diff options
Diffstat (limited to 'plugins/setoid_ring/Field_theory.v')
| -rw-r--r-- | plugins/setoid_ring/Field_theory.v | 523 |
1 files changed, 234 insertions, 289 deletions
diff --git a/plugins/setoid_ring/Field_theory.v b/plugins/setoid_ring/Field_theory.v index ccdec656..bc05c252 100644 --- a/plugins/setoid_ring/Field_theory.v +++ b/plugins/setoid_ring/Field_theory.v @@ -1,13 +1,13 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) Require Ring. -Import Ring_polynom Ring_tac Ring_theory InitialRing Setoid List. +Import Ring_polynom Ring_tac Ring_theory InitialRing Setoid List Morphisms. Require Import ZArith_base. (*Require Import Omega.*) Set Implicit Arguments. @@ -27,7 +27,7 @@ Section MakeFieldPol. Notation "x == y" := (req x y) (at level 70, no associativity). (* Equality properties *) - Variable Rsth : Setoid_Theory R req. + Variable Rsth : Equivalence req. Variable Reqe : ring_eq_ext radd rmul ropp req. Variable SRinv_ext : forall p q, p == q -> / p == / q. @@ -75,7 +75,6 @@ Qed. (* Useful tactics *) - Add Setoid R req Rsth as R_set1. Add Morphism radd : radd_ext. exact (Radd_ext Reqe). Qed. Add Morphism rmul : rmul_ext. exact (Rmul_ext Reqe). Qed. Add Morphism ropp : ropp_ext. exact (Ropp_ext Reqe). Qed. @@ -96,7 +95,7 @@ Hint Resolve (ARadd_0_l ARth) (ARadd_comm ARth) (ARadd_assoc ARth) (ARsub_def ARth) . (* Power coefficients *) - Variable Cpow : Set. + Variable Cpow : Type. Variable Cp_phi : N -> Cpow. Variable rpow : R -> Cpow -> R. Variable pow_th : power_theory rI rmul req Cp_phi rpow. @@ -116,16 +115,17 @@ Notation NPphi_pow := (Pphi_pow rO rI radd rmul rsub ropp cO cI ceqb phi Cp_phi (* add abstract semi-ring to help with some proofs *) Add Ring Rring : (ARth_SRth ARth). +Local Hint Extern 2 (_ == _) => f_equiv. (* additional ring properties *) Lemma rsub_0_l : forall r, 0 - r == - r. -intros; rewrite (ARsub_def ARth) in |- *;ring. +intros; rewrite (ARsub_def ARth);ring. Qed. Lemma rsub_0_r : forall r, r - 0 == r. -intros; rewrite (ARsub_def ARth) in |- *. -rewrite (ARopp_zero Rsth Reqe ARth) in |- *; ring. +intros; rewrite (ARsub_def ARth). +rewrite (ARopp_zero Rsth Reqe ARth); ring. Qed. (*************************************************************************** @@ -135,42 +135,40 @@ Qed. ***************************************************************************) Theorem rdiv_simpl: forall p q, ~ q == 0 -> q * (p / q) == p. +Proof. intros p q H. -rewrite rdiv_def in |- *. +rewrite rdiv_def. transitivity (/ q * q * p); [ ring | idtac ]. -rewrite rinv_l in |- *; auto. +rewrite rinv_l; auto. Qed. Hint Resolve rdiv_simpl . -Theorem SRdiv_ext: - forall p1 p2, p1 == p2 -> forall q1 q2, q1 == q2 -> p1 / q1 == p2 / q2. -intros p1 p2 H q1 q2 H0. +Instance SRdiv_ext: Proper (req ==> req ==> req) rdiv. +Proof. +intros p1 p2 Ep q1 q2 Eq. transitivity (p1 * / q1); auto. transitivity (p2 * / q2); auto. Qed. -Hint Resolve SRdiv_ext . - - Add Morphism rdiv : rdiv_ext. exact SRdiv_ext. Qed. +Hint Resolve SRdiv_ext. Lemma rmul_reg_l : forall p q1 q2, ~ p == 0 -> p * q1 == p * q2 -> q1 == q2. -intros. -rewrite <- (@rdiv_simpl q1 p) in |- *; trivial. -rewrite <- (@rdiv_simpl q2 p) in |- *; trivial. -repeat rewrite rdiv_def in |- *. -repeat rewrite (ARmul_assoc ARth) in |- *. -auto. +Proof. +intros p q1 q2 H EQ. +rewrite <- (@rdiv_simpl q1 p) by trivial. +rewrite <- (@rdiv_simpl q2 p) by trivial. +rewrite !rdiv_def, !(ARmul_assoc ARth). +now rewrite EQ. Qed. Theorem field_is_integral_domain : forall r1 r2, ~ r1 == 0 -> ~ r2 == 0 -> ~ r1 * r2 == 0. Proof. -red in |- *; intros. -apply H0. +intros r1 r2 H1 H2. contradict H2. transitivity (1 * r2); auto. transitivity (/ r1 * r1 * r2); auto. -rewrite <- (ARmul_assoc ARth) in |- *. -rewrite H1 in |- *. +rewrite <- (ARmul_assoc ARth). +rewrite H2. apply ARmul_0_r with (1 := Rsth) (2 := ARth). Qed. @@ -179,15 +177,15 @@ Theorem ropp_neq_0 : forall r, intros. setoid_replace (- r) with (- (1) * r). apply field_is_integral_domain; trivial. - rewrite <- (ARopp_mul_l ARth) in |- *. - rewrite (ARmul_1_l ARth) in |- *. + rewrite <- (ARopp_mul_l ARth). + rewrite (ARmul_1_l ARth). reflexivity. Qed. Theorem rdiv_r_r : forall r, ~ r == 0 -> r / r == 1. intros. -rewrite (AFdiv_def AFth) in |- *. -rewrite (ARmul_comm ARth) in |- *. +rewrite (AFdiv_def AFth). +rewrite (ARmul_comm ARth). apply (AFinv_l AFth). trivial. Qed. @@ -203,14 +201,14 @@ Theorem rdiv2: r1 / r2 + r3 / r4 == (r1 * r4 + r3 * r2) / (r2 * r4). Proof. intros r1 r2 r3 r4 H H0. -assert (~ r2 * r4 == 0) by complete (apply field_is_integral_domain; trivial). +assert (~ r2 * r4 == 0) by (apply field_is_integral_domain; trivial). apply rmul_reg_l with (r2 * r4); trivial. -rewrite rdiv_simpl in |- *; trivial. -rewrite (ARdistr_r Rsth Reqe ARth) in |- *. +rewrite rdiv_simpl; trivial. +rewrite (ARdistr_r Rsth Reqe ARth). apply (Radd_ext Reqe). - transitivity (r2 * (r1 / r2) * r4); [ ring | auto ]. - transitivity (r2 * (r4 * (r3 / r4))); auto. - transitivity (r2 * r3); auto. +- transitivity (r2 * (r1 / r2) * r4); [ ring | auto ]. +- transitivity (r2 * (r4 * (r3 / r4))); auto. + transitivity (r2 * r3); auto. Qed. @@ -225,35 +223,36 @@ assert (HH1: ~ r2 == 0) by (intros HH; case H; rewrite HH; ring). assert (HH2: ~ r5 == 0) by (intros HH; case H; rewrite HH; ring). assert (HH3: ~ r4 == 0) by (intros HH; case H0; rewrite HH; ring). assert (HH4: ~ r2 * (r4 * r5) == 0) - by complete (repeat apply field_is_integral_domain; trivial). + by (repeat apply field_is_integral_domain; trivial). apply rmul_reg_l with (r2 * (r4 * r5)); trivial. -rewrite rdiv_simpl in |- *; trivial. -rewrite (ARdistr_r Rsth Reqe ARth) in |- *. +rewrite rdiv_simpl; trivial. +rewrite (ARdistr_r Rsth Reqe ARth). apply (Radd_ext Reqe). transitivity ((r2 * r5) * (r1 / (r2 * r5)) * r4); [ ring | auto ]. transitivity ((r4 * r5) * (r3 / (r4 * r5)) * r2); [ ring | auto ]. Qed. Theorem rdiv5: forall r1 r2, - (r1 / r2) == - r1 / r2. +Proof. intros r1 r2. transitivity (- (r1 * / r2)); auto. transitivity (- r1 * / r2); auto. Qed. Hint Resolve rdiv5 . -Theorem rdiv3: - forall r1 r2 r3 r4, +Theorem rdiv3 r1 r2 r3 r4 : ~ r2 == 0 -> ~ r4 == 0 -> r1 / r2 - r3 / r4 == (r1 * r4 - r3 * r2) / (r2 * r4). -intros r1 r2 r3 r4 H H0. +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)); auto. +transitivity ((r1 * r4 + - r3 * r2) / (r2 * r4)). apply rdiv2; auto. -apply SRdiv_ext; auto. -transitivity (r1 * r4 + - (r3 * r2)); symmetry; auto. +f_equiv. +transitivity (r1 * r4 + - (r3 * r2)); auto. Qed. @@ -279,13 +278,13 @@ intros r1 r2 H H0. assert (~ r1 / r2 == 0) as Hk. intros H1; case H. transitivity (r2 * (r1 / r2)); auto. - rewrite H1 in |- *; ring. + rewrite H1; ring. apply rmul_reg_l with (r1 / r2); auto. transitivity (/ (r1 / r2) * (r1 / r2)); auto. transitivity 1; auto. - repeat rewrite rdiv_def in |- *. + repeat rewrite rdiv_def. transitivity (/ r1 * r1 * (/ r2 * r2)); [ idtac | ring ]. - repeat rewrite rinv_l in |- *; auto. + repeat rewrite rinv_l; auto. Qed. Hint Resolve rdiv6 . @@ -296,11 +295,11 @@ Hint Resolve rdiv6 . (r1 / r2) * (r3 / r4) == (r1 * r3) / (r2 * r4). Proof. intros r1 r2 r3 r4 H H0. -assert (~ r2 * r4 == 0) by complete (apply field_is_integral_domain; trivial). +assert (~ r2 * r4 == 0) by (apply field_is_integral_domain; trivial). apply rmul_reg_l with (r2 * r4); trivial. -rewrite rdiv_simpl in |- *; trivial. +rewrite rdiv_simpl; trivial. transitivity (r2 * (r1 / r2) * (r4 * (r3 / r4))); [ ring | idtac ]. -repeat rewrite rdiv_simpl in |- *; trivial. +repeat rewrite rdiv_simpl; trivial. Qed. Theorem rdiv4b: @@ -334,8 +333,8 @@ Theorem rdiv7: (r1 / r2) / (r3 / r4) == (r1 * r4) / (r2 * r3). Proof. intros. -rewrite (rdiv_def (r1 / r2)) in |- *. -rewrite rdiv6 in |- *; trivial. +rewrite (rdiv_def (r1 / r2)). +rewrite rdiv6; trivial. apply rdiv4; trivial. Qed. @@ -373,14 +372,14 @@ Theorem cross_product_eq : forall r1 r2 r3 r4, ~ r2 == 0 -> ~ r4 == 0 -> r1 * r4 == r3 * r2 -> r1 / r2 == r3 / r4. intros. transitivity (r1 / r2 * (r4 / r4)). - rewrite rdiv_r_r in |- *; trivial. - symmetry in |- *. + rewrite rdiv_r_r; trivial. + symmetry . apply (ARmul_1_r Rsth ARth). - rewrite rdiv4 in |- *; trivial. - rewrite H1 in |- *. - rewrite (ARmul_comm ARth r2 r4) in |- *. - rewrite <- rdiv4 in |- *; trivial. - rewrite rdiv_r_r in |- * by trivial. + rewrite rdiv4; trivial. + rewrite H1. + rewrite (ARmul_comm ARth r2 r4). + rewrite <- rdiv4; trivial. + rewrite rdiv_r_r by trivial. apply (ARmul_1_r Rsth ARth). Qed. @@ -390,52 +389,16 @@ Qed. ***************************************************************************) -Fixpoint positive_eq (p1 p2 : positive) {struct p1} : bool := - match p1, p2 with - xH, xH => true - | xO p3, xO p4 => positive_eq p3 p4 - | xI p3, xI p4 => positive_eq p3 p4 - | _, _ => false - end. - -Theorem positive_eq_correct: - forall p1 p2, if positive_eq p1 p2 then p1 = p2 else p1 <> p2. -intros p1; elim p1; - (try (intros p2; case p2; simpl; auto; intros; discriminate)). -intros p3 rec p2; case p2; simpl; auto; (try (intros; discriminate)); intros p4. -generalize (rec p4); case (positive_eq p3 p4); auto. -intros H1; apply f_equal with ( f := xI ); auto. -intros H1 H2; case H1; injection H2; auto. -intros p3 rec p2; case p2; simpl; auto; (try (intros; discriminate)); intros p4. -generalize (rec p4); case (positive_eq p3 p4); auto. -intros H1; apply f_equal with ( f := xO ); auto. -intros H1 H2; case H1; injection H2; auto. -Qed. - -Definition