 2 = x ^ 3 + A * x + B -> (- y) ^ 2 = x ^ 3 + A * x + B. Proof. intros x y H. Time field [H]. Qed. (**********************************************************) ( ) ( Opposite function ) ( ) (**********************************************************)	Lemma	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	opp_lem
opp : elt -> elt. refine (fun p => match p with inf_elt => inf_elt \| @curve_elt x y H => @curve_elt x (-y) _ end). apply opp_lem; auto. Defined.	Definition	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	opp
opp_opp : forall p, opp (opp p) = p. Proof. intros p; case p; simpl; auto; intros; apply curve_elt_irr; ring. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	opp_opp
curve_elt_opp : forall x1 x2 y1 y2 H1 H2, x1 = x2 -> @curve_elt x1 y1 H1 = @curve_elt x2 y2 H2 \/ @curve_elt x1 y1 H1 = opp (@curve_elt x2 y2 H2). intros x1 x2 y1 y2 H1 H2 H3. case (@Kmult_integral (y1 - y2) (y1 + y2)); try intros H4. ring_simplify. ring [H1 H2 H3]. left; apply curve_elt_irr; auto. apply Keq_minus_eq; auto. right; unfold opp; apply curve_elt_irr; auto. apply Keq_minus_eq; rewrite <- H4; ring. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	curve_elt_opp
add_lem1 : forall x1 y1, y1 <> 0 -> y1 ^ 2 = x1 ^ 3 + A * x1 + B -> let l := (3 * x1 * x1 + A) / (2 * y1) in let x3 := l ^ 2 - 2 * x1 in (- y1 - l * (x3 - x1)) ^ 2 = x3 ^ 3 + A * x3 + B. Proof. intros x1 y1 H H1 l x3; unfold x3, l. Time field [H1]. split; auto. Qed.	Lemma	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	add_lem1
add_lem2 : forall x1 y1 x2 y2, x1 <> x2 -> y1 ^ 2 = x1 ^ 3 + A * x1 + B -> y2 ^ 2 = x2 ^ 3 + A * x2 + B -> let l := (y2 - y1) / (x2 - x1) in let x3 := l ^ 2 - x1 - x2 in (- y1 - l * (x3 - x1)) ^ 2 = x3 ^ 3 + A * x3 + B. Proof. intros x1 y1 x2 y2 H H1 H2 l x3; unfold x3, l. Time field [H1 H2]; auto. Qed.	Lemma	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	add_lem2
add_zero : forall x1 x2 y1 y2, x1 = x2 -> y1 ^ 2 = x1 ^ 3 + A * x1 + B -> y2 ^ 2 = x2 ^ 3 + A * x2 + B -> y1 <> -y2 -> y1 = y2. Proof. intros x1 x2 y1 y2 H H1 H2 H3; subst x2. case (@Kmult_integral (y1 - y2) (y1 + y2)); try (intros H4; apply Keq_minus_eq; auto). ring [H1 H2]. case H3; apply Keq_minus_eq; rewrite <- H4; ring. Qed.	Lemma	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	add_zero
add_zero_diff : forall x1 x2 y1 y2, x1 = x2 -> y1 ^ 2 = x1 ^ 3 + A * x1 + B -> y2 ^ 2 = x2 ^ 3 + A * x2 + B -> y1 <> -y2 -> y1 <>0. Proof. intros x1 x2 y1 y2 H H1 H2 H3 H4. assert (H5:= add_zero H H1 H2 H3). case H3; rewrite <- H5; ring [H4]. Qed. (**********************************************************) ( ) ( Addition ) ( ) (**********************************************************)	Lemma	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	add_zero_diff
add : elt -> elt -> elt. refine (fun p1 p2 => match p1 with inf_elt => p2 \| @curve_elt x1 y1 H1 => match p2 with inf_elt => p1 \| @curve_elt x2 y2 H2 => if x1 ?= x2 then (* we have p1 = p2 or p1 = - p2 ) if (y1 ?= -y2) then ( we do p - p ) inf_elt else ( we do the tangent ) let l := (3x1x1 + A)/(2y1) in let x3 := l^2 - 2 * x1 in @curve_elt x3 (-y1 - l * (x3 - x1)) _ else (* general case ) let l := (y2 - y1)/(x2 - x1) in let x3 := l ^ 2 - x1 -x2 in @curve_elt x3 (-y1 - l (x3 - x1)) _ end end). apply (@add_lem1 x1 y1); auto. apply (@add_zero_diff x1 x2 y1 y2); auto. apply (@add_lem2 x1 y1 x2 y2); auto. Defined. (**********************************************************) ( ) ( Direct case predicate for add ) ( ) (**********************************************************)	Definition	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	add
kauto := auto; match goal with H: ~ ?A, H1: ?A \|- _ => case H; auto end. (* A little tactic to split kdec *)	Ltac	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	kauto
ksplit := let h := fresh "KD" in case Kdec; intros h; try (case h; kauto; fail).	Ltac	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	ksplit
