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.
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Open Local Scope nat_scope.
Implicit Types m n : nat.
Inductive even : nat -> Prop :=
| even_O : even 0
| even_S : forall n, odd n -> even (S n)
with odd : nat -> Prop :=
odd_S : forall n, even n -> odd (S n).
Hint Constructors even: arith.
Hint Constructors odd: arith.
Lemma even_or_odd : forall n, even n \/ odd n.
Proof.
induction n.
auto with arith.
elim IHn; auto with arith.
Qed.
Lemma even_odd_dec : forall n, {even n} + {odd n}.
Proof.
induction n.
auto with arith.
elim IHn; auto with arith.
Qed.
Lemma not_even_and_odd : forall n, even n -> odd n -> False.
Proof.
induction n.
intros. inversion H0.
intros. inversion H. inversion H0. auto with arith.
Qed.
Lemma even_plus_aux :
forall n m,
(odd (n + m) <-> odd n /\ even m \/ even n /\ odd m) /\
(even (n + m) <-> even n /\ even m \/ odd n /\ odd m).
Proof.
intros n; elim n; simpl in |- *; 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 : forall n m, even n -> even m -> even (n + m).
Proof.
intros n m; case (even_plus_aux n m).
intros H H0; case H0; auto.
Qed.
Lemma odd_even_plus : forall n m, odd n -> odd m -> even (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 : forall n m, even (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 : forall n m, even (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 : forall n m, even (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 : forall n m, even (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.
Hint Resolve even_even_plus odd_even_plus: arith.
Lemma odd_plus_l : forall n m, odd n -> even m -> odd (n + m).
Proof.
intros n m; case (even_plus_aux n m).
intros H; case H; auto.
Qed.
Lemma odd_plus_r : forall n m, even n -> odd m -> odd (n + m).
Proof.
intros n m; case (even_plus_aux n m).
intros H; case H; auto.
Qed.
Lemma odd_plus_even_inv_l : forall n m, odd (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 : forall n m, odd (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 : forall n m, odd (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 : forall n m, odd (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.
Hint Resolve odd_plus_l odd_plus_r: arith.
Lemma even_mult_aux :
forall n m,
(odd (n * m) <-> odd n /\ odd m) /\ (even (n * m) <-> even n \/ even m).
Proof.
intros n; elim n; simpl in |- *; 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 (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 (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 (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 (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 : forall n m, even n -> even (n * m).
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.
intros H H0; case H0; auto.
Qed.
Hint Resolve even_mult_l even_mult_r: arith.
Lemma even_mult_inv_r : forall n m, even (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 : forall n m, even (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 : 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'.
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 : forall n m, odd (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.