N_eq n1 n2 := - match n1, n2 with - | N0, N0 => true - | Npos p1, Npos p2 => positive_eq p1 p2 - | _, _ => false - end. - -Lemma N_eq_correct : forall n1 n2, if N_eq n1 n2 then n1 = n2 else n1 <> n2. -Proof. - intros [ |p1] [ |p2];simpl;trivial;try(intro H;discriminate H;fail). - assert (H:=positive_eq_correct p1 p2);destruct (positive_eq p1 p2); - [rewrite H;trivial | intro H1;injection H1;subst;apply H;trivial]. -Qed. - (* equality test *) Fixpoint PExpr_eq (e1 e2 : PExpr C) {struct e1} : bool := match e1, e2 with PEc c1, PEc c2 => ceqb c1 c2 - | PEX p1, PEX p2 => positive_eq p1 p2 + | PEX p1, PEX p2 => Pos.eqb p1 p2 | PEadd e3 e5, PEadd e4 e6 => if PExpr_eq e3 e4 then PExpr_eq e5 e6 else false | PEsub e3 e5, PEsub e4 e6 => if PExpr_eq e3 e4 then PExpr_eq e5 e6 else false | PEmul e3 e5, PEmul e4 e6 => if PExpr_eq e3 e4 then PExpr_eq e5 e6 else false | PEopp e3, PEopp e4 => PExpr_eq e3 e4 - | PEpow e3 n3, PEpow e4 n4 => if N_eq n3 n4 then PExpr_eq e3 e4 else false + | PEpow e3 n3, PEpow e4 n4 => if N.eqb n3 n4 then PExpr_eq e3 e4 else false | _, _ => false end. @@ -446,22 +409,14 @@ Qed. Add Morphism (pow_N rI rmul) with signature req ==> eq ==> req as pow_N_morph. intros x y H [|p];simpl;auto. apply pow_morph;trivial. Qed. -(* -Lemma rpow_morph : forall x y n, x == y ->rpow x (Cp_phi n) == rpow y (Cp_phi n). -Proof. - intros; repeat rewrite pow_th.(rpow_pow_N). - destruct n;simpl. apply eq_refl. - induction p;simpl;try rewrite IHp;try rewrite H; apply eq_refl. -Qed. -*) + Theorem PExpr_eq_semi_correct: forall l e1 e2, PExpr_eq e1 e2 = true -> NPEeval l e1 == NPEeval l e2. intros l e1; elim e1. intros c1; intros e2; elim e2; simpl; (try (intros; discriminate)). intros c2; apply (morph_eq CRmorph). intros p1; intros e2; elim e2; simpl; (try (intros; discriminate)). -intros p2; generalize (positive_eq_correct p1 p2); case (positive_eq p1 p2); - (try (intros; discriminate)); intros H; rewrite H; auto. +intros p2; case Pos.eqb_spec; intros; now subst. intros e3 rec1 e5 rec2 e2; case e2; simpl; (try (intros; discriminate)). intros e4 e6; generalize (rec1 e4); case (PExpr_eq e3 e4); (try (intros; discriminate)); generalize (rec2 e6); case (PExpr_eq e5 e6); @@ -478,9 +433,8 @@ intros e3 rec e2; (case e2; simpl; (try (intros; discriminate))). intros e4; generalize (rec e4); case (PExpr_eq e3 e4); (try (intros; discriminate)); auto. intros e3 rec n3 e2;(case e2;simpl;(try (intros;discriminate))). -intros e4 n4;generalize (N_eq_correct n3 n4);destruct (N_eq n3 n4); -intros;try discriminate. -repeat rewrite pow_th.(rpow_pow_N);rewrite H;rewrite (rec _ H0);auto. +intros e4 n4; case N.eqb_spec; try discriminate; intros EQ H; subst. +repeat rewrite pow_th.(rpow_pow_N). rewrite (rec _ H);auto. Qed. (* add *) @@ -497,8 +451,8 @@ Theorem NPEadd_correct: forall l e1 e2, NPEeval l (NPEadd e1 e2) == NPEeval l (PEadd e1 e2). Proof. intros l e1 e2. -destruct e1; destruct e2; simpl in |- *; try reflexivity; try apply ceqb_rect; - try (intro eq_c; rewrite eq_c in |- *); simpl in |- *;try apply eq_refl; +destruct e1; destruct e2; simpl; try reflexivity; try apply ceqb_rect; + try (intro eq_c; rewrite eq_c); simpl;try apply eq_refl; try (ring [(morph0 CRmorph)]). apply (morph_add CRmorph). Qed. @@ -507,7 +461,7 @@ Definition NPEpow x n := match n with | N0 => PEc cI | Npos p => - if positive_eq p xH then x else + if Pos.eqb p xH then x else match x with | PEc c => if ceqb c cI then PEc cI else if ceqb c cO then PEc cO else PEc (pow_pos cmul c p) @@ -520,10 +474,10 @@ Theorem NPEpow_correct : forall l e n, Proof. destruct n;simpl. rewrite pow_th.(rpow_pow_N);simpl;auto. - generalize (positive_eq_correct p xH). - destruct (positive_eq p 1);intros. - rewrite H;rewrite pow_th.(rpow_pow_N). trivial. - clear H;destruct e;simpl;auto. + fold (p =? 1)%positive. + case Pos.eqb_spec; intros H; (rewrite H || clear H). + now rewrite pow_th.(rpow_pow_N). + destruct e;simpl;auto. repeat apply ceqb_rect;simpl;intros;rewrite pow_th.(rpow_pow_N);simpl. symmetry;induction p;simpl;trivial; ring [IHp H CRmorph.(morph1)]. symmetry; induction p;simpl;trivial;ring [IHp CRmorph.(morph0)]. @@ -539,7 +493,7 @@ Fixpoint NPEmul (x y : PExpr C) {struct x} : PExpr C := | _, PEc c => if ceqb c cI then x else if ceqb c cO then PEc cO else PEmul x y | PEpow e1 n1, PEpow e2 n2 => - if N_eq n1 n2 then NPEpow (NPEmul e1 e2) n1 else PEmul x y + if N.eqb n1 n2 then NPEpow (NPEmul e1 e2) n1 else PEmul x y | _, _ => PEmul x y end. @@ -549,15 +503,15 @@ Qed. Theorem NPEmul_correct : forall l e1 e2, NPEeval l (NPEmul e1 e2) == NPEeval l (PEmul e1 e2). -induction e1;destruct e2; simpl in |- *;try reflexivity; +induction e1;destruct e2; simpl;try reflexivity; repeat apply ceqb_rect; - try (intro eq_c; rewrite eq_c in |- *); simpl in |- *; try reflexivity; + try (intro eq_c; rewrite eq_c); simpl; try reflexivity; try ring [(morph0 CRmorph) (morph1 CRmorph)]. apply (morph_mul CRmorph). -assert (H:=N_eq_correct n n0);destruct (N_eq n n0). +case N.eqb_spec; intros H; try rewrite <- H; clear H. rewrite NPEpow_correct. simpl. repeat rewrite pow_th.