Coq.IntMap.Addr

# Module Coq.IntMap.Addr

``

Representation of adresses by the `positive` type of binary numbers

`

`Require` Bool.
`Require` ZArith.

`Inductive` ad : Set :=
ad_z : ad
| ad_x : positive -> ad.

`Lemma` ad_sum : (a:ad) {p:positive | a=(ad_x p)}+{a=ad_z}.
`Proof`.
NewDestruct a; Auto.
Left; Exists p; Trivial.
`Qed`.

`Fixpoint` p_xor [p:positive] : positive -> ad :=
[p2] Cases p of
xH => Cases p2 of
xH => ad_z
| (xO p'2) => (ad_x (xI p'2))
| (xI p'2) => (ad_x (xO p'2))
end
| (xO p') => Cases p2 of
xH => (ad_x (xI p'))
| (xO p'2) => Cases (p_xor p' p'2) of
ad_z => ad_z
| (ad_x p'') => (ad_x (xO p''))
end
| (xI p'2) => Cases (p_xor p' p'2) of
ad_z => (ad_x xH)
| (ad_x p'') => (ad_x (xI p''))
end
end
| (xI p') => Cases p2 of
xH => (ad_x (xO p'))
| (xO p'2) => Cases (p_xor p' p'2) of
ad_z => (ad_x xH)
| (ad_x p'') => (ad_x (xI p''))
end
| (xI p'2) => Cases (p_xor p' p'2) of
ad_z => ad_z
| (ad_x p'') => (ad_x (xO p''))
end
end
end.

`Definition` ad_xor := [a,a':ad]
Cases a of
ad_z => a'
| (ad_x p) => Cases a' of
ad_z => a
| (ad_x p') => (p_xor p p')
end
end.

`Lemma` ad_xor_neutral_left : (a:ad) (ad_xor ad_z a)=a.
`Proof`.
Trivial.
`Qed`.

`Lemma` ad_xor_neutral_right : (a:ad) (ad_xor a ad_z)=a.
`Proof`.
Induction a; Trivial.
`Qed`.

`Lemma` ad_xor_comm : (a,a':ad) (ad_xor a a')=(ad_xor a' a).
`Proof`.
NewDestruct a; NewDestruct a'; Simpl; Auto.
Generalize p0; Clear p0; Induction p; Simpl; Auto.
NewDestruct p0; Simpl; Trivial; Intros.
Rewrite Hrecp; Trivial.
Rewrite Hrecp; Trivial.
NewDestruct p0; Simpl; Trivial; Intros.
Rewrite Hrecp; Trivial.
Rewrite Hrecp; Trivial.
Induction p0; Simpl; Auto.
`Qed`.

`Lemma` ad_xor_nilpotent : (a:ad) (ad_xor a a)=ad_z.
`Proof`.
Induction a; Trivial.
Induction p; Trivial.
Simpl; Intros. Rewrite H; Reflexivity.
Simpl; Intros. Rewrite H; Reflexivity.
`Qed`.

`Fixpoint` ad_bit_1 [p:positive] : nat -> bool :=
Cases p of
xH => [n:nat] Cases n of
O => true
| (S _) => false
end
| (xO p) => [n:nat] Cases n of
O => false
| (S n') => (ad_bit_1 p n')
end
| (xI p) => [n:nat] Cases n of
O => true
| (S n') => (ad_bit_1 p n')
end
end.

`Definition` ad_bit := [a:ad]
Cases a of
ad_z => [_:nat] false
| (ad_x p) => (ad_bit_1 p)
end.

`Definition` eqf := [f,g:nat->bool] (n:nat) (f n)=(g n).

`Lemma` ad_faithful_1 : (a:ad) (eqf (ad_bit ad_z) (ad_bit a)) -> ad_z=a.
`Proof`.
Induction a. Trivial.
Induction p. Intros. Absurd ad_z=(ad_x p0). Discriminate.
Exact (H [n:nat](H0 (S n))).
Intros. Absurd ad_z=(ad_x p0). Discriminate.
Exact (H [n:nat](H0 (S n))).
Intros. Absurd false=true. Discriminate.
Exact (H O).
`Qed`.

`Lemma` ad_faithful_2 : (a:ad) (eqf (ad_bit (ad_x xH)) (ad_bit a)) -> (ad_x xH)=a.
`Proof`.
Induction a. Intros. Absurd true=false. Discriminate.
Exact (H O).
Induction p. Intros. Absurd ad_z=(ad_x p0). Discriminate.
Exact (ad_faithful_1 (ad_x p0) [n:nat](H0 (S n))).
Intros. Absurd true=false. Discriminate.
Exact (H0 O).
Trivial.
`Qed`.

