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author | desmettr <desmettr@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2002-10-04 12:26:35 +0000 |
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committer | desmettr <desmettr@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2002-10-04 12:26:35 +0000 |
commit | 9098fae00ac832c95e56135604cb96e71c0d646c (patch) | |
tree | eac2d679928cde8ba7d9f1a03bcbf445d06af4aa /theories/Reals | |
parent | bfc3d11db7fec29dee03bffce8f1ad9bbc81a73a (diff) |
Ajout du lemme derivable_pt_lim_power
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@3089 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Reals')
-rw-r--r-- | theories/Reals/Rpower.v | 19 |
1 files changed, 19 insertions, 0 deletions
diff --git a/theories/Reals/Rpower.v b/theories/Reals/Rpower.v index b3ae325b9..17aec980e 100644 --- a/theories/Reals/Rpower.v +++ b/theories/Reals/Rpower.v @@ -545,4 +545,23 @@ Apply D_in_ext with f := [x : R] (Rmult z R1). Apply Rmult_1r. Apply (Dmult_const [x : ?] True [x : ?] x [x : ?] R1); Apply Dx. Assert H0 := (derivable_pt_lim_D_in exp exp ``z*(ln y)``); Elim H0; Clear H0; Intros _ H0; Apply H0; Apply derivable_pt_lim_exp. +Qed. + +Theorem derivable_pt_lim_power: (x, y : R) (Rlt R0 x) -> (derivable_pt_lim [x : ?] (Rpower x y) x (Rmult y (Rpower x (Rminus y R1)))). +Intros x y H. +Unfold Rminus; Rewrite Rpower_plus. +Rewrite Rpower_Ropp. +Rewrite Rpower_1; Auto. +Rewrite <- Rmult_assoc. +Unfold Rpower. +Apply derivable_pt_lim_comp with f1 := ln f2 := [x : ?] (exp (Rmult y x)). +Apply derivable_pt_lim_ln; Assumption. +Rewrite (Rmult_sym y). +Apply derivable_pt_lim_comp with f1 := [x : ?] (Rmult y x) f2 := exp. +Pattern 2 y; Replace y with (Rplus (Rmult R0 (ln x)) (Rmult y R1)). +Apply derivable_pt_lim_mult with f1 := [x : R] y f2 := [x : R] x. +Apply derivable_pt_lim_const with a := y. +Apply derivable_pt_lim_id. +Ring. +Apply derivable_pt_lim_exp. Qed.
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