Extension de Even et Div2

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@2367 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 261 insertions, 0 deletions

diff --git a/theories/Arith/Even.v b/theories/Arith/Even.v
index 0f1ab8559..13153bafb 100644
--- a/theories/Arith/Even.v
+++ b/theories/Arith/Even.v
@@ -43,3 +43,264 @@ Intros. Inversion H0.
 Intros. Inversion H. Inversion H0. Auto with arith.
 Save.
 
+Lemma even_plus_aux:
+  (n,m:nat)
+   (iff (odd (plus n m)) (odd n) /\ (even m) \/ (even n) /\ (odd m)) /\
+   (iff (even (plus n m)) (even n) /\ (even m) \/ (odd n) /\ (odd m)).
+Proof.
+Intros n; Elim n; Simpl; Auto with arith.
+Intros m; Split; Auto.
+Split.
+Intros H; Right; Split; Auto with arith.
+Intros H'; Case H'; Auto with arith.
+Intros H'0; Elim H'0; Intros H'1 H'2; Inversion H'1.
+Intros H; Elim H; Auto.
+Split; Auto with arith.
+Intros H'; Elim H'; Auto with arith.
+Intros H; Elim H; Auto.
+Intros H'0; Elim H'0; Intros H'1 H'2; Inversion H'1.
+Intros n0 H' m; Elim (H' m); Intros H'1 H'2; Elim H'1; Intros E1 E2; Elim H'2;
+ Intros E3 E4; Clear H'1 H'2.
+Split; Split.
+Intros H'0; Case E3.
+Inversion H'0; Auto.
+Intros H; Elim H; Intros H0 H1; Clear H; Auto with arith.
+Intros H; Elim H; Intros H0 H1; Clear H; Auto with arith.
+Intros H'0; Case H'0; Intros C0; Case C0; Intros C1 C2.
+Apply odd_S.
+Apply E4; Left; Split; Auto with arith.
+Inversion C1; Auto.
+Apply odd_S.
+Apply E4; Right; Split; Auto with arith.
+Inversion C1; Auto.
+Intros H'0.
+Case E1.
+Inversion H'0; Auto.
+Intros H; Elim H; Intros H0 H1; Clear H; Auto with arith.
+Intros H; Elim H; Intros H0 H1; Clear H; Auto with arith.
+Intros H'0; Case H'0; Intros C0; Case C0; Intros C1 C2.
+Apply even_S.
+Apply E2; Left; Split; Auto with arith.
+Inversion C1; Auto.
+Apply even_S.
+Apply E2; Right; Split; Auto with arith.
+Inversion C1; Auto.
+Qed.
+ 
+Lemma even_even_plus : (n,m:nat) (even n) -> (even m) -> (even (plus n m)).
+Proof.
+Intros n m; Case (even_plus_aux n m).
+Intros H H0; Case H0; Auto.
+Qed.
+ 
+Lemma odd_even_plus : (n,m:nat) (odd n) -> (odd m) -> (even (plus n m)).
+Proof.
+Intros n m; Case (even_plus_aux n m).
+Intros H H0; Case H0; Auto.
+Qed.
+ 
+Lemma even_plus_even_inv_r :
+  (n,m:nat) (even (plus n m)) -> (even n) -> (even m).
+Proof.
+Intros n m H; Case (even_plus_aux n m).
+Intros H' H'0; Elim H'0.
+Intros H'1; Case H'1; Auto.
+Intros H0; Elim H0; Auto.
+Intros H0 H1 H2; Case (not_even_and_odd n); Auto.
+Case H0; Auto.
+Qed.
+ 
+Lemma even_plus_even_inv_l :
+  (n,m:nat) (even (plus n m)) -> (even m) -> (even n).
+Proof.
+Intros n m H; Case (even_plus_aux n m).
+Intros H' H'0; Elim H'0.
+Intros H'1; Case H'1; Auto.
+Intros H0; Elim H0; Auto.
+Intros H0 H1 H2; Case (not_even_and_odd m); Auto.
+Case H0; Auto.
+Qed.
