diff options
Diffstat (limited to 'plugins/setoid_ring/Field_theory.v')
| -rw-r--r-- | plugins/setoid_ring/Field_theory.v | 158 |
1 files changed, 157 insertions, 1 deletions
diff --git a/plugins/setoid_ring/Field_theory.v b/plugins/setoid_ring/Field_theory.v index 3622c7fe9..2b9dce1b0 100644 --- a/plugins/setoid_ring/Field_theory.v +++ b/plugins/setoid_ring/Field_theory.v @@ -10,6 +10,7 @@ Require Ring. Import Ring_polynom Ring_tac Ring_theory InitialRing Setoid List Morphisms. Require Import ZArith_base. Set Implicit Arguments. +(* Set Universe Polymorphism. *) Section MakeFieldPol. @@ -278,6 +279,21 @@ apply radd_ext. [ ring | now rewrite rdiv_simpl ]. Qed. +Theorem rdiv3 r1 r2 r3 r4 : + ~ r2 == 0 -> + ~ r4 == 0 -> + r1 / r2 - r3 / r4 == (r1 * r4 - r3 * r2) / (r2 * r4). +Proof. +intros H2 H4. +assert (~ r2 * r4 == 0) by (apply field_is_integral_domain; trivial). +transitivity (r1 / r2 + - (r3 / r4)); auto. +transitivity (r1 / r2 + - r3 / r4); auto. +transitivity ((r1 * r4 + - r3 * r2) / (r2 * r4)). +apply rdiv2; auto. +f_equiv. +transitivity (r1 * r4 + - (r3 * r2)); auto. +Qed. + Theorem rdiv5 a b : - (a / b) == - a / b. Proof. now rewrite !rdiv_def, ropp_mul_l. @@ -696,6 +712,7 @@ Fixpoint PEsimp (e : PExpr C) : PExpr C := | _ => e end%poly. +<<<<<<< .merge_file_5Z3Qpn Theorem PEsimp_ok e : (PEsimp e === e)%poly. Proof. induction e; simpl. @@ -708,6 +725,32 @@ induction e; simpl. - rewrite NPEmul_ok. now f_equiv. - rewrite NPEopp_ok. now f_equiv. - rewrite NPEpow_ok. now f_equiv. +======= +Theorem PExpr_simp_correct: + forall l e, NPEeval l (PExpr_simp e) == NPEeval l e. +clear eq_sym. +intros l e; elim e; simpl; auto. +intros e1 He1 e2 He2. +transitivity (NPEeval l (PEadd (PExpr_simp e1) (PExpr_simp e2))); auto. +apply NPEadd_correct. +simpl; auto. +intros e1 He1 e2 He2. +transitivity (NPEeval l (PEsub (PExpr_simp e1) (PExpr_simp e2))). auto. +apply NPEsub_correct. +simpl; auto. +intros e1 He1 e2 He2. +transitivity (NPEeval l (PEmul (PExpr_simp e1) (PExpr_simp e2))); auto. +apply NPEmul_correct. +simpl; auto. +intros e1 He1. +transitivity (NPEeval l (PEopp (PExpr_simp e1))); auto. +apply NPEopp_correct. +simpl; auto. +intros e1 He1 n;simpl. +rewrite NPEpow_correct;simpl. +repeat rewrite pow_th.(rpow_pow_N). +rewrite He1;auto. +>>>>>>> .merge_file_U4r9lJ Qed. @@ -961,6 +1004,7 @@ Fixpoint split_aux e1 p e2 {struct e1}: rsplit := end end%poly. +<<<<<<< .merge_file_5Z3Qpn Lemma split_aux_ok1 e1 p e2 : (let res := match isIn e1 p e2 1 with | Some (N0,e3) => mk_rsplit 1 (e1 ^^ Npos p) e3 @@ -971,6 +1015,20 @@ Lemma split_aux_ok1 e1 p e2 : e1 ^ Npos p === left res * common res /\ e2 === right res * common res)%poly. Proof. +======= +Lemma split_aux_correct_1 : forall l e1 p e2, + let res := match isIn e1 p e2 xH with + | Some (N0,e3) => mk_rsplit (PEc cI) (NPEpow e1 (Npos p)) e3 + | Some (Npos q, e3) => mk_rsplit (NPEpow e1 (Npos q)) (NPEpow e1 (Npos (p - q))) e3 + | None => mk_rsplit (NPEpow e1 (Npos p)) (PEc cI) e2 + end in + NPEeval l (PEpow e1 (Npos p)) == NPEeval l (NPEmul (left res) (common res)) + /\ + NPEeval l e2 == NPEeval l (NPEmul (right res) (common res)). +Proof. + intros. unfold res. clear res; generalize (isIn_correct l e1 p e2 xH). + destruct (isIn e1 p e2 1). destruct p0. +>>>>>>> .merge_file_U4r9lJ Opaque NPEpow NPEmul. intros. unfold res;clear res; generalize (isIn_ok e1 p e2 xH). destruct (isIn e1 p e2 1) as [([|p'],e')|]; simpl. @@ -1090,6 +1148,7 @@ Eval compute Theorem Pcond_Fnorm l e : PCond l (condition (Fnorm e)) -> ~ (denum (Fnorm e))@l == 0. Proof. +<<<<<<< .merge_file_5Z3Qpn induction e; simpl condition; rewrite ?PCond_cons, ?PCond_app; simpl denum; intros (Hc1,Hc2) || intros Hc; rewrite ?NPEmul_ok. - simpl. rewrite phi_1; exact rI_neq_rO. @@ -1112,6 +1171,93 @@ induction e; simpl condition; rewrite ?PCond_cons, ?PCond_app; + apply split_nz_r, Hc1. - rewrite NPEpow_ok. apply PEpow_nz, IHe, Hc. Qed. +======= + induction p;simpl. + intro Hp;assert (H1 := @rmul_reg_l _ (pow_pos rmul x p * pow_pos rmul x p) 0 H). + apply IHp. + rewrite (@rmul_reg_l _ (pow_pos rmul x p) 0 IHp). + reflexivity. + rewrite H1. ring. rewrite Hp;ring. + intro Hp;apply IHp. rewrite (@rmul_reg_l _ (pow_pos rmul x p) 0 IHp). + reflexivity. rewrite Hp;ring. trivial. +Qed. + +Theorem Pcond_Fnorm: + forall l e, + PCond l (condition (Fnorm e)) -> ~ NPEeval l ((Fnorm e).