diff options
Diffstat (limited to 'theories/Reals/Rsqrt_def.v')
| -rw-r--r-- | theories/Reals/Rsqrt_def.v | 12 |
1 files changed, 6 insertions, 6 deletions
diff --git a/theories/Reals/Rsqrt_def.v b/theories/Reals/Rsqrt_def.v index 0a3af6ca..33c20355 100644 --- a/theories/Reals/Rsqrt_def.v +++ b/theories/Reals/Rsqrt_def.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Rsqrt_def.v 10710 2008-03-23 09:24:09Z herbelin $ i*) +(*i $Id$ i*) Require Import Sumbool. Require Import Rbase. @@ -23,7 +23,7 @@ Boxed Fixpoint Dichotomy_lb (x y:R) (P:R -> bool) (N:nat) {struct N} : R := let up := Dichotomy_ub x y P n in let z := (down + up) / 2 in if P z then down else z end - + with Dichotomy_ub (x y:R) (P:R -> bool) (N:nat) {struct N} : R := match N with | O => y @@ -471,8 +471,8 @@ Proof. intros. cut (x <= y). intro. - generalize (dicho_lb_cv x y (fun z:R => cond_positivity (f z)) H3). - generalize (dicho_up_cv x y (fun z:R => cond_positivity (f z)) H3). + generalize (dicho_lb_cv x y (fun z:R => cond_positivity (f z)) H3). + generalize (dicho_up_cv x y (fun z:R => cond_positivity (f z)) H3). intros X X0. elim X; intros. elim X0; intros. @@ -667,7 +667,7 @@ Proof. apply Ropp_0_gt_lt_contravar; assumption. Qed. -(** We can now define the square root function as the reciprocal +(** We can now define the square root function as the reciprocal transformation of the square root function *) Lemma Rsqrt_exists : forall y:R, 0 <= y -> { z:R | 0 <= z /\ y = Rsqr z }. @@ -698,7 +698,7 @@ Proof. rewrite Rsqr_1. apply Rplus_le_reg_l with y. rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *; - rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; + rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; left; assumption. exists 1. split. |