+ 
+Lemma even_plus_odd_inv_r : (n,m:nat) (even (plus n m)) -> (odd n) -> (odd m).
+Proof.
+Intros n m H; Case (even_plus_aux n m).
+Intros H' H'0; Elim H'0.
+Intros H'1; Case H'1; Auto.
+Intros H0 H1 H2; Case (not_even_and_odd n); Auto.
+Case H0; Auto.
+Intros H0; Case H0; Auto.
+Qed.
+ 
+Lemma even_plus_odd_inv_l : (n,m:nat) (even (plus n m)) -> (odd m) -> (odd n).
+Proof.
+Intros n m H; Case (even_plus_aux n m).
+Intros H' H'0; Elim H'0.
+Intros H'1; Case H'1; Auto.
+Intros H0 H1 H2; Case (not_even_and_odd m); Auto.
+Case H0; Auto.
+Intros H0; Case H0; Auto.
+Qed.
+Hints Resolve even_even_plus odd_even_plus :arith.
+ 
+Lemma odd_plus_l : (n,m:nat) (odd n) -> (even m) -> (odd (plus n m)).
+Proof.
+Intros n m; Case (even_plus_aux n m).
+Intros H; Case H; Auto.
+Qed.
+ 
+Lemma odd_plus_r : (n,m:nat) (even n) -> (odd m) -> (odd (plus n m)).
+Proof.
+Intros n m; Case (even_plus_aux n m).
+Intros H; Case H; Auto.
+Qed.
+ 
+Lemma odd_plus_even_inv_l : (n,m:nat) (odd (plus n m)) -> (odd m) -> (even n).
+Proof.
+Intros n m H; Case (even_plus_aux n m).
+Intros H' H'0; Elim H'.
+Intros H'1; Case H'1; Auto.
+Intros H0 H1 H2; Case (not_even_and_odd m); Auto.
+Case H0; Auto.
+Intros H0; Case H0; Auto.
+Qed.
+ 
+Lemma odd_plus_even_inv_r : (n,m:nat) (odd (plus n m)) -> (odd n) -> (even m).
+Proof.
+Intros n m H; Case (even_plus_aux n m).
+Intros H' H'0; Elim H'.
+Intros H'1; Case H'1; Auto.
+Intros H0; Case H0; Auto.
+Intros H0 H1 H2; Case (not_even_and_odd n); Auto.
+Case H0; Auto.
+Qed.
+ 
+Lemma odd_plus_odd_inv_l : (n,m:nat) (odd (plus n m)) -> (even m) -> (odd n).
+Proof.
+Intros n m H; Case (even_plus_aux n m).
+Intros H' H'0; Elim H'.
+Intros H'1; Case H'1; Auto.
+Intros H0; Case H0; Auto.
+Intros H0 H1 H2; Case (not_even_and_odd m); Auto.
+Case H0; Auto.
+Qed.
+ 
+Lemma odd_plus_odd_inv_r : (n,m:nat) (odd (plus n m)) -> (even n) -> (odd m).
+Proof.
+Intros n m H; Case (even_plus_aux n m).
+Intros H' H'0; Elim H'.
+Intros H'1; Case H'1; Auto.
+Intros H0 H1 H2; Case (not_even_and_odd n); Auto.
+Case H0; Auto.
+Intros H0; Case H0; Auto.
+Qed.
+Hints Resolve odd_plus_l odd_plus_r :arith.
+ 
+Lemma even_mult_aux :
+  (n,m:nat)
+   (iff (odd (mult n m)) (odd n) /\ (odd m)) /\
+   (iff (even (mult n m)) (even n) \/ (even m)).
+Proof.
+Intros n; Elim n; Simpl; Auto with arith.
+Intros m; Split; Split; Auto with arith.
+Intros H'; Inversion H'.
+Intros H'; Elim H'; Auto.
+Intros n0 H' m; Split; Split; Auto with arith.
+Intros H'0.
+Elim (even_plus_aux m (mult n0 m)); Intros H'3 H'4; Case H'3; Intros H'1 H'2;
+ Case H'1; Auto.
+Intros H'5; Elim H'5; Intros H'6 H'7; Auto with arith.
