diff options
| author |  glondu <glondu@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2009-09-17 15:58:14 +0000 |
|---|---|---|
| committer | glondu <glondu@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2009-09-17 15:58:14 +0000 |
| commit | 61ccbc81a2f3b4662ed4a2bad9d07d2003dda3a2 (patch) | |
| tree | 961cc88c714aa91a0276ea9fbf8bc53b2b9d5c28 /theories/Reals/PartSum.v | |
| parent | 6d3fbdf36c6a47b49c2a4b16f498972c93c07574 (diff) | |
Delete trailing whitespaces in all *.{v,ml*} files
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12337 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Reals/PartSum.v')
| -rw-r--r-- | theories/Reals/PartSum.v | 16 |
1 files changed, 8 insertions, 8 deletions
diff --git a/theories/Reals/PartSum.v b/theories/Reals/PartSum.v index 40972fbcf..6a33b8092 100644 --- a/theories/Reals/PartSum.v +++ b/theories/Reals/PartSum.v @@ -31,7 +31,7 @@ Lemma tech2 : forall (An:nat -> R) (m n:nat), (m < n)%nat -> sum_f_R0 An n = - sum_f_R0 An m + sum_f_R0 (fun i:nat => An (S m + i)%nat) (n - S m). + sum_f_R0 An m + sum_f_R0 (fun i:nat => An (S m + i)%nat) (n - S m). Proof. intros; induction n as [| n Hrecn]. elim (lt_n_O _ H). @@ -155,7 +155,7 @@ Lemma tech12 : Proof. intros; unfold Pser in |- *; unfold infinite_sum in |- *; unfold Un_cv in H; assumption. -Qed. +Qed. Lemma scal_sum : forall (An:nat -> R) (N:nat) (x:R), @@ -256,12 +256,12 @@ Qed. Lemma minus_sum : forall (An Bn:nat -> R) (N:nat), - sum_f_R0 (fun i:nat => An i - Bn i) N = sum_f_R0 An N - sum_f_R0 Bn N. + sum_f_R0 (fun i:nat => An i - Bn i) N = sum_f_R0 An N - sum_f_R0 Bn N. Proof. - intros; induction N as [| N HrecN]. - simpl in |- *; ring. - do 3 rewrite tech5; rewrite HrecN; ring. -Qed. + intros; induction N as [| N HrecN]. + simpl in |- *; ring. + do 3 rewrite tech5; rewrite HrecN; ring. +Qed. Lemma sum_decomposition : forall (An:nat -> R) (N:nat), @@ -346,7 +346,7 @@ Qed. (**********) Lemma Rabs_triang_gen : forall (An:nat -> R) (N:nat), - Rabs (sum_f_R0 An N) <= sum_f_R0 (fun i:nat => Rabs (An i)) N. + Rabs (sum_f_R0 An N) <= sum_f_R0 (fun i:nat => Rabs (An i)) N. Proof. intros. induction N as [| N HrecN]. |