add_case : forall P, (forall p, P inf_elt p p) -> (forall p, P p inf_elt p) -> (forall p, P p (opp p) inf_elt) -> (forall p1 x1 y1 H1 p2 x2 y2 H2 l, p1 = (@curve_elt x1 y1 H1) -> p2 = (@curve_elt x2 y2 H2) -> p2 = add p1 p1 -> y1<>0 -> l = (3 * x1 * x1 + A) / (2 * y1) -> x2 = l ^ 2 - 2 * x1 -> y2 = - y1 - l * (x2 - x1) -> P p1 p1 p2) -> (forall p1 x1 y1 H1 p2 x2 y2 H2 p3 x3 y3 H3 l, p1 = (@curve_elt x1 y1 H1) -> p2 = (@curve_elt x2 y2 H2) -> p3 = (@curve_elt x3 y3 H3) -> p3 = add p1 p2 -> x1 <> x2 -> l = (y2 - y1) / (x2 - x1) -> x3 = l ^ 2 - x1 - x2 -> y3 = -y1 - l * (x3 - x1) -> P p1 p2 p3)-> forall p q, P p q (add p q). Proof. intros P H1 H2 H3 H4 H5 p q; case p; case q; auto. intros x2 y2 e2 x1 y1 e1; unfold add. repeat ksplit. replace (@curve_elt x2 y2 e2) with (opp (@curve_elt x1 y1 e1)); auto; simpl. apply curve_elt_irr; auto; ring [KD0]. assert (HH: y1 <> 0). apply (@add_zero_diff x1 x2 y1 y2); auto. replace (@curve_elt x2 y2 e2) with (@curve_elt x1 y1 e1); auto. eapply H4; eauto; simpl; repeat ksplit; try apply curve_elt_irr; auto. case HH; apply Keq_opp_is_zero; auto. apply curve_elt_irr; auto. case (@Kmult_integral (y1 - y2) (y1 + y2)); try intros HH1. ring [e1 e2 KD]. apply Keq_minus_eq; auto. case KD0; apply Keq_minus_eq; ring_simplify; auto. eapply H5; eauto; simpl; repeat ksplit. apply curve_elt_irr; auto. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	add_case
add_casew : forall P, (forall p, P inf_elt p p) -> (forall p, P p inf_elt p) -> (forall p, P p (opp p) inf_elt) -> (forall p1 x1 y1 H1 p2 x2 y2 H2 p3 x3 y3 H3 l, p1 = (@curve_elt x1 y1 H1) -> p2 = (@curve_elt x2 y2 H2) -> p3 = (@curve_elt x3 y3 H3) -> p3 = add p1 p2 -> p1 <> opp p2 -> ((x1 = x2 /\ y1 = y2 /\ l = (3 * x1 * x1 + A) / (2 * y1)) \/ (x1 <> x2 /\ l = (y2 - y1) / (x2 - x1)) ) -> x3 = l ^ 2 - x1 - x2 -> y3 = -y1 - l * (x3 - x1) -> P p1 p2 p3)-> forall p q, P p q (add p q). intros; apply add_case; auto. intros; eapply X2; eauto. rewrite H; simpl; intros tmp; case H4; injection tmp; apply Keq_opp_is_zero. ring [H6]. intros; eapply X2; eauto. rewrite H; rewrite H0; simpl; intros tmp; case H6; injection tmp; auto. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	add_casew
is_tangent p1 p2 := p1 <> inf_elt /\ p1 = p2 /\ p1 <> opp p2.	Definition	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	is_tangent
is_generic p1 p2 := p1 <> inf_elt /\ p2 <> inf_elt /\ p1 <> p2 /\ p1 <> opp p2.	Definition	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	is_generic
is_gotan p1 p2 := p1 <> inf_elt /\ p2 <> inf_elt /\ p1 <> opp p2.	Definition	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	is_gotan
kcase X Y := pattern X, Y, (add X Y); apply add_case; auto; clear X Y.	Ltac	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	kcase
kcasew X Y := pattern X, Y, (add X Y); apply add_casew; auto; clear X Y. (**********************************************************) ( ) ( Generic case for associativity ) ( (A + B) + C = A + (B + C) ) ( ) (**********************************************************)	Ltac	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	kcasew
spec1_assoc : forall p1 p2 p3, is_generic p1 p2 -> is_generic p2 p3 -> is_generic (add p1 p2) p3 -> is_generic p1 (add p2 p3) -> add p1 (add p2 p3) = add (add p1 p2) p3. intros p1 p2; kcase p1 p2. intros p p3 _ _ (HH, _); case HH; auto. intros p3 _ _ _ p4 _ _ _ _ _ _ _ _ _ _ _ p5 (_, (_, (HH, _))); case HH; auto. intros p1 x1 y1 H1 p2 x2 y2 H2 p4 x4 y4 H4 l Hp Hp2 Hp4 Hp4b Hx Hl Hx4 Hy4 p3. generalize Hp2 Hp4b; clear Hp2 Hp4b; kcase p2 p3. intros; discriminate. intros p _ _ _ (_,(HH, _)); case HH; auto. intros p _ _ _ (_,(_,(_,HH))); case HH; rewrite opp_opp; auto. intros p2 _ _ _ p3 _ _ _ _ _ _ _ _ _ _ _ _ _ _ (_, (_, (HH, _))); case HH; auto. intros p2 x2b y2b H2b p3 x3 y3 H3 p5 x5 y5 H5 l1. intros Hp2; pattern p2 at 2; rewrite Hp2; clear Hp2. intros Hp3 Hp5 Hp5b Hd Hl1 Hx5 Hy5 tmp. injection tmp; intros; subst y2b x2b; clear tmp H2b. generalize Hp Hp5 Hp5b Hp4b H6 H9; clear Hp Hp5 Hp5b Hp4b H6 H9. kcase p1 p5. intros; discriminate. intros; discriminate. intros p _ _ _ _ _ (_,(_,(_,HH))); case HH; rewrite opp_opp; auto. intros p1 _ _ _ p5 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (_,(_,(HH,_))); case HH; auto. intros p1 x1b y1b H1b. intros p5b x5b y5b H5b p6 x6 y6 H6 l2. intros Hp1; pattern p1 at 2; rewrite Hp1; clear Hp1. intros Hp5; pattern p5b at 2; rewrite Hp5; clear Hp5. intros Hp6 _ Hd2 Hl2 Hx6 Hy6. intros tmp; injection tmp; intros HH1 HH2; subst y1b x1b; clear tmp H1b. intros tmp; injection tmp; intros HH1 HH2; subst y5b x5b; clear tmp H5b. intros _ Hp4b _ _. generalize Hp3 Hp4 Hp4b H7 H8; clear Hp3 Hp4 Hp4b H7 H8. kcase p4 p3. intros; discriminate. intros; discriminate. intros p _ _ _ _ (_, (_, (_,HH))); case HH; rewrite opp_opp; auto. intros p3 _ _ _ p4 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (_,(_,(HH, _))); case HH; auto. intros p4b x4b y4b H4b p3b x3b y3b H3b p7 x7 y7 H7 l3. intros Hp4b; pattern p4b at 2; rewrite Hp4b; clear Hp4b. intros Hp3b; pattern p3b at 2; rewrite Hp3b; clear