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author | glondu <glondu@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2009-09-17 15:58:14 +0000 |
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committer | glondu <glondu@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2009-09-17 15:58:14 +0000 |
commit | 61ccbc81a2f3b4662ed4a2bad9d07d2003dda3a2 (patch) | |
tree | 961cc88c714aa91a0276ea9fbf8bc53b2b9d5c28 /theories/Arith/Even.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/Arith/Even.v')
-rw-r--r-- | theories/Arith/Even.v | 20 |
1 files changed, 10 insertions, 10 deletions
diff --git a/theories/Arith/Even.v b/theories/Arith/Even.v index d2a4006a0..eaa1bb2d6 100644 --- a/theories/Arith/Even.v +++ b/theories/Arith/Even.v @@ -17,7 +17,7 @@ Open Local Scope nat_scope. Implicit Types m n : nat. -(** * Definition of [even] and [odd], and basic facts *) +(** * Definition of [even] and [odd], and basic facts *) Inductive even : nat -> Prop := | even_O : even 0 @@ -52,9 +52,9 @@ Qed. (** * Facts about [even] & [odd] wrt. [plus] *) -Lemma even_plus_split : forall n m, +Lemma even_plus_split : forall n m, (even (n + m) -> even n /\ even m \/ odd n /\ odd m) -with odd_plus_split : forall n m, +with odd_plus_split : forall n m, odd (n + m) -> odd n /\ even m \/ even n /\ odd m. Proof. intros. clear even_plus_split. destruct n; simpl in *. @@ -95,7 +95,7 @@ Proof. intros n m H; destruct (even_plus_split n m) as [[]|[]]; auto. intro; destruct (not_even_and_odd n); auto. Qed. - + Lemma even_plus_even_inv_l : forall n m, even (n + m) -> even m -> even n. Proof. intros n m H; destruct (even_plus_split n m) as [[]|[]]; auto. @@ -120,13 +120,13 @@ Proof. intros n m H; destruct (odd_plus_split n m) as [[]|[]]; auto. intro; destruct (not_even_and_odd m); auto. Qed. - + Lemma odd_plus_even_inv_r : forall n m, odd (n + m) -> odd n -> even m. Proof. intros n m H; destruct (odd_plus_split n m) as [[]|[]]; auto. intro; destruct (not_even_and_odd n); auto. Qed. - + Lemma odd_plus_odd_inv_l : forall n m, odd (n + m) -> even m -> odd n. Proof. intros n m H; destruct (odd_plus_split n m) as [[]|[]]; auto. @@ -203,7 +203,7 @@ Proof. intros n m; case (even_mult_aux n m); auto. intros H H0; case H0; auto. Qed. - + Lemma even_mult_r : forall n m, even m -> even (n * m). Proof. intros n m; case (even_mult_aux n m); auto. @@ -219,7 +219,7 @@ Proof. intros H'3; elim H'3; auto. intros H; case (not_even_and_odd n); auto. Qed. - + Lemma even_mult_inv_l : forall n m, even (n * m) -> odd m -> even n. Proof. intros n m H' H'0. @@ -228,13 +228,13 @@ Proof. intros H'3; elim H'3; auto. intros H; case (not_even_and_odd m); auto. Qed. - + Lemma odd_mult : forall n m, odd n -> odd m -> odd (n * m). Proof. intros n m; case (even_mult_aux n m); intros H; case H; auto. Qed. Hint Resolve even_mult_l even_mult_r odd_mult: arith. - + Lemma odd_mult_inv_l : forall n m, odd (n * m) -> odd n. Proof. intros n m H'. |