Representation of adresses by the positive type of binary numbers
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Require Import Bool.
Require Import ZArith.
Inductive ad : Set :=
| ad_z : ad
| ad_x : positive -> ad.
Lemma ad_sum : forall a:ad, {p : positive | a = ad_x p} + {a = ad_z}.
Proof.
destruct a; auto.
left; exists p; trivial.
Qed.
Fixpoint p_xor (p p2:positive) {struct p} : ad :=
match p with
| xH =>
match p2 with
| xH => ad_z
| xO p'2 => ad_x (xI p'2)
| xI p'2 => ad_x (xO p'2)
end
| xO p' =>
match p2 with
| xH => ad_x (xI p')
| xO p'2 =>
match p_xor p' p'2 with
| ad_z => ad_z
| ad_x p'' => ad_x (xO p'')
end
| xI p'2 =>
match p_xor p' p'2 with
| ad_z => ad_x 1
| ad_x p'' => ad_x (xI p'')
end
end
| xI p' =>
match p2 with
| xH => ad_x (xO p')
| xO p'2 =>
match p_xor p' p'2 with
| ad_z => ad_x 1
| ad_x p'' => ad_x (xI p'')
end
| xI p'2 =>
match p_xor p' p'2 with
| ad_z => ad_z
| ad_x p'' => ad_x (xO p'')
end
end
end.
Definition ad_xor (a a':ad) :=
match a with
| ad_z => a'
| ad_x p => match a' with
| ad_z => a
| ad_x p' => p_xor p p'
end
end.
Lemma ad_xor_neutral_left : forall a:ad, ad_xor ad_z a = a.
Proof.
trivial.
Qed.
Lemma ad_xor_neutral_right : forall a:ad, ad_xor a ad_z = a.
Proof.
destruct a; trivial.
Qed.
Lemma ad_xor_comm : forall a a':ad, ad_xor a a' = ad_xor a' a.
Proof.
destruct a; destruct a'; simpl in |- *; auto.
generalize p0; clear p0; induction p as [p Hrecp| p Hrecp| ]; simpl in |- *;
auto.
destruct p0; simpl in |- *; trivial; intros.
rewrite Hrecp; trivial.
rewrite Hrecp; trivial.
destruct p0; simpl in |- *; trivial; intros.
rewrite Hrecp; trivial.
rewrite Hrecp; trivial.
destruct p0 as [p| p| ]; simpl in |- *; auto.
Qed.
Lemma ad_xor_nilpotent : forall a:ad, ad_xor a a = ad_z.
Proof.
destruct a; trivial.
simpl in |- *. induction p as [p IHp| p IHp| ]; trivial.
simpl in |- *. rewrite IHp; reflexivity.
simpl in |- *. rewrite IHp; reflexivity.
Qed.
Fixpoint ad_bit_1 (p:positive) : nat -> bool :=
match p with
| xH => fun n:nat => match n with
| O => true
| S _ => false
end
| xO p =>
fun n:nat => match n with
| O => false
| S n' => ad_bit_1 p n'
end
| xI p => fun n:nat => match n with
| O => true
| S n' => ad_bit_1 p n'
end
end.
Definition ad_bit (a:ad) :=
match a with
| ad_z => fun _:nat => false
| ad_x p => ad_bit_1 p
end.
Definition eqf (f g:nat -> bool) := forall n:nat, f n = g n.
Lemma ad_faithful_1 : forall a:ad, eqf (ad_bit ad_z) (ad_bit a) -> ad_z = a.
Proof.
destruct a. trivial.
induction p as [p IHp| p IHp| ]; intro H. absurd (ad_z = ad_x p). discriminate.
exact (IHp (fun n:nat => H (S n))).
absurd (ad_z = ad_x p). discriminate.
exact (IHp (fun n:nat => H (S n))).
absurd (false = true). discriminate.
exact (H 0).
Qed.
Lemma ad_faithful_2 :
forall a:ad, eqf (ad_bit (ad_x 1)) (ad_bit a) -> ad_x 1 = a.