Hp3b. intros Hp7 _ Hd3 Hl3 Hx7 Hy7. intros tmp; injection tmp; intros HH1 HH2; subst y3b x3b; clear tmp H3b. intros tmp; injection tmp; intros HH1 HH2; subst y4b x4b; clear tmp H4b. intros _ _ _. subst p6 p7; apply curve_elt_irr; clear H6 H7; apply Keq_minus_eq; clear H4 H5; subst. Time field [H1 H2 H3]; auto; repeat split; auto. intros VV; field_simplify_eq[H1 H2] in VV. case Hd3; symmetry; apply Keq_minus_eq; field_simplify_eq [H1 H2]; auto. intros VV; field_simplify_eq[H1 H2] in VV. case Hd2; symmetry; apply Keq_minus_eq; field_simplify_eq [H1 H2]; auto. Time field [H1 H2 H3]; auto; repeat split; auto. intros VV; field_simplify_eq[H1 H2] in VV. case Hd3; symmetry; apply Keq_minus_eq; field_simplify_eq [H1 H2]; auto. intros VV; field_simplify_eq[H1 H2] in VV. case Hd2; symmetry; apply Keq_minus_eq; field_simplify_eq [H1 H2]; auto. Qed. (**********************************************************) ( ) ( Tangent case for associativity ) ( A + (B + B) = (A + B) + B ) ( ) (**********************************************************)	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	spec1_assoc
spec2_assoc : forall p1 p2 p3, is_generic p1 p2 -> is_tangent p2 p3 -> is_generic (add p1 p2) p3 -> is_generic p1 (add p2 p3) -> add p1 (add p2 p3) = add (add p1 p2) p3. intros p1 p2; kcase p1 p2. intros p p3 _ _ (HH, _); case HH; auto. intros p3 _ _ _ p4 _ _ _ _ _ _ _ _ _ _ _ p5 (_, (_, (HH, _))); case HH; auto. intros p1 x1 y1 H1 p2 x2 y2 H2 p4 x4 y4 H4 l Hp Hp2 Hp4 Hp4b Hx Hl Hx4 Hy4 p3. generalize Hp2 Hp4b; clear Hp2 Hp4b. kcase p2 p3. intros; discriminate. intros p _ _ _ _ (_, (HH, _)); case HH; auto. intros p _ _ _ _ _ (_, (HH, _)); case HH; auto. intros p2 x2b y2b H2b p5 x5 y5 H5 l1. intros Hp2b. intros Hp5 Hp5b Hd Hl1 Hx5 Hy5 Hp2. rewrite Hp2 in Hp2b. injection Hp2b; intros HH HH1; subst y2b x2b; clear Hp2b. generalize Hp Hp5 Hp5b; clear Hp Hp5 Hp5b. kcase p1 p5. intros; discriminate. intros; discriminate. intros p _ _ _ _ _ _ _ (_,(_,(_,HH))); case HH; rewrite opp_opp; auto. intros p1 _ _ _ p5 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (_,(_,(HH,_))); case HH; auto. intros p1 x1b y1b H1b. intros p5b x5b y5b H5b p6 x6 y6 H6 l2. intros Hp1; pattern p1 at 2; rewrite Hp1; clear Hp1. intros Hp5; pattern p5b at 2; rewrite Hp5; clear Hp5. intros Hp6 _ Hd2 Hl2 Hx6 Hy6. intros tmp; injection tmp; intros HH1 HH2; subst y1b x1b; clear tmp H1b. intros tmp; injection tmp; intros HH1 HH2; subst y5b x5b; clear tmp H5b. intros _ Hp4b _ _. generalize Hp2 Hp4 Hp4b; clear Hp2 Hp4 Hp4b. kcase p4 p2. intros; discriminate. intros; discriminate. intros p _ _ _ (_, (_, (_,HH))); case HH; rewrite opp_opp; auto. intros p3 _ _ _ p4 _ _ _ _ _ _ _ _ _ _ _ _ _ _ (_,(_,(HH, _))); case HH; auto. intros p4b x4b y4b H4b p3b x3b y3b H3b p7 x7 y7 H7 l3. intros Hp4b; pattern p4b at 2; rewrite Hp4b; clear Hp4b. intros Hp3b; pattern p3b at 2; rewrite Hp3b; clear Hp3b. intros Hp7 _ Hd3 Hl3 Hx7 Hy7. intros tmp; injection tmp; intros HH1 HH2; subst y3b x3b; clear tmp H3b. intros tmp; injection tmp; intros HH1 HH2; subst y4b x4b; clear tmp H4b. intros _ _ _. subst p6 p7; apply curve_elt_irr; clear H6 H7 H2b; apply Keq_minus_eq; clear H4 H5; subst. Time field [H1 H2]; auto; repeat split; auto. intros VV; field_simplify_eq[H1 H2] in VV. case Hd3; symmetry; apply Keq_minus_eq; field_simplify_eq [H1 H2]; auto. intros VV; field_simplify_eq[H1 H2] in VV. case Hd2; symmetry; apply Keq_minus_eq; field_simplify_eq [H1 H2]; auto. Time field [H1 H2]; auto; repeat split; auto. intros VV; field_simplify_eq[H1 H2] in VV. case Hd3; symmetry; apply Keq_minus_eq; field_simplify_eq [H1 H2]; auto. intros VV; field_simplify_eq[H1 H2] in VV. case Hd2; symmetry; apply Keq_minus_eq; field_simplify_eq [H1 H2]; auto. intros p3 x3 y3 H3 p5 x5 y5 H5 p6 x6 y6 H6 l1 Hp3 Hp5 _ _ Hd _ _ _ _ _ _. rewrite Hp3; rewrite Hp5; intros (_, (HH,_)); case Hd; injection HH; auto. Time Qed. (**********************************************************) ( ) ( Identity case for associativity ) ( (A + A) + (A + A) = A + (A + (A + A)) ) ( ) (**********************************************************)	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	spec2_assoc