(rpow_pow_N). -rewrite IHe1;rewrite <- H;destruct n;simpl;try ring. +rewrite IHe1; destruct n;simpl;try ring. apply pow_pos_mul. simpl;auto. Qed. @@ -575,9 +529,9 @@ Definition NPEsub e1 e2 := Theorem NPEsub_correct: forall l e1 e2, NPEeval l (NPEsub e1 e2) == NPEeval l (PEsub e1 e2). intros l e1 e2. -destruct e1; destruct e2; simpl in |- *; try reflexivity; try apply ceqb_rect; - try (intro eq_c; rewrite eq_c in |- *); simpl in |- *; - try rewrite (morph0 CRmorph) in |- *; try reflexivity; +destruct e1; destruct e2; simpl; try reflexivity; try apply ceqb_rect; + try (intro eq_c; rewrite eq_c); simpl; + try rewrite (morph0 CRmorph); try reflexivity; try (symmetry; apply rsub_0_l); try (symmetry; apply rsub_0_r). apply (morph_sub CRmorph). Qed. @@ -697,8 +651,8 @@ destruct H; trivial. Qed. Theorem PCond_app_inv_l: forall l l1 l2, PCond l (l1 ++ l2) -> PCond l l1. -intros l l1 l2; elim l1; simpl app in |- *. - simpl in |- *; auto. +intros l l1 l2; elim l1; simpl app. + simpl; auto. destruct l0; simpl in *. destruct l2; firstorder. firstorder. @@ -713,8 +667,8 @@ Qed. Definition absurd_PCond := cons (PEc cO) nil. Lemma absurd_PCond_bottom : forall l, ~ PCond l absurd_PCond. -unfold absurd_PCond in |- *; simpl in |- *. -red in |- *; intros. +unfold absurd_PCond; simpl. +red; intros. apply H. apply (morph0 CRmorph). Qed. @@ -743,10 +697,10 @@ Fixpoint isIn (e1:PExpr C) (p1:positive) end end | PEpow e3 N0 => None - | PEpow e3 (Npos p3) => isIn e1 p1 e3 (Pmult p3 p2) + | PEpow e3 (Npos p3) => isIn e1 p1 e3 (Pos.mul p3 p2) | _ => if PExpr_eq e1 e2 then - match Zminus (Zpos p1) (Zpos p2) with + match Z.pos_sub p1 p2 with | Zpos p => Some (Npos p, PEc cI) | Z0 => Some (N0, PEc cI) | Zneg p => Some (N0, NPEpow e2 (Npos p)) @@ -757,13 +711,19 @@ Fixpoint isIn (e1:PExpr C) (p1:positive) Definition ZtoN z := match z with Zpos p => Npos p | _ => N0 end. Definition NtoZ n := match n with Npos p => Zpos p | _ => Z0 end. - Notation pow_pos_plus := (Ring_theory.pow_pos_Pplus _ Rsth Reqe.(Rmul_ext) - ARth.(ARmul_comm) ARth.(ARmul_assoc)). + Notation pow_pos_add := + (Ring_theory.pow_pos_add Rsth Reqe.(Rmul_ext) ARth.(ARmul_assoc)). + + Lemma Z_pos_sub_gt p q : (p > q)%positive -> + Z.pos_sub p q = Zpos (p - q). + Proof. intros; now apply Z.pos_sub_gt, Pos.gt_lt. Qed. + + Ltac simpl_pos_sub := rewrite ?Z_pos_sub_gt in * by assumption. Lemma isIn_correct_aux : forall l e1 e2 p1 p2, match (if PExpr_eq e1 e2 then - match Zminus (Zpos p1) (Zpos p2) with + match Z.sub (Zpos p1) (Zpos p2) with | Zpos p => Some (Npos p, PEc cI) | Z0 => Some (N0, PEc cI) | Zneg p => Some (N0, NPEpow e2 (Npos p)) @@ -779,37 +739,29 @@ Fixpoint isIn (e1:PExpr C) (p1:positive) Proof. intros l e1 e2 p1 p2; generalize (PExpr_eq_semi_correct l e1 e2); case (PExpr_eq e1 e2); simpl; auto; intros H. - case_eq ((p1 ?= p2)%positive Eq);intros;simpl. - repeat rewrite pow_th.(rpow_pow_N);simpl. split. 2:refine (refl_equal _). - rewrite (Pcompare_Eq_eq _ _ H0). - rewrite H by trivial. ring [ (morph1 CRmorph)]. - fold (NPEpow e2 (Npos (p2 - p1))). - rewrite NPEpow_correct;simpl. - repeat rewrite pow_th.(rpow_pow_N);simpl. - rewrite H;trivial. split. 2:refine (refl_equal _). - rewrite <- pow_pos_plus; rewrite Pplus_minus;auto. apply ZC2;trivial. - repeat rewrite pow_th.(rpow_pow_N);simpl. - rewrite H;trivial. - change (ZtoN - match (p1 ?= p1 - p2)%positive Eq with - | Eq => 0 - | Lt => Zneg (p1 - p2 - p1) - | Gt => Zpos (p1 - (p1 - p2)) - end) with (ZtoN (Zpos p1 - Zpos (p1 -p2))). - replace (Zpos (p1 - p2)) with (Zpos p1 - Zpos p2)%Z. - split. - repeat rewrite Zth.(Rsub_def). rewrite (Ring_theory.Ropp_add Zsth Zeqe Zth). - rewrite Zplus_assoc. simpl. rewrite Pcompare_refl. simpl. - ring [ (morph1 CRmorph)]. - assert (Zpos p1 > 0 /\ Zpos p2 > 0)%Z. split;refine (refl_equal _). - apply Zplus_gt_reg_l with (Zpos p2). - rewrite Zplus_minus. change (Zpos p2 + Zpos p1 > 0 + Zpos p1)%Z. - apply Zplus_gt_compat_r. refine (refl_equal _). - simpl;rewrite H0;trivial. + rewrite Z.pos_sub_spec. + case Pos.compare_spec;intros;simpl. + - repeat rewrite pow_th.(rpow_pow_N);simpl. split. 2:reflexivity. + subst. rewrite H by trivial. ring [ (morph1 CRmorph)]. + - fold (p2 - p1 =? 1)%positive. + fold (NPEpow e2 (Npos (p2 - p1))). + rewrite NPEpow_correct;simpl. + repeat rewrite pow_th.