`Lemma` ad_faithful_3 :
(a:ad) (p:positive)
((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`.
Induction 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. Intro. Unfold eqf in H0. Rewrite H0. Reflexivity.
Intro. Case p. Intros. Absurd false=true. Discriminate.
Exact (H0 O).
Intros. Rewrite (H p0 [n:nat](H0 (S n))). Reflexivity.
Intros. Absurd false=true. Discriminate.
Exact (H0 O).
`Qed`.

`Lemma` ad_faithful_4 :
(a:ad) (p:positive)
((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`.
Induction 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. Intro. Unfold eqf in H0. Rewrite H0. Reflexivity.
Intro. Case p. Intros. Rewrite (H p0 [n:nat](H0 (S n))). Reflexivity.
Intros. Absurd true=false. Discriminate.
Exact (H0 O).
Intros. Absurd ad_z=(ad_x p0). Discriminate.
Cut (eqf (ad_bit (ad_x xH)) (ad_bit (ad_x (xI p0)))).
Intro. Exact (ad_faithful_1 (ad_x p0) [n:nat](H1 (S n))).
Unfold eqf. Unfold eqf in H0. Intro. Rewrite H0. Reflexivity.
`Qed`.

`Lemma` ad_faithful : (a,a':ad) (eqf (ad_bit a) (ad_bit a')) -> a=a'.
`Proof`.
Induction a. Exact ad_faithful_1.
Induction p. Intros. Apply ad_faithful_4. Intros. Cut (ad_x p0)=(ad_x p').
Intro. Inversion H2. Reflexivity.
Exact (H (ad_x p') H1).
Assumption.
Intros. Apply ad_faithful_3. Intros. Cut (ad_x p0)=(ad_x p'). Intro. Inversion H2. Reflexivity.
Exact (H (ad_x p') H1).
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 : (a':ad) (ad_bit (ad_xor ad_z a') O)=(ad_bit a' O).
`Proof`.
Trivial.
`Qed`.

`Lemma` ad_xor_sem_2 : (a':ad) (ad_bit (ad_xor (ad_x xH) a') O)=(negb (ad_bit a' O)).
`Proof`.
Intro. Case a'. Trivial.
Simpl. Intro.
Case p; Trivial.
`Qed`.

`Lemma` ad_xor_sem_3 :
(p:positive) (a':ad) (ad_bit (ad_xor (ad_x (xO p)) a') O)=(ad_bit a' O).
`Proof`.
Intros. Case a'. Trivial.
Simpl. Intro.
Case p0; Trivial. Intro.
Case (p_xor p p1); Trivial.
Intro. Case (p_xor p p1); Trivial.
`Qed`.

`Lemma` ad_xor_sem_4 : (p:positive) (a':ad)
(ad_bit (ad_xor (ad_x (xI p)) a') O)=(negb (ad_bit a' O)).
`Proof`.
Intros. Case a'. Trivial.
Simpl. Intro. Case p0; Trivial. Intro.
Case (p_xor p p1); Trivial.
Intro.
Case (p_xor p p1); Trivial.
`Qed`.

`Lemma` ad_xor_sem_5 :
(a,a':ad) (ad_bit (ad_xor a a') O)=(adf_xor (ad_bit a) (ad_bit a') O).
`Proof`.
Induction a. Intro. Change (ad_bit a' O)=(xorb false (ad_bit a' O)). Rewrite false_xorb. Trivial.
Intro. Case p. Exact ad_xor_sem_4.
Intros. Change (ad_bit (ad_xor (ad_x (xO p0)) a') O)=(xorb false (ad_bit a' O)).
Rewrite false_xorb. Apply ad_xor_sem_3. Exact ad_xor_sem_2.
`Qed`.

`Lemma` ad_xor_sem_6 : (n:nat)
((a,a':ad) (ad_bit (ad_xor a a') n)=(adf_xor (ad_bit a) (ad_bit a') n)) ->
(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. Unfold 2 ad_bit. Rewrite false_xorb. Reflexivity.
Case a'. Unfold adf_xor. Unfold 3 ad_bit. 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).
Rewrite <- H. Simpl.
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).
Rewrite <- H. Simpl.
Case (p_xor p2 p1); Trivial.
Intro. Unfold adf_xor. Unfold 3 ad_bit. Unfold ad_bit_1. 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).
Rewrite <- H. Simpl.
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).
Rewrite <- H. Simpl.
Case (p_xor p2 p1); Trivial.
Intro. Unfold adf_xor. Unfold 3 ad_bit. Unfold ad_bit_1. Rewrite xorb_false. Reflexivity.
Unfold adf_xor. Unfold 2 ad_bit. Unfold ad_bit_1. Rewrite false_xorb. Simpl. Case p; Trivial.
`Qed`.