spec3_assoc : forall p1 p2 p3, is_generic p1 p2 -> is_tangent p2 p3 -> is_generic (add p1 p2) p3 -> is_tangent p1 (add p2 p3) -> add p1 (add p2 p3) = add (add p1 p2) p3. intros p1 p2. kcase p1 p2. intros p p3 _ _ (HH, _); case HH; auto. intros p3 _ _ _ p4 _ _ _ _ _ _ _ _ _ _ _ p5 (_, (_, (HH, _))); case HH; auto. intros p1 x1 y1 H1 p2 x2 y2 H2 p4 x4 y4 H4 l Hp Hp2 Hp4 Hp4b Hx Hl Hx4 Hy4 p3. generalize Hp2 Hp4b; clear Hp2 Hp4b. kcase p2 p3. intros; discriminate. intros p _ _ _ _ (_, (HH, _)); case HH; auto. intros p _ _ _ (_ ,(_ , HH)); case HH; rewrite opp_opp; auto. intros p2 x2b y2b H2b p5 x5 y5 H5 l1. intros Hp2b. intros Hp5 Hp5b Hd Hl1 Hx5 Hy5 Hp2. rewrite Hp2 in Hp2b. injection Hp2b; intros HH HH1; subst y2b x2b; clear Hp2b H2b. generalize Hp Hp5 Hp5b; clear Hp Hp5 Hp5b. kcase p1 p5. intros; discriminate. intros; discriminate. intros p _ _ _ _ _ _ _ (_, (_,HH)); case HH; rewrite opp_opp; auto. intros p1 x1b y1b H1b. intros p6 x6 y6 H6 l2. intros Hp1; pattern p1 at 3 4; rewrite Hp1; clear Hp1. intros Hp6 _ Hd2 Hl2 Hx6 Hy6. intros tmp; injection tmp; intros HH1 HH2; subst y1b x1b; clear tmp. intros tmp; injection tmp; intros HH1 HH2. subst y5 x5; clear tmp H5. rename HH1 into Hy1; rename HH2 into Hx1. generalize Hp2 Hp4; clear Hp2 Hp4. kcase p4 p2. intros; discriminate. intros; discriminate. intros p _ _ _ _ _ _ (_, (_, (_, HH))); case HH; rewrite opp_opp; auto. intros p3 _ _ _ p4 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (_,(_,(HH, _))); case HH; auto. intros p4b x4b y4b H4b p2b x2b y2b H2b. intros p7 x7 y7 H7 l3. intros Hp4b; pattern p4b at 2; rewrite Hp4b; clear Hp4b. intros Hp2b; pattern p2b at 2; rewrite Hp2b; clear Hp2b. intros Hp7 _ Hd1 Hl3 Hx7 Hy7. intros tmp; injection tmp; intros HH1 HH2; subst y2b x2b; clear tmp H2b. intros tmp; injection tmp; intros HH1 HH2; subst y4b x4b; clear tmp H4b. intros _ _ _ _ _ _. subst p6 p7; apply curve_elt_irr; clear H6 H7; apply Keq_minus_eq; clear H4 H1b; subst. Time field [H2]; auto; repeat split; auto; intros HH; field_simplify_eq in HH; auto. case Hx; symmetry; apply Keq_minus_eq. field_simplify_eq; auto. case Hd1; symmetry; apply Keq_minus_eq; field_simplify_eq; repeat split; auto. intros HH1; ring_simplify in HH1; auto. case Hx; symmetry; apply Keq_minus_eq. field_simplify_eq; auto. case Hd2; apply Keq_minus_eq; field_simplify_eq; auto. Time field [H2]; auto; repeat split; auto; intros HH; field_simplify_eq in HH; auto. case Hx; symmetry; apply Keq_minus_eq. field_simplify_eq; auto. case Hd1; symmetry; apply Keq_minus_eq; field_simplify_eq; repeat split; auto. intros HH1; ring_simplify in HH1; auto. case Hx; symmetry; apply Keq_minus_eq. field_simplify_eq; auto. case Hd2; apply Keq_minus_eq; field_simplify_eq; auto. intros p1b x1b y1b H1b. intros p5b x5b y5b H5b. intros p3 x3 y3 H3 l2. intros Hp1b; pattern p1b at 2 5; rewrite Hp1b; clear Hp1b. intros Hp5b; pattern p5b at 2 4; rewrite Hp5b; clear Hp5b. intros Hp3 _; rewrite Hp3; clear Hp3. intros Hx1 _ _ _. intros tmp; injection tmp; intros HH1 HH2; subst y1b x1b; clear tmp. intros tmp; injection tmp; intros HH1 HH2; subst y5b x5b; clear tmp. intros _ _ _ _ _ (_,(HH, _)); case Hx1; injection HH; auto. intros p2b x2b y2b H2b. intros p3 x3 y3 H3. intros p5 x5 y5 H5 l2. intros Hp2b; pattern p2b at 2 5; rewrite Hp2b; clear Hp2b. intros Hp3; rewrite Hp3; clear Hp3. intros _ _ Hx1 _ _ _. intros tmp; injection tmp; intros HH1 HH2; subst y2b x2b; clear tmp. intros _ _ (_,(HH, _)); case Hx1; injection HH; auto. Time Qed. (**********************************************************) ( ) ( inf_elt is the zero ) ( ) (**********************************************************)	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	spec3_assoc
add_0_l : forall p, add inf_elt p = p. Proof. auto. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	add_0_l
add_0_r : forall p, add p inf_elt = p. Proof. intros p; case p; auto. Qed. (**********************************************************) ( ) ( opp is the opposite ) ( ) (**********************************************************)	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	add_0_r
add_opp : forall p, add p (opp p) = inf_elt. Proof. intros p; case p; unfold add; simpl; auto. intros x1 y1 H1. repeat ksplit. case KD0; ring. Qed. (**********************************************************) ( ) ( Addition is commutative ) ( ) (**********************************************************)	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	add_opp
add_comm : forall p1 p2, add p1 p2 = add p2 p1. Proof. intros p1 p2; case p1. rewrite add_0_r; rewrite add_0_l; auto. intros x1 y1 H1; case p2. rewrite add_0_r; rewrite add_0_l; auto. intros x2 y2 H2; simpl; repeat ksplit. case KD2; ring [KD0]. case KD0; ring [KD2]. assert (H3 := add_zero KD H1 H2 KD0). apply curve_elt_irr; subst x2 y2; auto. case KD; auto. case KD; auto. apply curve_elt_irr; field; auto. Qed.	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	add_comm