+Split; Auto with arith.
+Case (H' m).
+Intros H'8 H'9; Case H'9.
+Intros H'10; Case H'10; Auto with arith.
+Intros H'11 H'12; Case (not_even_and_odd m); Auto with arith.
+Intros H'5; Elim H'5; Intros H'6 H'7; Case (not_even_and_odd (mult n0 m)); Auto.
+Case (H' m).
+Intros H'8 H'9; Case H'9; Auto.
+Intros H'0; Elim H'0; Intros H'1 H'2; Clear H'0.
+Elim (even_plus_aux m (mult n0 m)); Auto.
+Intros H'0 H'3.
+Elim H'0.
+Intros H'4 H'5; Apply H'5; Auto.
+Left; Split; Auto with arith.
+Case (H' m).
+Intros H'6 H'7; Elim H'7.
+Intros H'8 H'9; Apply H'9.
+Left.
+Inversion H'1; Auto.
+Intros H'0.
+Elim (even_plus_aux m (mult n0 m)); Intros H'3 H'4; Case H'4.
+Intros H'1 H'2.
+Elim H'1; Auto.
+Intros H; Case H; Auto.
+Intros H'5; Elim H'5; Intros H'6 H'7; Auto with arith.
+Left.
+Case (H' m).
+Intros H'8; Elim H'8.
+Intros H'9; Elim H'9; Auto with arith.
+Intros H'0; Elim H'0; Intros H'1.
+Case (even_or_odd m); Intros H'2.
+Apply even_even_plus; Auto.
+Case (H' m).
+Intros H H0; Case H0; Auto.
+Apply odd_even_plus; Auto.
+Inversion H'1; Case (H' m); Auto.
+Intros H1; Case H1; Auto.
+Apply even_even_plus; Auto.
+Case (H' m).
+Intros H H0; Case H0; Auto.
+Qed.
+ 
+Lemma even_mult_l : (n,m:nat) (even n) -> (even (mult n m)).
+Proof.
+Intros n m; Case (even_mult_aux n m); Auto.
+Intros H H0; Case H0; Auto.
+Qed.
+ 
+Lemma even_mult_r: (n,m:nat) (even m) -> (even (mult n m)).
+Proof.
+Intros n m; Case (even_mult_aux n m); Auto.
+Intros H H0; Case H0; Auto.
+Qed.
+Hints Resolve even_mult_l even_mult_r :arith.
+ 
+Lemma even_mult_inv_r: (n,m:nat) (even (mult n m)) -> (odd n) -> (even m).
+Proof.
+Intros n m H' H'0.
+Case (even_mult_aux n m).
+Intros H'1 H'2; Elim H'2.
+Intros H'3; Elim H'3; Auto.
+Intros H; Case (not_even_and_odd n); Auto.
+Qed.
+ 
+Lemma even_mult_inv_l : (n,m:nat) (even (mult n m)) -> (odd m) -> (even n).
+Proof.
+Intros n m H' H'0.
+Case (even_mult_aux n m).
+Intros H'1 H'2; Elim H'2.
+Intros H'3; Elim H'3; Auto.
+Intros H; Case (not_even_and_odd m); Auto.
+Qed.
+ 
+Lemma odd_mult : (n,m:nat) (odd n) -> (odd m) -> (odd (mult n m)).
+Proof.
+Intros n m; Case (even_mult_aux n m); Intros H; Case H; Auto.
+Qed.
+Hints Resolve even_mult_l even_mult_r odd_mult :arith.
+ 
+Lemma odd_mult_inv_l : (n,m:nat) (odd (mult n m)) -> (odd n).
+Proof.
+Intros n m H'.
+Case (even_mult_aux n m).
+Intros H'1 H'2; Elim H'1.
+Intros H'3; Elim H'3; Auto.
+Qed.
+ 
+Lemma odd_mult_inv_r : (n,m:nat) (odd (mult n m)) -> (odd m).
+Proof.
+Intros n m H'.
+Case (even_mult_aux n m).
+Intros H'1 H'2; Elim H'1.
+Intros H'3; Elim H'3; Auto.
+Qed.
+