aboutsummaryrefslogtreecommitdiffhomepage
path: root/theories/Arith/Even.v
diff options
context:
space:
mode:
authorGravatar glondu <glondu@85f007b7-540e-0410-9357-904b9bb8a0f7>2009-09-17 15:58:14 +0000
committerGravatar glondu <glondu@85f007b7-540e-0410-9357-904b9bb8a0f7>2009-09-17 15:58:14 +0000
commit61ccbc81a2f3b4662ed4a2bad9d07d2003dda3a2 (patch)
tree961cc88c714aa91a0276ea9fbf8bc53b2b9d5c28 /theories/Arith/Even.v
parent6d3fbdf36c6a47b49c2a4b16f498972c93c07574 (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.v20
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'.