aux1 : forall x1 y1 x2 y2, y1 ^ 2 = x1 ^ 3 + A * x1 + B -> y2 ^ 2 = x2 ^ 3 + A * x2 + B -> x1 <> x2 -> y2 = 0 -> ((y2 - y1) / (x2 - x1))^2 - x1 - x2 = x2 -> False. Proof. intros x1 y1 x2 y2 H H1 H2 H3 H4. subst y2. assert (Hu : x2 ^ 3 = -(A * x2 + B)). apply trans_equal with ((x2 ^ 3 + A * x2 + B) - (A * x2 + B)); try ring. rewrite <- H1; ring. assert (H5:= (Keq_minus_eq_inv H4)); clear H4. field_simplify_eq [H Hu] in H5; auto. generalize (Kmult_eq_compat_l x2 H5); rename H5 into H4. replace (x2 * 0) with 0; try ring; intros H5. field_simplify_eq [Hu] in H5. generalize H5; clear H5. match goal with \|- (?X = _ -> _) => replace X with ((x2 - x1) * (2* A x2 + 3 B)); try ring end. intros tmp; case (Kmult_integral tmp); clear tmp; intros HH2. case H2; apply sym_equal; apply Keq_minus_eq; auto. generalize (Kmult_eq_compat_l ((2 * A )^3) (sym_equal H1)). replace ((2 * A)^3 * 0 ^ 2) with 0; try (ring). intros H5; ring_simplify in H5; auto. match type of H5 with (?X + ?Y + _ = _) => let x := (constr:(2 * A * x2)) in ((replace Y with (x ^ 3) in H5; try ring); (replace X with (4 * A^3 * x) in H5; try ring); replace x with (-(3) * B) in H5) end. 2: apply sym_equal; apply Keq_minus_eq; apply trans_equal with (2:= HH2); ring. ring_simplify in H5; auto. match type of Eth.(NonSingular) with ?X <> 0 => case (@Kmult_integral (-B) X); try intros HH3; try (case NonSingular; auto; fail) end. rewrite <- H5; ring. assert (HH3b : B = 0). replace B with (-(-B)); try ring; rewrite HH3; ring. case (@Kmult_integral 2 (A * x2)); try intros HH4; auto. apply trans_equal with (2:= HH2). rewrite HH3b; ring. case (Kmult_integral HH4); try intros HH5; auto. case Eth.(NonSingular); rewrite HH3b; rewrite HH5; simpl; ring. ring_simplify [HH5] in H4; auto. case (@Kmult_integral A x1); try intros HH6; auto. apply trans_equal with (2 := H4); rewrite HH3b; ring. case Eth.(NonSingular); rewrite HH3b; rewrite HH6; ring. case H2; rewrite HH6; auto. Qed. (**********************************************************) ( ) ( There is only one zero ) ( ) (**********************************************************)	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	aux1
uniq_zero : forall p1 p2, add p1 p2 = p2 -> p1 = inf_elt. Proof. intros p1 p2; kcase p1 p2. intros p; case p; simpl; auto; intros; discriminate. intros. subst p1 p2; injection H8; intros H9 H10. generalize (Keq_minus_eq_inv H7); clear H7; intros H7. ring_simplify [H9 H10] in H7. case (Kmult_integral H7); auto; intros H11. case Kdiff_2_0; auto. case H4; auto. intros p1 x1 y1 H1 p2 x2 y2 H2 p3 x3 y3 H3 l Hp1 Hp2 Hp3 Hp3b Hd Hl Hx3 Hy3 Hp. apply False_ind. subst p2; subst p3; injection Hp; clear p1 Hp1 Hp Hp3b; intros. case (@aux1 x1 y1 x2 y2); auto. generalize (Keq_minus_eq_inv Hy3); rewrite Hl; rewrite H; rewrite H0; clear Hy3; intros Hy3. field_simplify_eq in Hy3; auto. assert (HH: 2 * y2 * (x2 - x1) = 0). rewrite Hy3; ring. clear Hy3; rename HH into Hy3. case (Kmult_integral Hy3); auto; clear Hy3; intros Hy3. case (Kmult_integral Hy3); auto; clear Hy3; intros Hy3. case Kdiff_2_0; auto. case Hd. symmetry; apply Keq_minus_eq; auto. rewrite <- Hl; rewrite <- Hx3; auto. Qed. (**********************************************************) ( ) ( There is only one opposite ) ( ) (**********************************************************)	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	uniq_zero
uniq_opp : forall p1 p2, add p1 p2 = inf_elt -> p2 = opp p1. Proof. intros p1 p2; kcase p1 p2. intros p H; rewrite H; auto. intros; subst; discriminate. intros; subst; discriminate. Qed. (**********************************************************) ( ) ( Opposite of zero is zero ) ( ) (**********************************************************)	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	uniq_opp
opp_0 : opp (inf_elt) = inf_elt. Proof. auto. Qed. (**********************************************************) ( ) ( Opposite of a sum is the sum of opposite ) ( ) (**********************************************************)	Theorem	src	[ "From Coq Require Import Arith_base.", "From Coq Require Import Field_tac.", "From Coq Require Import Ring.", "From Coq Require Import Eqdep_dec.", "From Coqprime Require Import FGroup.", "From Coq Require Import List.", "From Coqprime Require Import UList.", "From Coq Require Import ZArith." ]	src/Coqprime/elliptic/SMain.v	opp_0