(denum)) == 0. +intros l e; elim e. + simpl; intros _ _; rewrite (morph1 CRmorph); exact rI_neq_rO. + simpl; intros _ _; rewrite (morph1 CRmorph); exact rI_neq_rO. + intros e1 Hrec1 e2 Hrec2 Hcond. + simpl in Hcond. + simpl @denum. + rewrite NPEmul_correct. + simpl. + apply field_is_integral_domain. + intros HH; case Hrec1; auto. + apply PCond_app_inv_l with (1 := Hcond). + rewrite (split_correct_l l (denum (Fnorm e1)) (denum (Fnorm e2))). + rewrite NPEmul_correct; simpl; rewrite HH; ring. + intros HH; case Hrec2; auto. + apply PCond_app_inv_r with (1 := Hcond). + rewrite (split_correct_r l (denum (Fnorm e1)) (denum (Fnorm e2))); auto. + intros e1 Hrec1 e2 Hrec2 Hcond. + simpl @condition in Hcond. + simpl @denum. + rewrite NPEmul_correct. + simpl. + apply field_is_integral_domain. + intros HH; case Hrec1; auto. + apply PCond_app_inv_l with (1 := Hcond). + rewrite (split_correct_l l (denum (Fnorm e1)) (denum (Fnorm e2))). + rewrite NPEmul_correct; simpl; rewrite HH; ring. + intros HH; case Hrec2; auto. + apply PCond_app_inv_r with (1 := Hcond). + rewrite (split_correct_r l (denum (Fnorm e1)) (denum (Fnorm e2))); auto. + intros e1 Hrec1 e2 Hrec2 Hcond. + simpl in Hcond. + simpl @denum. + rewrite NPEmul_correct. + simpl. + apply field_is_integral_domain. + intros HH; apply Hrec1. + apply PCond_app_inv_l with (1 := Hcond). + rewrite (split_correct_r l (num (Fnorm e2)) (denum (Fnorm e1))). + rewrite NPEmul_correct; simpl; rewrite HH; ring. + intros HH; apply Hrec2. + apply PCond_app_inv_r with (1 := Hcond). + rewrite (split_correct_r l (num (Fnorm e1)) (denum (Fnorm e2))). + rewrite NPEmul_correct; simpl; rewrite HH; ring. + intros e1 Hrec1 Hcond. + simpl in Hcond. + simpl @denum. + auto. + intros e1 Hrec1 Hcond. + simpl in Hcond. + simpl @denum. + apply PCond_cons_inv_l with (1:=Hcond). + intros e1 Hrec1 e2 Hrec2 Hcond. + simpl in Hcond. + simpl @denum. + rewrite NPEmul_correct. + simpl. + apply field_is_integral_domain. + intros HH; apply Hrec1. + specialize PCond_cons_inv_r with (1:=Hcond); intro Hcond1. + apply PCond_app_inv_l with (1 := Hcond1). + rewrite (split_correct_l l (denum (Fnorm e1)) (denum (Fnorm e2))). + rewrite NPEmul_correct; simpl; rewrite HH; ring. + intros HH; apply PCond_cons_inv_l with (1:=Hcond). + rewrite (split_correct_r l (num (Fnorm e1)) (num (Fnorm e2))). + rewrite NPEmul_correct; simpl; rewrite HH; ring. + simpl;intros e1 Hrec1 n Hcond. + rewrite NPEpow_correct. + simpl;rewrite pow_th.(rpow_pow_N). + destruct n;simpl;intros. + apply AFth.(AF_1_neq_0). apply pow_pos_not_0;auto. +Qed. +Hint Resolve Pcond_Fnorm. +>>>>>>> .merge_file_U4r9lJ (*************************************************************************** @@ -1502,11 +1648,21 @@ Hypothesis ceqb_complete : forall c1 c2, [c1] == [c2] -> ceqb c1 c2 = true. Lemma ceqb_spec' c1 c2 : Bool.reflect ([c1] == [c2]) (ceqb c1 c2). Proof. +<<<<<<< .merge_file_5Z3Qpn assert (H := morph_eq CRmorph c1 c2). assert (H' := @ceqb_complete c1 c2). destruct (ceqb c1 c2); constructor. - now apply H. - intro E. specialize (H' E). discriminate. +======= +intros. +generalize (fun h => X (morph_eq CRmorph _ _ h)). +generalize (@ceqb_complete c1 c2). +case (c1 ?=! c2); auto; intros. +apply X0. +red; intro. +absurd (false = true); auto; discriminate. +>>>>>>> .merge_file_U4r9lJ Qed. Fixpoint Fcons1 (e:PExpr C) (l:list (PExpr C)) {struct e} : list (PExpr C) := @@ -1766,4 +1922,4 @@ End Field. End Complete. Arguments FEO [C]. -Arguments FEI [C].
\ No newline at end of file +Arguments FEI [C]. |