(rpow_pow_N);simpl. + rewrite H;trivial. split. 2:reflexivity. + rewrite <- pow_pos_add. now rewrite Pos.add_comm, Pos.sub_add. + - repeat rewrite pow_th.(rpow_pow_N);simpl. + rewrite H;trivial. + rewrite Z.pos_sub_gt by now apply Pos.sub_decr. + replace (p1 - (p1 - p2))%positive with p2; + [| rewrite Pos.sub_sub_distr, Pos.add_comm; + auto using Pos.add_sub, Pos.sub_decr ]. + split. + simpl. ring [ (morph1 CRmorph)]. + now apply Z.lt_gt, Pos.sub_decr. Qed. Lemma pow_pos_pow_pos : forall x p1 p2, pow_pos rmul (pow_pos rmul x p1) p2 == pow_pos rmul x (p1*p2). -induction p1;simpl;intros;repeat rewrite pow_pos_mul;repeat rewrite pow_pos_plus;simpl. +induction p1;simpl;intros;repeat rewrite pow_pos_mul;repeat rewrite pow_pos_add;simpl. ring [(IHp1 p2)]. ring [(IHp1 p2)]. auto. Qed. @@ -835,39 +787,39 @@ destruct n. destruct n;simpl. rewrite NPEmul_correct;repeat rewrite pow_th.(rpow_pow_N);simpl. intros (H1,H2) (H3,H4). - unfold Zgt in H2, H4;simpl in H2,H4. rewrite H4 in H3;simpl in H3. + simpl_pos_sub. simpl in H3. rewrite pow_pos_mul. rewrite H1;rewrite H3. assert (pow_pos rmul (NPEeval l e1) (p1 - p4) * NPEeval l p3 * (pow_pos rmul (NPEeval l e1) p4 * NPEeval l p5) == pow_pos rmul (NPEeval l e1) p4 * pow_pos rmul (NPEeval l e1) (p1 - p4) * NPEeval l p3 *NPEeval l p5) by ring. rewrite H;clear H. - rewrite <- pow_pos_plus. rewrite Pplus_minus. - split. symmetry;apply ARth.(ARmul_assoc). refine (refl_equal _). trivial. + rewrite <- pow_pos_add. + rewrite Pos.add_comm, Pos.sub_add by (now apply Z.gt_lt in H4). + split. symmetry;apply ARth.(ARmul_assoc). reflexivity. repeat rewrite pow_th.(rpow_pow_N);simpl. intros (H1,H2) (H3,H4). - unfold Zgt in H2, H4;simpl in H2,H4. rewrite H4 in H3;simpl in H3. - rewrite H2 in H1;simpl in H1. + simpl_pos_sub. simpl in H1, H3. assert (Zpos p1 > Zpos p6)%Z. apply Zgt_trans with (Zpos p4). exact H4. exact H2. - unfold Zgt in H;simpl in H;rewrite H. + simpl_pos_sub. split. 2:exact H. rewrite pow_pos_mul. simpl;rewrite H1;rewrite H3. assert (pow_pos rmul (NPEeval l e1) (p1 - p4) * NPEeval l p3 * (pow_pos rmul (NPEeval l e1) (p4 - p6) * NPEeval l p5) == pow_pos rmul (NPEeval l e1) (p1 - p4) * pow_pos rmul (NPEeval l e1) (p4 - p6) * NPEeval l p3 * NPEeval l p5) by ring. rewrite H0;clear H0. - rewrite <- pow_pos_plus. + rewrite <- pow_pos_add. replace (p1 - p4 + (p4 - p6))%positive with (p1 - p6)%positive. rewrite NPEmul_correct. simpl;ring. assert (Zpos p1 - Zpos p6 = Zpos p1 - Zpos p4 + (Zpos p4 - Zpos p6))%Z. change ((Zpos p1 - Zpos p6)%Z = (Zpos p1 + (- Zpos p4) + (Zpos p4 +(- Zpos p6)))%Z). - rewrite <- Zplus_assoc. rewrite (Zplus_assoc (- Zpos p4)). - simpl. rewrite Pcompare_refl. simpl. reflexivity. - unfold Zminus, Zopp in H0. simpl in H0. - rewrite H2 in H0;rewrite H4 in H0;rewrite H in H0. inversion H0;trivial. + rewrite <- Z.add_assoc. rewrite (Z.add_assoc (- Zpos p4)). + simpl. rewrite Z.pos_sub_diag. simpl. reflexivity. + unfold Z.sub, Z.opp in H0. simpl in H0. + simpl_pos_sub. inversion H0; trivial. simpl. repeat rewrite pow_th.(rpow_pow_N). - intros H1 (H2,H3). unfold Zgt in H3;simpl in H3. rewrite H3 in H2;rewrite H3. + intros H1 (H2,H3). simpl_pos_sub. rewrite NPEmul_correct;simpl;rewrite NPEpow_correct;simpl. simpl in H2. rewrite pow_th.(rpow_pow_N);simpl. rewrite pow_pos_mul. split. ring [H2]. exact H3. @@ -878,8 +830,7 @@ destruct n. rewrite NPEmul_correct;simpl;rewrite NPEpow_correct;simpl. repeat rewrite pow_th.(rpow_pow_N);simpl. rewrite pow_pos_mul. intros (H1, H2);rewrite H1;split. - unfold Zgt in H2;simpl in H2;rewrite H2;rewrite H2 in H1. - simpl in H1;ring [H1]. trivial. + simpl_pos_sub. simpl in H1;ring [H1]. trivial. trivial. destruct n. trivial. generalize (H p1 (p0*p2)%positive);clear H;destruct (isIn e1 p1 p (p0*p2)). destruct p3. @@ -910,7 +861,7 @@ Fixpoint split_aux (e1: PExpr C) (p:positive) (e2:PExpr C) {struct e1}: rsplit : (NPEmul (common r1) (common r2)) (right r2) | PEpow e3 N0 => mk_rsplit (PEc cI) (PEc cI) e2 - | PEpow e3 (Npos p3) => split_aux e3 (Pmult p3 p) e2 + | PEpow e3 (Npos p3) => split_aux e3 (Pos.mul p3 p) e2 | _ => match isIn e1 p e2 xH with | Some (N0,e3) => mk_rsplit (PEc cI) (NPEpow e1 (Npos p)) e3 @@ -937,9 +888,9 @@ Proof. repeat rewrite NPEpow_correct;simpl; repeat rewrite pow_th.(rpow_pow_N);simpl). intros (H, Hgt);split;try ring [H CRmorph.(morph1)]. - intros (H, Hgt). unfold Zgt in Hgt;simpl in Hgt;rewrite Hgt in H. - simpl in H;split;try ring [H]. - rewrite <- pow_pos_plus. rewrite Pplus_minus. reflexivity. trivial. + intros (H, Hgt). simpl_pos_sub. simpl in H;split;try ring [H]. + apply Z.gt_lt in Hgt. + now rewrite <- pow_pos_add, Pos.add_comm, Pos.sub_add. simpl;intros. repeat rewrite NPEmul_correct;simpl. rewrite NPEpow_correct;simpl. split;ring [CRmorph.