znz_inj : forall a b, a = b -> val a = val b. intros; subst; auto. Qed.

Theorem

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

znz_inj

Zeq_iok : forall x y, x = y -> Zeq_bool x y = true. intros x y H; subst. apply Zeq_is_eq_bool, eq_refl. Qed.

Theorem

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

Zeq_iok

modz : forall x, (x mod n) = (x mod n) mod n. intros x; rewrite Zmod_mod; auto with zarith. Qed.

Lemma

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

modz

zero := mkznz _ (modz 0).

Definition

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

zero

one := mkznz _ (modz 1).

Definition

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

one

add v1 v2 := mkznz _ (modz (val v1 + val v2)).

Definition

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

add

sub v1 v2 := mkznz _ (modz (val v1 - val v2)).

Definition

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

sub

mul v1 v2 := mkznz _ (modz (val v1 * val v2)).

Definition

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

mul

opp v := mkznz _ (modz (-val v)).

Definition

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

opp

zirr : forall x1 x2 H1 H2, x1 = x2 -> mkznz x1 H1 = mkznz x2 H2. Proof. intros x1 x2 H1 H2 H3. subst x1. rewrite (fun H => eq_proofs_unicity H H1 H2); auto. intros x y; case (Z.eq_dec x y); auto. Qed.

