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Diffstat (limited to 'theories7/Arith/Even.v')
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diff --git a/theories7/Arith/Even.v b/theories7/Arith/Even.v deleted file mode 100644 index bcc413f5..00000000 --- a/theories7/Arith/Even.v +++ /dev/null @@ -1,310 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(*i $Id: Even.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*) - -(** Here we define the predicates [even] and [odd] by mutual induction - and we prove the decidability and the exclusion of those predicates. - The main results about parity are proved in the module Div2. *) - -V7only [Import nat_scope.]. -Open Local Scope nat_scope. - -Implicit Variables Type m,n:nat. - -Inductive even : nat->Prop := - even_O : (even O) - | even_S : (n:nat)(odd n)->(even (S n)) -with odd : nat->Prop := - odd_S : (n:nat)(even n)->(odd (S n)). - -Hint constr_even : arith := Constructors even. -Hint constr_odd : arith := Constructors odd. - -Lemma even_or_odd : (n:nat) (even n)\/(odd n). -Proof. -NewInduction n. -Auto with arith. -Elim IHn; Auto with arith. -Qed. - -Lemma even_odd_dec : (n:nat) { (even n) }+{ (odd n) }. -Proof. -NewInduction n. -Auto with arith. -Elim IHn; Auto with arith. -Qed. - -Lemma not_even_and_odd : (n:nat) (even n) -> (odd n) -> False. -Proof. -NewInduction n. -Intros. Inversion H0. -Intros. Inversion H. Inversion H0. Auto with arith. -Qed. - -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. - |