`Lemma` ad_xor_semantics :
(a,a':ad) (eqf (ad_bit (ad_xor a a')) (adf_xor (ad_bit a) (ad_bit a'))).
`Proof`.
Unfold eqf. Intros. Generalize a a'. Elim n. Exact ad_xor_sem_5.
Exact ad_xor_sem_6.
`Qed`.

`Lemma` eqf_sym : (f,f':nat->bool) (eqf f f') -> (eqf f' f).
`Proof`.
Unfold eqf. Intros. Rewrite H. Reflexivity.
`Qed`.

`Lemma` eqf_refl : (f:nat->bool) (eqf f f).
`Proof`.
Unfold eqf. Trivial.
`Qed`.

`Lemma` eqf_trans : (f,f',f'':nat->bool) (eqf f f') -> (eqf f' f'') -> (eqf f f'').
`Proof`.
Unfold eqf. Intros. Rewrite H. Exact (H0 n).
`Qed`.

`Lemma` adf_xor_eq : (f,f':nat->bool) (eqf (adf_xor f f') [n:nat] false) -> (eqf f f').
`Proof`.
Unfold eqf. Unfold adf_xor. Intros. Apply xorb_eq. Apply H.
`Qed`.

`Lemma` ad_xor_eq : (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. Trivial.
`Qed`.

`Lemma` adf_xor_assoc : (f,f',f'':nat->bool)
(eqf (adf_xor (adf_xor f f') f'') (adf_xor f (adf_xor f' f''))).
`Proof`.
Unfold eqf. Unfold adf_xor. Intros. Apply xorb_assoc.
`Qed`.

`Lemma` eqf_xor_1 : (f,f',f'',f''':nat->bool) (eqf f f') -> (eqf f'' f''') ->
(eqf (adf_xor f f'') (adf_xor f' f''')).
`Proof`.
Unfold eqf. Intros. Unfold adf_xor. Rewrite H. Rewrite H0. Reflexivity.
`Qed`.

`Lemma` ad_xor_assoc :
(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]
Cases a of
ad_z => ad_z
| (ad_x p) => (ad_x (xO p))
end.

`Definition` ad_double_plus_un := [a:ad]
Cases a of
ad_z => (ad_x xH)
| (ad_x p) => (ad_x (xI p))
end.

`Definition` ad_div_2 := [a:ad]
Cases a of
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 : (a:ad) (ad_div_2 (ad_double a))=a.
`Proof`.
Induction a; Trivial.
`Qed`.

`Lemma` ad_double_plus_un_div_2 : (a:ad) (ad_div_2 (ad_double_plus_un a))=a.
`Proof`.
Induction a; Trivial.
`Qed`.

`Lemma` ad_double_inj : (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 :
(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]
Cases a of
ad_z => false
| (ad_x (xO _)) => false
| _ => true
end.

`Lemma` ad_double_bit_0 : (a:ad) (ad_bit_0 (ad_double a))=false.
`Proof`.
Induction a; Trivial.
`Qed`.

`Lemma` ad_double_plus_un_bit_0 : (a:ad) (ad_bit_0 (ad_double_plus_un a))=true.
`Proof`.
Induction a; Trivial.
`Qed`.

`Lemma` ad_div_2_double : (a:ad) (ad_bit_0 a)=false -> (ad_double (ad_div_2 a))=a.
`Proof`.
Induction a. Trivial. Induction p. Intros. Discriminate H0.
Intros. Reflexivity.
Intro. Discriminate H.
`Qed`.

`Lemma` ad_div_2_double_plus_un :
(a:ad) (ad_bit_0 a)=true -> (ad_double_plus_un (ad_div_2 a))=a.
`Proof`.
Induction a. Intro. Discriminate H.
Induction p. Intros. Reflexivity.
Intros. Discriminate H0.
Intro. Reflexivity.
`Qed`.

`Lemma` ad_bit_0_correct : (a:ad) (ad_bit a O)=(ad_bit_0 a).
`Proof`.
Induction a; Trivial.
Induction p; Trivial.
`Qed`.

`Lemma` ad_div_2_correct : (a:ad) (n:nat) (ad_bit (ad_div_2 a) n)=(ad_bit a (S n)).
`Proof`.
Induction a; Trivial.
Induction p; Trivial.
`Qed`.

`Lemma` ad_xor_bit_0 :
(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' O).
Unfold adf_xor. Rewrite ad_bit_0_correct. Rewrite ad_bit_0_correct. Reflexivity.
`Qed`.

`Lemma` ad_xor_div_2 :
(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. 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. Rewrite ad_div_2_correct. Rewrite ad_div_2_correct.
Reflexivity.
`Qed`.

`Lemma` ad_neg_bit_0 : (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 :
(a,a':ad) (ad_xor a a')=(ad_x xH) -> (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 : (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 : (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`.

`

Index