Theorem

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

zirr

znz1 : forall x, x mod 1 = 0. intros x; apply Zdivide_mod; auto with zarith. Qed.

Lemma

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

znz1

RZnZ : ring_theory zero one add mul sub opp (@eq znz). split; auto. intros p; case p; intros x H; refine (zirr _ _ _ _ _); simpl; auto. intros [x Hx] [y Hy]. refine (zirr _ _ _ _ _); simpl. rewrite Zplus_comm; auto. intros [x Hx] [y Hy] [z Hz]. refine (zirr _ _ _ _ _); simpl. rewrite Zplus_mod; auto. rewrite (Zplus_mod((x + y) mod n)); auto. repeat rewrite Zmod_mod; auto. repeat rewrite <- Zplus_mod; auto; rewrite Zplus_assoc; auto. intros p; case p; intros x H. refine (zirr _ _ _ _ _); simpl. case (Zle_lt_or_eq 1 n); auto with zarith; intros Hz. rewrite (Zmod_small 1); auto with zarith. rewrite Zmult_1_l; auto. clear p; subst n; rewrite znz1; rewrite H; rewrite znz1; auto. intros [x Hx] [y Hy]. refine (zirr _ _ _ _ _); simpl. rewrite Zmult_comm; auto. intros [x Hx] [y Hy] [z Hz]. refine (zirr _ _ _ _ _); simpl. rewrite Zmult_mod; auto. rewrite (Zmult_mod ((x * y) mod n)); auto. repeat rewrite Zmod_mod; auto. repeat rewrite <- Zmult_mod; auto; rewrite Zmult_assoc; auto. intros [x Hx] [y Hy] [z Hz]. refine (zirr _ _ _ _ _); simpl. rewrite Zmult_mod; auto. rewrite Zmod_mod; auto. rewrite <- Zmult_mod; auto. rewrite (Zplus_mod ((x*z) mod n)); auto. repeat rewrite Zmod_mod; auto. rewrite <- Zplus_mod; auto. apply f_equal2 with (f := Z.modulo); auto; ring. intros [x Hx] [y Hy]. refine (zirr _ _ _ _ _); simpl. rewrite Zplus_mod; auto. repeat rewrite Zmod_mod; auto. rewrite <- Zplus_mod; auto. intros [x Hx]. refine (zirr _ _ _ _ _); simpl. rewrite Zplus_mod; auto. repeat rewrite Zmod_mod; auto. rewrite <- Zplus_mod; auto. apply f_equal2 with (f := Z.modulo); auto; ring. Defined.

Definition

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

RZnZ

Ring RZnZ : RZnZ. (* It is finite *)

Add

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

Ring

mklist (n: nat): list nat := match n with O => nil | (S n) => cons n (mklist n) end.

Fixpoint

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

mklist

mklist_length : forall n1, length (mklist n1) = n1. Proof. intros n1; elim n1; simpl; auto; clear n1. Qed.

Lemma

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

mklist_length

mklist_lt : forall n1 x, (In x (mklist n1)) -> (x < n1)%nat. intros n1; elim n1; simpl; auto; clear n1. intros x H; case H. intros n1 Hrec x [H1 | H1]; try subst x; auto with arith. Qed.

Theorem

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

mklist_lt

lt_mklist_lt : forall n1 x, (x < n1)%nat -> (In x (mklist n1)). intros n1 x H; elim H; simpl; auto. Qed.

Theorem

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

lt_mklist_lt

uniq_mklist : forall m, ulist (mklist m). intros m; elim m; simpl; auto; clear m. intros m H; constructor; auto. intros H1; absurd (m < m)%nat; auto with arith. apply mklist_lt; auto. Qed.

Theorem

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

uniq_mklist

nat_z_kt : forall x, (x < Z.abs_nat n)%nat -> (Z_of_nat x) = (Z_of_nat x) mod n. Proof. intros x H; rewrite Zmod_small; lia. Qed.

Theorem

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

nat_z_kt

mkzlist : forall (l: list nat), (forall x, In x l -> (x < Z.abs_nat n)%nat) -> list znz. fix mkzlist 1; intros l; case l. intros; exact nil. intros n1 l1 Hn. assert (F1: forall x, In x l1 -> (x < Z.abs_nat n)%nat). intros; apply Hn; simpl; auto. assert (F2: (n1 < Z.abs_nat n)%nat). apply Hn; simpl; auto. exact (cons (mkznz _ (nat_z_kt _ F2)) (mkzlist _ F1)). Defined.

Definition

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

mkzlist

mkzlist_length : forall l H, length (mkzlist l H) = length l. Proof. intros l; elim l; simpl; auto; clear l. Qed.

Lemma

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

mkzlist_length

in_mkzlist : forall l a Ha Hl, In (mkznz (Z_of_nat a) Ha) (mkzlist l Hl) -> In a l. intros l1; elim l1; simpl; auto; clear l1. intros a1 l1 Hrec1 a2 l2 Hl2 [H4 | H4]. generalize (znz_inj _ _ H4); simpl; clear H4; intros H4; left. rewrite <- (Zabs_nat_Z_of_nat a1); rewrite H4; rewrite Zabs_nat_Z_of_nat; auto. right; apply (Hrec1 _ _ _ H4). Qed.