Proof.
destruct a. intros. absurd (true = false). discriminate.
exact (H 0).
destruct p. intro H. absurd (ad_z = ad_x p). discriminate.
exact (ad_faithful_1 (ad_x p) (fun n:nat => H (S n))).
intros. absurd (true = false). discriminate.
exact (H 0).
trivial.
Qed.
Lemma ad_faithful_3 :
forall (a:ad) (p:positive),
(forall p':positive, eqf (ad_bit (ad_x p)) (ad_bit (ad_x p')) -> p = p') ->
eqf (ad_bit (ad_x (xO p))) (ad_bit a) -> ad_x (xO p) = a.
Proof.
destruct a. intros. cut (eqf (ad_bit ad_z) (ad_bit (ad_x (xO p)))).
intro. rewrite (ad_faithful_1 (ad_x (xO p)) H1). reflexivity.
unfold eqf in |- *. intro. unfold eqf in H0. rewrite H0. reflexivity.
case p. intros. absurd (false = true). discriminate.
exact (H0 0).
intros. rewrite (H p0 (fun n:nat => H0 (S n))). reflexivity.
intros. absurd (false = true). discriminate.
exact (H0 0).
Qed.
Lemma ad_faithful_4 :
forall (a:ad) (p:positive),
(forall p':positive, eqf (ad_bit (ad_x p)) (ad_bit (ad_x p')) -> p = p') ->
eqf (ad_bit (ad_x (xI p))) (ad_bit a) -> ad_x (xI p) = a.
Proof.
destruct a. intros. cut (eqf (ad_bit ad_z) (ad_bit (ad_x (xI p)))).
intro. rewrite (ad_faithful_1 (ad_x (xI p)) H1). reflexivity.
unfold eqf in |- *. intro. unfold eqf in H0. rewrite H0. reflexivity.
case p. intros. rewrite (H p0 (fun n:nat => H0 (S n))). reflexivity.
intros. absurd (true = false). discriminate.
exact (H0 0).
intros. absurd (ad_z = ad_x p0). discriminate.
cut (eqf (ad_bit (ad_x 1)) (ad_bit (ad_x (xI p0)))).
intro. exact (ad_faithful_1 (ad_x p0) (fun n:nat => H1 (S n))).
unfold eqf in |- *. unfold eqf in H0. intro. rewrite H0. reflexivity.
Qed.
Lemma ad_faithful : forall a a':ad, eqf (ad_bit a) (ad_bit a') -> a = a'.
Proof.
destruct a. exact ad_faithful_1.
induction p. intros a' H. apply ad_faithful_4. intros. cut (ad_x p = ad_x p').
intro. inversion H1. reflexivity.
exact (IHp (ad_x p') H0).
assumption.
intros. apply ad_faithful_3. intros. cut (ad_x p = ad_x p'). intro. inversion H1. reflexivity.
exact (IHp (ad_x p') H0).
assumption.
exact ad_faithful_2.
Qed.
Definition adf_xor (f g:nat -> bool) (n:nat) := xorb (f n) (g n).
Lemma ad_xor_sem_1 : forall a':ad, ad_bit (ad_xor ad_z a') 0 = ad_bit a' 0.
Proof.
trivial.
Qed.
Lemma ad_xor_sem_2 :
forall a':ad, ad_bit (ad_xor (ad_x 1) a') 0 = negb (ad_bit a' 0).
Proof.
intro. case a'. trivial.
simpl in |- *. intro.
case p; trivial.
Qed.
Lemma ad_xor_sem_3 :
forall (p:positive) (a':ad),
ad_bit (ad_xor (ad_x (xO p)) a') 0 = ad_bit a' 0.
Proof.
intros. case a'. trivial.
simpl in |- *. intro.
case p0; trivial. intro.
case (p_xor p p1); trivial.
intro. case (p_xor p p1); trivial.
Qed.