(morph1)]. Qed. @@ -1061,13 +1012,13 @@ Theorem Pcond_Fnorm: forall l e, PCond l (condition (Fnorm e)) -> ~ NPEeval l (denum (Fnorm e)) == 0. intros l e; elim e. - simpl in |- *; intros _ _; rewrite (morph1 CRmorph) in |- *; exact rI_neq_rO. - simpl in |- *; intros _ _; rewrite (morph1 CRmorph) in |- *; exact rI_neq_rO. + 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 condition in Hcond. - simpl denum in |- *. - rewrite NPEmul_correct in |- *. - simpl in |- *. + simpl denum. + rewrite NPEmul_correct. + simpl. apply field_is_integral_domain. intros HH; case Hrec1; auto. apply PCond_app_inv_l with (1 := Hcond). @@ -1078,9 +1029,9 @@ intros l e; elim e. rewrite (split_correct_r l (denum (Fnorm e1)) (denum (Fnorm e2))); auto. intros e1 Hrec1 e2 Hrec2 Hcond. simpl condition in Hcond. - simpl denum in |- *. - rewrite NPEmul_correct in |- *. - simpl in |- *. + simpl denum. + rewrite NPEmul_correct. + simpl. apply field_is_integral_domain. intros HH; case Hrec1; auto. apply PCond_app_inv_l with (1 := Hcond). @@ -1091,9 +1042,9 @@ intros l e; elim e. rewrite (split_correct_r l (denum (Fnorm e1)) (denum (Fnorm e2))); auto. intros e1 Hrec1 e2 Hrec2 Hcond. simpl condition in Hcond. - simpl denum in |- *. - rewrite NPEmul_correct in |- *. - simpl in |- *. + simpl denum. + rewrite NPEmul_correct. + simpl. apply field_is_integral_domain. intros HH; apply Hrec1. apply PCond_app_inv_l with (1 := Hcond). @@ -1105,17 +1056,17 @@ intros l e; elim e. rewrite NPEmul_correct; simpl; rewrite HH; ring. intros e1 Hrec1 Hcond. simpl condition in Hcond. - simpl denum in |- *. + simpl denum. auto. intros e1 Hrec1 Hcond. simpl condition in Hcond. - simpl denum in |- *. + simpl denum. apply PCond_cons_inv_l with (1:=Hcond). intros e1 Hrec1 e2 Hrec2 Hcond. simpl condition in Hcond. - simpl denum in |- *. - rewrite NPEmul_correct in |- *. - simpl in |- *. + 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. @@ -1258,9 +1209,9 @@ Theorem Fnorm_crossproduct: PCond l (condition nfe1 ++ condition nfe2) -> FEeval l fe1 == FEeval l fe2. intros l fe1 fe2 nfe1 nfe2 Hcrossprod Hcond; subst nfe1 nfe2. -rewrite Fnorm_FEeval_PEeval in |- * by +rewrite Fnorm_FEeval_PEeval by apply PCond_app_inv_l with (1 := Hcond). - rewrite Fnorm_FEeval_PEeval in |- * by + rewrite Fnorm_FEeval_PEeval by apply PCond_app_inv_r with (1 := Hcond). apply cross_product_eq; trivial. apply Pcond_Fnorm. @@ -1355,9 +1306,9 @@ apply Fnorm_crossproduct; trivial. match goal with [ |- NPEeval l ?x == NPEeval l ?y] => rewrite (ring_rw_correct Rsth Reqe ARth CRmorph pow_th cdiv_th get_sign_spec - O nil l I (refl_equal nil) x (refl_equal (Nnorm O nil x))); + O nil l I Logic.eq_refl x Logic.eq_refl); rewrite (ring_rw_correct Rsth Reqe ARth CRmorph pow_th cdiv_th get_sign_spec - O nil l I (refl_equal nil) y (refl_equal (Nnorm O nil y))) + O nil l I Logic.eq_refl y Logic.eq_refl) end. trivial. Qed. @@ -1377,28 +1328,28 @@ Proof. intros n l lpe fe1 fe2 Hlpe lmp Hlmp nfe1 eq1 nfe2 eq2 den eq3 Hcrossprod Hcond; subst nfe1 nfe2 den lmp. apply Fnorm_crossproduct; trivial. -simpl in |- *. -rewrite (split_correct_l l (denum (Fnorm fe1)) (denum (Fnorm fe2))) in |- *. -rewrite (split_correct_r l (denum (Fnorm fe1)) (denum (Fnorm fe2))) in |- *. -rewrite NPEmul_correct in |- *. -rewrite NPEmul_correct in |- *. -simpl in |- *. -repeat rewrite (ARmul_assoc ARth) in |- *. +simpl. +rewrite (split_correct_l l (denum (Fnorm fe1)) (denum (Fnorm fe2))). +rewrite (split_correct_r l (denum (Fnorm fe1)) (denum (Fnorm fe2))). +rewrite NPEmul_correct. +rewrite NPEmul_correct. +simpl. +repeat rewrite (ARmul_assoc ARth). rewrite <-( let x := PEmul (num (Fnorm fe1)) (rsplit_right (split (denum (Fnorm fe1)) (denum (Fnorm fe2)))) in ring_rw_correct Rsth Reqe ARth CRmorph pow_th cdiv_th get_sign_spec n lpe l - Hlpe (refl_equal (Nmk_monpol_list lpe)) - x (refl_equal (Nnorm n (Nmk_monpol_list lpe) x))) in Hcrossprod. + Hlpe Logic.eq_refl + x Logic.eq_refl) in Hcrossprod. rewrite <-( let x := (PEmul (num (Fnorm fe2)) (rsplit_left (split (denum (Fnorm fe1)) (denum (Fnorm fe2))))) in ring_rw_correct Rsth Reqe ARth CRmorph pow_th cdiv_th get_sign_spec n lpe l - Hlpe (refl_equal (Nmk_monpol_list lpe)) - x (refl_equal (Nnorm n (Nmk_monpol_list lpe) x))) in Hcrossprod. + Hlpe Logic.eq_refl + x Logic.eq_refl) in Hcrossprod. simpl in Hcrossprod. -rewrite Hcrossprod in |- *. +rewrite Hcrossprod. reflexivity. Qed. @@ -1417,28 +1368,28 @@ Proof. intros n l lpe fe1 fe2 Hlpe lmp Hlmp nfe1 eq1 nfe2 eq2 den eq3 Hcrossprod Hcond; subst nfe1 nfe2 den lmp. apply Fnorm_crossproduct; trivial. -simpl in |- *. -rewrite (split_correct_l l (denum (Fnorm fe1)) (denum (Fnorm fe2))) in |- *. -rewrite (split_correct_r l (denum (Fnorm fe1)) (denum (Fnorm fe2))) in |- *. -rewrite NPEmul_correct in |- *. -rewrite NPEmul_correct in |- *. -simpl in |- *. -repeat rewrite (ARmul_assoc ARth) in |- *. +simpl. +rewrite (split_correct_l l (denum (Fnorm fe1)) (denum (Fnorm fe2))). +rewrite (split_correct_r l (denum (Fnorm fe1)) (denum (Fnorm fe2))). +rewrite NPEmul_correct. +rewrite NPEmul_correct. +simpl. +repeat rewrite (ARmul_assoc ARth). rewrite <-( let x := PEmul (num (Fnorm fe1)) (rsplit_right (split (denum (Fnorm fe1)) (denum (Fnorm fe2)))) in ring_rw_pow_correct Rsth Reqe ARth CRmorph pow_th cdiv_th get_sign_spec n lpe l - Hlpe (refl_equal (Nmk_monpol_list lpe)) - x (refl_equal (Nnorm n (Nmk_monpol_list lpe) x))) in Hcrossprod. + Hlpe Logic.eq_refl + x Logic.eq_refl) in Hcrossprod. rewrite <-( let x := (PEmul (num (Fnorm fe2)) (rsplit_left (split (denum (Fnorm fe1)) (denum (Fnorm fe2))))) in ring_rw_pow_correct Rsth Reqe ARth CRmorph pow_th cdiv_th get_sign_spec n lpe l - Hlpe (refl_equal (Nmk_monpol_list lpe)) - x (refl_equal (Nnorm n (Nmk_monpol_list lpe) x))) in Hcrossprod. + Hlpe Logic.eq_refl + x Logic.eq_refl) in Hcrossprod. simpl in Hcrossprod. -rewrite Hcrossprod in |- *. +rewrite Hcrossprod. reflexivity. Qed. @@ -1558,7 +1509,7 @@ Fixpoint Fapp (l m:list (PExpr C)) {struct l} : list (PExpr C) := Lemma fcons_correct : forall l l1, PCond l (Fapp l1 nil) -> PCond l l1. -induction l1; simpl in |- *; intros. +induction l1; simpl; intros. trivial. elim PCond_fcons_inv with (1 := H); intros. destruct l1; auto. @@ -1639,7 +1590,7 @@ intros l a; elim a; try (intros; apply PFcons0_fcons_inv; auto; fail). simpl in H1. case (H _ H1); intros H2 H3. case (H0 _ H3); intros H4 H5; split; auto. - simpl in |- *. + simpl. apply field_is_integral_domain; trivial. simpl;intros. rewrite pow_th.(rpow_pow_N). destruct (H _ H0);split;auto. @@ -1667,7 +1618,7 @@ generalize (fun h => X (morph_eq CRmorph c1 c2 h)). generalize (@ceqb_complete c1 c2). case (c1 ?=! c2); auto; intros. apply X0. -red in |- *; intro. +red; intro. absurd (false = true); auto; discriminate. Qed. @@ -1683,18 +1634,18 @@ Fixpoint Fcons1 (e:PExpr C) (l:list (PExpr C)) {struct e} : list (PExpr C) := Theorem PFcons1_fcons_inv: forall l a l1, PCond l (Fcons1 a l1) -> ~ NPEeval l a == 0 /\ PCond l l1. intros l a; elim a; try (intros; apply PFcons0_fcons_inv; auto; fail). - simpl in |- *; intros c l1. + simpl; intros c l1. apply ceqb_rect_complete; intros. elim (@absurd_PCond_bottom l H0). split; trivial. - rewrite <- (morph0 CRmorph) in |- *; trivial. + rewrite <- (morph0 CRmorph); trivial. intros p H p0 H0 l1 H1. simpl in H1. case (H _ H1); intros H2 H3. case (H0 _ H3); intros H4 H5; split; auto. - simpl in |- *. + simpl. apply field_is_integral_domain; trivial. - simpl in |- *; intros p H l1. + simpl; intros p H l1. apply ceqb_rect_complete; intros. elim (@absurd_PCond_bottom l H1). destruct (H _ H1). @@ -1713,7 +1664,7 @@ Definition Fcons2 e l := Fcons1 (PExpr_simp e) l. Theorem PFcons2_fcons_inv: forall l a l1, PCond l (Fcons2 a l1) -> ~ NPEeval l a == 0 /\ PCond l l1. -unfold Fcons2 in |- *; intros l a l1 H; split; +unfold Fcons2; intros l a l1 H; split; case (PFcons1_fcons_inv l (PExpr_simp a) l1); auto. intros H1 H2 H3; case H1. transitivity (NPEeval l a); trivial. @@ -1792,61 +1743,55 @@ Hypothesis gen_phiPOS_not_0 : forall p, ~ gen_phiPOS1 rI radd rmul p == 0. Lemma add_inj_r : forall p x y, gen_phiPOS1 rI radd rmul p + x == gen_phiPOS1 rI radd rmul p + y -> x==y. intros p x y. -elim p using Pind; simpl in |- *; intros. +elim p using Pos.peano_ind; simpl; intros. apply S_inj; trivial. apply H. apply S_inj. - repeat rewrite (ARadd_assoc ARth) in |- *. - rewrite <- (ARgen_phiPOS_Psucc Rsth Reqe ARth) in |- *; trivial. + repeat rewrite (ARadd_assoc ARth). + rewrite <- (ARgen_phiPOS_Psucc Rsth Reqe ARth); trivial. Qed. Lemma gen_phiPOS_inj : forall x y, gen_phiPOS rI radd rmul x == gen_phiPOS rI radd rmul y -> x = y. intros x y. -repeat rewrite <- (same_gen Rsth Reqe ARth) in |- *. -ElimPcompare x y; intro. +repeat rewrite <- (same_gen Rsth Reqe ARth). +case (Pos.compare_spec x y). + intros. + trivial. intros. - apply Pcompare_Eq_eq; trivial. - intro. elim gen_phiPOS_not_0 with (y - x)%positive. apply add_inj_r with x. - symmetry in |- *. - rewrite (ARadd_0_r Rsth ARth) in |- *. - rewrite <- (ARgen_phiPOS_add Rsth Reqe ARth) in |- *. - rewrite Pplus_minus in |- *; trivial. - change Eq with (CompOpp Eq) in |- *. - rewrite <- Pcompare_antisym in |- *; trivial. - rewrite H in |- *; trivial. - intro. + symmetry. + rewrite (ARadd_0_r Rsth ARth). + rewrite <- (ARgen_phiPOS_add Rsth Reqe ARth). + now rewrite Pos.add_comm, Pos.sub_add. + intros. elim