Theorem

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

in_mkzlist

mkzlist_in : forall l a Ha Hl, In (Z.abs_nat a) l -> In (mkznz a Ha) (mkzlist l Hl). intros l1; elim l1; simpl; auto; clear l1. intros a1 l1 Hrec1 a2 l2 Hl2 [H4 | H4]; auto. left; apply zirr; auto. rewrite H4; rewrite inj_Zabs_nat; auto. rewrite Z.abs_eq; auto with zarith. case (Z_mod_lt a2 n); auto with zarith. Qed.

Theorem

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

mkzlist_in

mkzlist_uniq : forall l H, ulist l -> ulist (mkzlist l H). intros l H H1; generalize H; elim H1; simpl; auto; clear l H H1. intros a l H1 H2 Hrec H3; constructor; auto. intros HH; case H1; generalize HH; clear HH H1. assert (F1: forall l a Ha Hl, In (mkznz (Z_of_nat a) Ha) (mkzlist l Hl) -> In a l); auto. intros l1; elim l1; simpl; auto; clear l1. intros a1 l1 Hrec1 a2 l2 Hl2 [H4 | H4]. generalize (znz_inj _ _ H4); simpl; clear H4; intros H4; left. rewrite <- (Zabs_nat_Z_of_nat a1); rewrite H4; rewrite Zabs_nat_Z_of_nat; auto. right; apply (Hrec1 _ _ _ H4). apply in_mkzlist. Qed.

Theorem

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

mkzlist_uniq

all_znz : list znz := (mkzlist (mklist (Z.abs_nat n)) (mklist_lt _)).

Definition

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

all_znz

all_znz_length : length all_znz = (Z.abs_nat n). Proof. unfold all_znz; rewrite mkzlist_length. rewrite mklist_length; auto. Qed.

Lemma

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

all_znz_length

uniq_all_znz : ulist all_znz. unfold all_znz; apply mkzlist_uniq. apply uniq_mklist. Qed.

Theorem

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

uniq_all_znz

in_all_znz : forall z, In z all_znz. intros (z1, Hz1). unfold all_znz; apply mkzlist_in. apply lt_mklist_lt. case (Z_mod_lt z1 n). auto with zarith. rewrite <- Hz1; intros H1 H2. case (Nat.le_gt_cases (Z.abs_nat n) (Z.abs_nat z1)); auto; intros H3. absurd (z1 < n); auto; apply Zle_not_lt. rewrite <- Z.abs_eq; auto. rewrite <- inj_Zabs_nat; auto. rewrite <- (Z.abs_eq n) by auto with zarith. rewrite <- (inj_Zabs_nat n); auto. apply inj_le; auto. Qed.

Theorem

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

in_all_znz

p_pos : 0 < p. generalize (prime_ge_2 _ p_prime); auto with zarith. Qed.

Theorem

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

p_pos

inv v := mkznz _ _ (modz p (fst (fst (Zegcd (val p v) p)))).

Definition

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

inv

div v1 v2 := mul _ v1 (inv v2).

Definition

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

div

FZpZ : field_theory (zero _) (one _) (add _) (mul _) (sub _) (opp _) div inv (@eq (znz p)). assert (Hmp := p_pos). split; auto. exact (RZnZ _ p_pos). intros H; injection H; repeat rewrite Zmod_small; auto with zarith. generalize (prime_ge_2 _ p_prime); auto with zarith. intros (n, Hn); unfold zero, one, inv, mul; simpl. intros H; apply zirr. generalize (Zegcd_is_egcd n p); case Zegcd; intros (u,v) w (Hu, (Hv, Hw)); simpl. assert (F1: rel_prime n p). apply rel_prime_le_prime; auto. rewrite Hn; auto. case (Z_mod_lt n p); try (intros H1 H2; split); auto with zarith. case (Zle_lt_or_eq _ _ H1); auto with zarith. rewrite <- Hn; intros H3; case H; apply zirr; rewrite <- H3; auto. red in F1. case (Zis_gcd_unique _ _ _ _ Hv F1); auto with zarith; intros; subst w. rewrite <- H0. rewrite Zmult_mod; repeat rewrite Zmod_mod; try rewrite <- Zmult_mod; auto. rewrite Z_mod_plus; auto with zarith. Defined.

Definition

src

[ "From Coq Require Import ZArith Znumtheory.", "From Coq Require Import Eqdep_dec.", "From Coq Require Import List.", "From Coq Require Import Lia.", "From Coqprime Require Import UList.", "From Coq Require Import Field.", "From Coqprime Require Import Pmod." ]

src/Coqprime/elliptic/GZnZ.v

FZpZ

ell_theory : Prop := mk_ell_theory { (* field properties *) Kfth : field_theory kO kI kplus kmul ksub kopp kdiv kinv (@eq K); NonSingular: 4 * A * A * A + 27 * B * B <> 0; (* Characteristic greater than 2 *) one_not_zero: 1 <> 0; two_not_zero: 2 <> 0; is_zero_correct: forall k, is_zero k = true <-> k = 0 }. Variable Eth: ell_theory.

Record

src