Lemma ad_xor_sem_4 :
forall (p:positive) (a':ad),
ad_bit (ad_xor (ad_x (xI p)) a') 0 = negb (ad_bit a' 0).
Proof.
intros. case a'. trivial.
simpl in |- *. intro. case p0; trivial. intro.
case (p_xor p p1); trivial.
intro.
case (p_xor p p1); trivial.
Qed.
Lemma ad_xor_sem_5 :
forall a a':ad, ad_bit (ad_xor a a') 0 = adf_xor (ad_bit a) (ad_bit a') 0.
Proof.
destruct a. intro. change (ad_bit a' 0 = xorb false (ad_bit a' 0)) in |- *. rewrite false_xorb. trivial.
case p. exact ad_xor_sem_4.
intros. change (ad_bit (ad_xor (ad_x (xO p0)) a') 0 = xorb false (ad_bit a' 0))
in |- *.
rewrite false_xorb. apply ad_xor_sem_3. exact ad_xor_sem_2.
Qed.
Lemma ad_xor_sem_6 :
forall n:nat,
(forall a a':ad, ad_bit (ad_xor a a') n = adf_xor (ad_bit a) (ad_bit a') n) ->
forall a a':ad,
ad_bit (ad_xor a a') (S n) = adf_xor (ad_bit a) (ad_bit a') (S n).
Proof.
intros. case a. unfold adf_xor in |- *. unfold ad_bit at 2 in |- *. rewrite false_xorb. reflexivity.
case a'. unfold adf_xor in |- *. unfold ad_bit at 3 in |- *. intro. rewrite xorb_false. reflexivity.
intros. case p0. case p. intros.
change
(ad_bit (ad_xor (ad_x (xI p2)) (ad_x (xI p1))) (S n) =
adf_xor (ad_bit (ad_x p2)) (ad_bit (ad_x p1)) n)
in |- *.
rewrite <- H. simpl in |- *.
case (p_xor p2 p1); trivial.
intros.
change
(ad_bit (ad_xor (ad_x (xI p2)) (ad_x (xO p1))) (S n) =
adf_xor (ad_bit (ad_x p2)) (ad_bit (ad_x p1)) n)
in |- *.
rewrite <- H. simpl in |- *.
case (p_xor p2 p1); trivial.
intro. unfold adf_xor in |- *. unfold ad_bit at 3 in |- *. unfold ad_bit_1 in |- *. rewrite xorb_false. reflexivity.
case p. intros.
change
(ad_bit (ad_xor (ad_x (xO p2)) (ad_x (xI p1))) (S n) =
adf_xor (ad_bit (ad_x p2)) (ad_bit (ad_x p1)) n)
in |- *.
rewrite <- H. simpl in |- *.
case (p_xor p2 p1); trivial.
intros.
change
(ad_bit (ad_xor (ad_x (xO p2)) (ad_x (xO p1))) (S n) =
adf_xor (ad_bit (ad_x p2)) (ad_bit (ad_x p1)) n)
in |- *.
rewrite <- H. simpl in |- *.
case (p_xor p2 p1); trivial.
intro. unfold adf_xor in |- *. unfold ad_bit at 3 in |- *. unfold ad_bit_1 in |- *. rewrite xorb_false. reflexivity.
unfold adf_xor in |- *. unfold ad_bit at 2 in |- *. unfold ad_bit_1 in |- *. rewrite false_xorb. simpl in |- *. case p; trivial.
Qed.
Lemma ad_xor_semantics :
forall a a':ad, eqf (ad_bit (ad_xor a a')) (adf_xor (ad_bit a) (ad_bit a')).
Proof.
unfold eqf in |- *. intros. generalize a a'. elim n. exact ad_xor_sem_5.
exact ad_xor_sem_6.
Qed.
Lemma eqf_sym : forall f f':nat -> bool, eqf f f' -> eqf f' f.
Proof.
unfold eqf in |- *. intros. rewrite H. reflexivity.
Qed.
Lemma eqf_refl : forall f:nat -> bool, eqf f f.