gen_phiPOS_not_0 with (x - y)%positive. apply add_inj_r with y. - rewrite (ARadd_0_r Rsth ARth) in |- *. - rewrite <- (ARgen_phiPOS_add Rsth Reqe ARth) in |- *. - rewrite Pplus_minus in |- *; trivial. + rewrite (ARadd_0_r Rsth ARth). + rewrite <- (ARgen_phiPOS_add Rsth Reqe ARth). + now rewrite Pos.add_comm, Pos.sub_add. Qed. Lemma gen_phiN_inj : forall x y, gen_phiN rO rI radd rmul x == gen_phiN rO rI radd rmul y -> x = y. -destruct x; destruct y; simpl in |- *; intros; trivial. +destruct x; destruct y; simpl; intros; trivial. elim gen_phiPOS_not_0 with p. - symmetry in |- *. - rewrite (same_gen Rsth Reqe ARth) in |- *; trivial. + symmetry . + rewrite (same_gen Rsth Reqe ARth); trivial. elim gen_phiPOS_not_0 with p. - rewrite (same_gen Rsth Reqe ARth) in |- *; trivial. - rewrite gen_phiPOS_inj with (1 := H) in |- *; trivial. + rewrite (same_gen Rsth Reqe ARth); trivial. + rewrite gen_phiPOS_inj with (1 := H); trivial. Qed. Lemma gen_phiN_complete : forall x y, gen_phiN rO rI radd rmul x == gen_phiN rO rI radd rmul y -> - Neq_bool x y = true. -intros. - replace y with x. - unfold Neq_bool in |- *. - rewrite Ncompare_refl in |- *; trivial. - apply gen_phiN_inj; trivial. + N.eqb x y = true. +Proof. +intros. now apply N.eqb_eq, gen_phiN_inj. Qed. End AlmostField. @@ -1864,17 +1809,17 @@ Section Field. Lemma ring_S_inj : forall x y, 1+x==1+y -> x==y. intros. transitivity (x + (1 + - (1))). - rewrite (Ropp_def Rth) in |- *. - symmetry in |- *. + rewrite (Ropp_def Rth). + symmetry . apply (ARadd_0_r Rsth ARth). transitivity (y + (1 + - (1))). - repeat rewrite <- (ARplus_assoc ARth) in |- *. - repeat rewrite (ARadd_assoc ARth) in |- *. + repeat rewrite <- (ARplus_assoc ARth). + repeat rewrite (ARadd_assoc ARth). apply (Radd_ext Reqe). - repeat rewrite <- (ARadd_comm ARth 1) in |- *. + repeat rewrite <- (ARadd_comm ARth 1). trivial. reflexivity. - rewrite (Ropp_def Rth) in |- *. + rewrite (Ropp_def Rth). apply (ARadd_0_r Rsth ARth). Qed. @@ -1886,14 +1831,14 @@ Let gen_phiPOS_inject := Lemma gen_phiPOS_discr_sgn : forall x y, ~ gen_phiPOS rI radd rmul x == - gen_phiPOS rI radd rmul y. -red in |- *; intros. +red; intros. apply gen_phiPOS_not_0 with (y + x)%positive. -rewrite (ARgen_phiPOS_add Rsth Reqe ARth) in |- *. +rewrite (ARgen_phiPOS_add Rsth Reqe ARth). transitivity (gen_phiPOS1 1 radd rmul y + - gen_phiPOS1 1 radd rmul y). apply (Radd_ext Reqe); trivial. reflexivity. - rewrite (same_gen Rsth Reqe ARth) in |- *. - rewrite (same_gen Rsth Reqe ARth) in |- *. + rewrite (same_gen Rsth Reqe ARth). + rewrite (same_gen Rsth Reqe ARth). trivial. apply (Ropp_def Rth). Qed. @@ -1901,33 +1846,33 @@ Qed. Lemma gen_phiZ_inj : forall x y, gen_phiZ rO rI radd rmul ropp x == gen_phiZ rO rI radd rmul ropp y -> x = y. -destruct x; destruct y; simpl in |- *; intros. +destruct x; destruct y; simpl; intros. trivial. elim gen_phiPOS_not_0 with p. - rewrite (same_gen Rsth Reqe ARth) in |- *. - symmetry in |- *; trivial. + rewrite (same_gen Rsth Reqe ARth). + symmetry ; trivial. elim gen_phiPOS_not_0 with p. - rewrite (same_gen Rsth Reqe ARth) in |- *. - rewrite <- (Ropp_opp Rsth Reqe Rth (gen_phiPOS 1 radd rmul p)) in |- *. - rewrite <- H in |- *. + rewrite (same_gen Rsth Reqe ARth). + rewrite <- (Ropp_opp Rsth Reqe Rth (gen_phiPOS 1 radd rmul p)). + rewrite <- H. apply (ARopp_zero Rsth Reqe ARth). elim gen_phiPOS_not_0 with p. - rewrite (same_gen Rsth Reqe ARth) in |- *. + rewrite (same_gen Rsth Reqe ARth). trivial. - rewrite gen_phiPOS_inject with (1 := H) in |- *; trivial. + rewrite gen_phiPOS_inject with (1 := H); trivial. elim gen_phiPOS_discr_sgn with (1 := H). elim gen_phiPOS_not_0 with p. - rewrite (same_gen Rsth Reqe ARth) in |- *. - rewrite <- (Ropp_opp Rsth Reqe Rth (gen_phiPOS 1 radd rmul p)) in |- *. - rewrite H in |- *. + rewrite (same_gen Rsth Reqe ARth). + rewrite <- (Ropp_opp Rsth Reqe Rth (gen_phiPOS 1 radd rmul p)). + rewrite H. apply (ARopp_zero Rsth Reqe ARth). elim gen_phiPOS_discr_sgn with p0 p. - symmetry in |- *; trivial. + symmetry ; trivial. replace p0 with p; trivial. apply gen_phiPOS_inject. - rewrite <- (Ropp_opp Rsth Reqe Rth (gen_phiPOS 1 radd rmul p)) in |- *. - rewrite <- (Ropp_opp Rsth Reqe Rth (gen_phiPOS 1 radd rmul p0)) in |- *. - rewrite H in |- *; trivial. + rewrite <- (Ropp_opp Rsth Reqe Rth (gen_phiPOS 1 radd rmul p)). + rewrite <- (Ropp_opp Rsth Reqe Rth (gen_phiPOS 1 radd rmul p0)). + rewrite H; trivial. reflexivity. Qed. @@ -1936,8 +1881,8 @@ Lemma gen_phiZ_complete : forall x y, Zeq_bool x y = true. intros. replace y with x. - unfold Zeq_bool in |- *. - rewrite Zcompare_refl in |- *; trivial. + unfold Zeq_bool. + rewrite Z.compare_refl; trivial. apply gen_phiZ_inj; trivial. Qed. |