diff options
Diffstat (limited to 'plugins/setoid_ring/RealField.v')
| -rw-r--r-- | plugins/setoid_ring/RealField.v | 64 |
1 files changed, 32 insertions, 32 deletions
diff --git a/plugins/setoid_ring/RealField.v b/plugins/setoid_ring/RealField.v index 56473adb..29372212 100644 --- a/plugins/setoid_ring/RealField.v +++ b/plugins/setoid_ring/RealField.v @@ -5,21 +5,21 @@ Require Import Rdefinitions. Require Import Rpow_def. Require Import Raxioms. -Open Local Scope R_scope. +Local Open Scope R_scope. Lemma RTheory : ring_theory 0 1 Rplus Rmult Rminus Ropp (eq (A:=R)). Proof. constructor. intro; apply Rplus_0_l. exact Rplus_comm. - symmetry in |- *; apply Rplus_assoc. + symmetry ; apply Rplus_assoc. intro; apply Rmult_1_l. exact Rmult_comm. - symmetry in |- *; apply Rmult_assoc. + symmetry ; apply Rmult_assoc. intros m n p. - rewrite Rmult_comm in |- *. - rewrite (Rmult_comm n p) in |- *. - rewrite (Rmult_comm m p) in |- *. + rewrite Rmult_comm. + rewrite (Rmult_comm n p). + rewrite (Rmult_comm m p). apply Rmult_plus_distr_l. reflexivity. exact Rplus_opp_r. @@ -42,17 +42,17 @@ destruct H0. apply Rlt_trans with (IZR (up x)); trivial. replace (IZR (up x)) with (x + (IZR (up x) - x))%R. apply Rplus_lt_compat_l; trivial. - unfold Rminus in |- *. - rewrite (Rplus_comm (IZR (up x)) (- x)) in |- *. - rewrite <- Rplus_assoc in |- *. - rewrite Rplus_opp_r in |- *. + unfold Rminus. + rewrite (Rplus_comm (IZR (up x)) (- x)). + rewrite <- Rplus_assoc. + rewrite Rplus_opp_r. apply Rplus_0_l. elim H0. - unfold Rminus in |- *. - rewrite (Rplus_comm (IZR (up x)) (- x)) in |- *. - rewrite <- Rplus_assoc in |- *. - rewrite Rplus_opp_r in |- *. - rewrite Rplus_0_l in |- *; trivial. + unfold Rminus. + rewrite (Rplus_comm (IZR (up x)) (- x)). + rewrite <- Rplus_assoc. + rewrite Rplus_opp_r. + rewrite Rplus_0_l; trivial. Qed. Notation Rset := (Eqsth R). @@ -61,7 +61,7 @@ Notation Rext := (Eq_ext Rplus Rmult Ropp). Lemma Rlt_0_2 : 0 < 2. apply Rlt_trans with (0 + 1). apply Rlt_n_Sn. - rewrite Rplus_comm in |- *. + rewrite Rplus_comm. apply Rplus_lt_compat_l. replace 1 with (0 + 1). apply Rlt_n_Sn. @@ -69,19 +69,19 @@ apply Rlt_trans with (0 + 1). Qed. Lemma Rgen_phiPOS : forall x, InitialRing.gen_phiPOS1 1 Rplus Rmult x > 0. -unfold Rgt in |- *. -induction x; simpl in |- *; intros. +unfold Rgt. +induction x; simpl; intros. apply Rlt_trans with (1 + 0). - rewrite Rplus_comm in |- *. + rewrite Rplus_comm. apply Rlt_n_Sn. apply Rplus_lt_compat_l. - rewrite <- (Rmul_0_l Rset Rext RTheory 2) in |- *. - rewrite Rmult_comm in |- *. + rewrite <- (Rmul_0_l Rset Rext RTheory 2). + rewrite Rmult_comm. apply Rmult_lt_compat_l. apply Rlt_0_2. trivial. - rewrite <- (Rmul_0_l Rset Rext RTheory 2) in |- *. - rewrite Rmult_comm in |- *. + rewrite <- (Rmul_0_l Rset Rext RTheory 2). + rewrite Rmult_comm. apply Rmult_lt_compat_l. apply Rlt_0_2. trivial. @@ -93,9 +93,9 @@ Qed. Lemma Rgen_phiPOS_not_0 : forall x, InitialRing.gen_phiPOS1 1 Rplus Rmult x <> 0. -red in |- *; intros. +red; intros. specialize (Rgen_phiPOS x). -rewrite H in |- *; intro. +rewrite H; intro. apply (Rlt_asym 0 0); trivial. Qed. @@ -107,23 +107,23 @@ Proof gen_phiZ_complete Rset Rext Rfield Rgen_phiPOS_not_0. Lemma Rdef_pow_add : forall (x:R) (n m:nat), pow x (n + m) = pow x n * pow x m. Proof. - intros x n; elim n; simpl in |- *; auto with real. + intros x n; elim n; simpl; auto with real. intros n0 H' m; rewrite H'; auto with real. Qed. -Lemma R_power_theory : power_theory 1%R Rmult (eq (A:=R)) nat_of_N pow. +Lemma R_power_theory : power_theory 1%R Rmult (@eq R) N.to_nat pow. Proof. constructor. destruct n. reflexivity. - simpl. induction p;simpl. - rewrite ZL6. rewrite Rdef_pow_add;rewrite IHp. reflexivity. - unfold nat_of_P;simpl;rewrite ZL6;rewrite Rdef_pow_add;rewrite IHp;trivial. - rewrite Rmult_comm;apply Rmult_1_l. + simpl. induction p. + - rewrite Pos2Nat.inj_xI. simpl. now rewrite plus_0_r, Rdef_pow_add, IHp. + - rewrite Pos2Nat.inj_xO. simpl. now rewrite plus_0_r, Rdef_pow_add, IHp. + - simpl. rewrite Rmult_comm;apply Rmult_1_l. Qed. Ltac Rpow_tac t := match isnatcst t with | false => constr:(InitialRing.NotConstant) - | _ => constr:(N_of_nat t) + | _ => constr:(N.of_nat t) end. Add Field RField : Rfield |