Proof.
unfold eqf in |- *. trivial.
Qed.
Lemma eqf_trans :
forall f f' f'':nat -> bool, eqf f f' -> eqf f' f'' -> eqf f f''.
Proof.
unfold eqf in |- *. intros. rewrite H. exact (H0 n).
Qed.
Lemma adf_xor_eq :
forall f f':nat -> bool, eqf (adf_xor f f') (fun n:nat => false) -> eqf f f'.
Proof.
unfold eqf in |- *. unfold adf_xor in |- *. intros. apply xorb_eq. apply H.
Qed.
Lemma ad_xor_eq : forall a a':ad, ad_xor a a' = ad_z -> a = a'.
Proof.
intros. apply ad_faithful. apply adf_xor_eq. apply eqf_trans with (f':= ad_bit (ad_xor a a')).
apply eqf_sym. apply ad_xor_semantics.
rewrite H. unfold eqf in |- *. trivial.
Qed.
Lemma adf_xor_assoc :
forall f f' f'':nat -> bool,
eqf (adf_xor (adf_xor f f') f'') (adf_xor f (adf_xor f' f'')).
Proof.
unfold eqf in |- *. unfold adf_xor in |- *. intros. apply xorb_assoc.
Qed.
Lemma eqf_xor_1 :
forall f f' f'' f''':nat -> bool,
eqf f f' -> eqf f'' f''' -> eqf (adf_xor f f'') (adf_xor f' f''').
Proof.
unfold eqf in |- *. intros. unfold adf_xor in |- *. rewrite H. rewrite H0. reflexivity.
Qed.
Lemma ad_xor_assoc :
forall a a' a'':ad, ad_xor (ad_xor a a') a'' = ad_xor a (ad_xor a' a'').
Proof.
intros. apply ad_faithful.
apply eqf_trans with
(f':= adf_xor (adf_xor (ad_bit a) (ad_bit a')) (ad_bit a'')).
apply eqf_trans with (f':= adf_xor (ad_bit (ad_xor a a')) (ad_bit a'')).
apply ad_xor_semantics.
apply eqf_xor_1. apply ad_xor_semantics.
apply eqf_refl.
apply eqf_trans with
(f':= adf_xor (ad_bit a) (adf_xor (ad_bit a') (ad_bit a''))).
apply adf_xor_assoc.
apply eqf_trans with (f':= adf_xor (ad_bit a) (ad_bit (ad_xor a' a''))).
apply eqf_xor_1. apply eqf_refl.
apply eqf_sym. apply ad_xor_semantics.
apply eqf_sym. apply ad_xor_semantics.
Qed.
Definition ad_double (a:ad) :=
match a with
| ad_z => ad_z
| ad_x p => ad_x (xO p)
end.
Definition ad_double_plus_un (a:ad) :=
match a with
| ad_z => ad_x 1
| ad_x p => ad_x (xI p)
end.
Definition ad_div_2 (a:ad) :=
match a with
| ad_z => ad_z
| ad_x xH => ad_z
| ad_x (xO p) => ad_x p
| ad_x (xI p) => ad_x p
end.
Lemma ad_double_div_2 : forall a:ad, ad_div_2 (ad_double a) = a.
Proof.
destruct a; trivial.
Qed.
Lemma ad_double_plus_un_div_2 :
forall a:ad, ad_div_2 (ad_double_plus_un a) = a.
Proof.
destruct a; trivial.
Qed.
Lemma ad_double_inj : forall a0 a1:ad, ad_double a0 = ad_double a1 -> a0 = a1.
Proof.
intros. rewrite <- (ad_double_div_2 a0). rewrite H. apply ad_double_div_2.
Qed.
Lemma ad_double_plus_un_inj :
forall a0 a1:ad, ad_double_plus_un a0 = ad_double_plus_un a1 -> a0 = a1.
Proof.
intros. rewrite <- (ad_double_plus_un_div_2 a0). rewrite H. apply ad_double_plus_un_div_2.
Qed.
Definition ad_bit_0 (a:ad) :=
match a with
| ad_z => false
| ad_x (xO _) => false
| _ => true
end.
Lemma ad_double_bit_0 : forall a:ad, ad_bit_0 (ad_double a) = false.
Proof.
destruct a; trivial.
Qed.
Lemma ad_double_plus_un_bit_0 :
forall a:ad, ad_bit_0 (ad_double_plus_un a) = true.
Proof.
destruct a; trivial.
Qed.
Lemma ad_div_2_double :
forall a:ad, ad_bit_0 a = false -> ad_double (ad_div_2 a) = a.
Proof.
destruct a. trivial. destruct p. intro H. discriminate H.
intros. reflexivity.
intro H. discriminate H.
Qed.
Lemma ad_div_2_double_plus_un :
forall a:ad, ad_bit_0 a = true -> ad_double_plus_un (ad_div_2 a) = a.
Proof.
destruct a. intro. discriminate H.
destruct p. intros. reflexivity.
intro H. discriminate H.
intro. reflexivity.
Qed.
Lemma ad_bit_0_correct : forall a:ad, ad_bit a 0 = ad_bit_0 a.
Proof.
destruct a; trivial.
destruct p; trivial.
Qed.
Lemma ad_div_2_correct :
forall (a:ad) (n:nat), ad_bit (ad_div_2 a) n = ad_bit a (S n).
Proof.
destruct a; trivial.
destruct p; trivial.
Qed.
Lemma ad_xor_bit_0 :
forall a a':ad, ad_bit_0 (ad_xor a a') = xorb (ad_bit_0 a) (ad_bit_0 a').
Proof.
intros. rewrite <- ad_bit_0_correct. rewrite (ad_xor_semantics a a' 0).
unfold adf_xor in |- *. rewrite ad_bit_0_correct. rewrite ad_bit_0_correct. reflexivity.
Qed.
Lemma ad_xor_div_2 :
forall a a':ad, ad_div_2 (ad_xor a a') = ad_xor (ad_div_2 a) (ad_div_2 a').
Proof.
intros. apply ad_faithful. unfold eqf in |- *. intro.
rewrite (ad_xor_semantics (ad_div_2 a) (ad_div_2 a') n).
rewrite ad_div_2_correct.
rewrite (ad_xor_semantics a a' (S n)).
unfold adf_xor in |- *. rewrite ad_div_2_correct. rewrite ad_div_2_correct.
reflexivity.
Qed.
Lemma ad_neg_bit_0 :
forall a a':ad,
ad_bit_0 (ad_xor a a') = true -> ad_bit_0 a = negb (ad_bit_0 a').
Proof.
intros. rewrite <- true_xorb. rewrite <- H. rewrite ad_xor_bit_0.
rewrite xorb_assoc. rewrite xorb_nilpotent. rewrite xorb_false. reflexivity.
Qed.
Lemma ad_neg_bit_0_1 :
forall a a':ad, ad_xor a a' = ad_x 1 -> ad_bit_0 a = negb (ad_bit_0 a').
Proof.
intros. apply ad_neg_bit_0. rewrite H. reflexivity.
Qed.
Lemma ad_neg_bit_0_2 :
forall (a a':ad) (p:positive),
ad_xor a a' = ad_x (xI p) -> ad_bit_0 a = negb (ad_bit_0 a').
Proof.
intros. apply ad_neg_bit_0. rewrite H. reflexivity.
Qed.
Lemma ad_same_bit_0 :
forall (a a':ad) (p:positive),
ad_xor a a' = ad_x (xO p) -> ad_bit_0 a = ad_bit_0 a'.
Proof.
intros. rewrite <- (xorb_false (ad_bit_0 a)). cut (ad_bit_0 (ad_x (xO p)) = false).
intro. rewrite <- H0. rewrite <- H. rewrite ad_xor_bit_0. rewrite <- xorb_assoc.
rewrite xorb_nilpotent. rewrite false_xorb. reflexivity.
reflexivity.
Qed.