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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** Bitwise operations for ZArith *)
Require Import Bool BinPos BinNat BinInt Znat Zeven Zpow_def
Zorder Zcompare.
Local Open Scope Z_scope.
Notation Ztestbit := Z.testbit (only parsing).
Notation Zshiftr := Z.shiftr (only parsing).
Notation Zshiftl := Z.shiftl (only parsing).
Notation Zor := Z.lor (only parsing).
Notation Zand := Z.land (only parsing).
Notation Zdiff := Z.ldiff (only parsing).
Notation Zxor := Z.lxor (only parsing).
(** Conversions between [Ztestbit] and [Ntestbit] *)
Lemma Ztestbit_of_N : forall a n,
Ztestbit (Z_of_N a) (Z_of_N n) = N.testbit a n.
Proof.
intros [ |a] [ |n]; simpl; trivial. now destruct a.
Qed.
Lemma Ztestbit_of_N' : forall a n, 0<=n ->
Ztestbit (Z_of_N a) n = N.testbit a (Zabs_N n).
Proof.
intros. now rewrite <- Ztestbit_of_N, Z_of_N_abs, Zabs_eq.
Qed.
Lemma Ztestbit_Zpos : forall a n, 0<=n ->
Ztestbit (Zpos a) n = N.testbit (Npos a) (Zabs_N n).
Proof.
intros. now rewrite <- Ztestbit_of_N'.
Qed.
Lemma Ztestbit_Zneg : forall a n, 0<=n ->
Ztestbit (Zneg a) n = negb (N.testbit (Pos.pred_N a) (Zabs_N n)).
Proof.
intros a n Hn.
rewrite <- Ztestbit_of_N' by trivial.
destruct n as [ |n|n];
[ | simpl; now destruct (Ppred_N a) | now destruct Hn].
unfold Ztestbit.
replace (Z_of_N (Pos.pred_N a)) with (Zpred (Zpos a))
by (destruct a; trivial).
now rewrite Zodd_bool_pred, <- Zodd_even_bool.
Qed.
(** Proofs of specifications *)
Lemma Zdiv2_spec : forall a, Zdiv2 a = Zshiftr a 1.
Proof.
reflexivity.
Qed.
Lemma Ztestbit_0_l : forall n, Ztestbit 0 n = false.
Proof.
now destruct n.
Qed.
Lemma Ztestbit_neg_r : forall a n, n<0 -> Ztestbit a n = false.
Proof.
now destruct n.
Qed.
Lemma Ztestbit_odd_0 a : Ztestbit (2*a+1) 0 = true.
Proof.
now destruct a as [|a|[a|a|]].
Qed.
Lemma Ztestbit_even_0 a : Ztestbit (2*a) 0 = false.
Proof.
now destruct a.
Qed.
Lemma Ztestbit_odd_succ a n : 0<=n ->
Ztestbit (2*a+1) (Zsucc n) = Ztestbit a n.
Proof.
destruct n as [|n|n]; (now destruct 1) || intros _.
destruct a as [|[a|a|]|[a|a|]]; simpl; trivial. now destruct a.
unfold Ztestbit. rewrite <- Zpos_succ_morphism.
destruct a as [|a|[a|a|]]; simpl; trivial;
rewrite ?Pos.pred_N_succ; now destruct n.
Qed.
Lemma Ztestbit_even_succ a n : 0<=n ->
Ztestbit (2*a) (Zsucc n) = Ztestbit a n.
Proof.
destruct n as [|n|n]; (now destruct 1) || intros _.
destruct a as [|[a|a|]|[a|a|]]; simpl; trivial. now destruct a.
unfold Ztestbit. rewrite <- Zpos_succ_morphism.
destruct a as [|a|[a|a|]]; simpl; trivial;
rewrite ?Pos.pred_N_succ; now destruct n.
Qed.
Lemma Ppred_div2_up : forall p,
Pos.pred_N (Pos.div2_up p) = N.div2 (Pos.pred_N p).
Proof.
intros [p|p| ]; trivial.
simpl. apply Pos.pred_N_succ.
destruct p; simpl; trivial.
Qed.
Lemma Z_of_N_div2 : forall n, Z_of_N (N.div2 n) = Zdiv2 (Z_of_N n).
Proof.
intros [|p]; trivial. now destruct p.
Qed.
Lemma Z_of_N_quot2 : forall n, Z_of_N (N.div2 n) = Zquot2 (Z_of_N n).
Proof.
intros [|p]; trivial. now destruct p.
Qed.
(** Auxiliary results about right shift on positive numbers *)
Lemma Ppred_Pshiftl_low : forall p n m, (m<n)%N ->
N.testbit (Pos.pred_N (Pos.shiftl p n)) m = true.
Proof.
induction n using N.peano_ind.
now destruct m.
intros m H. unfold Pos.shiftl.
destruct n as [|n]; simpl in *.
destruct m. now destruct p. elim (Pos.nlt_1_r _ H).
rewrite Pos.iter_succ. simpl.
set (u:=Pos.iter n xO p) in *; clearbody u.
destruct m as [|m]. now destruct u.
rewrite <- (IHn (Pos.pred_N m)).
rewrite <- (N.testbit_odd_succ _ (Pos.pred_N m)).
rewrite N.succ_pos_pred. now destruct u.
apply N.le_0_l.
apply N.succ_lt_mono. now rewrite N.succ_pos_pred.
Qed.
Lemma Ppred_Pshiftl_high : forall p n m, (n<=m)%N ->
N.testbit (Pos.pred_N (Pos.shiftl p n)) m =
N.testbit (N.shiftl (Pos.pred_N p) n) m.
Proof.
induction n using N.peano_ind; intros m H.
unfold N.shiftl. simpl. now destruct (Pos.pred_N p).
rewrite N.shiftl_succ_r.
destruct n as [|n].
destruct m as [|m]. now destruct H. now destruct p.
destruct m as [|m]. now destruct H.
rewrite <- (N.succ_pos_pred m).
rewrite N.double_spec, N.testbit_even_succ by apply N.le_0_l.
rewrite <- IHn.
rewrite N.testbit_succ_r_div2 by apply N.le_0_l.
f_equal. simpl. rewrite Pos.iter_succ.
now destruct (Pos.iter n xO p).
apply N.succ_le_mono. now rewrite N.succ_pos_pred.
Qed.
(** Correctness proofs about [Zshiftr] and [Zshiftl] *)
Lemma Zshiftr_spec_aux : forall a n m, 0<=n -> 0<=m ->
Ztestbit (Zshiftr a n) m = Ztestbit a (m+n).
Proof.
intros a n m Hn Hm. unfold Zshiftr.
destruct n as [ |n|n]; (now destruct Hn) || clear Hn; simpl.
now rewrite Zplus_0_r.
destruct a as [ |a|a].
(* a = 0 *)
replace (iter_pos n _ Zdiv2 0) with 0
by (apply iter_pos_invariant; intros; subst; trivial).
now rewrite 2 Ztestbit_0_l.
(* a > 0 *)
rewrite <- (Z_of_N_pos a) at 1.
rewrite <- (iter_pos_swap_gen _ _ _ Ndiv2) by exact Z_of_N_div2.
rewrite Ztestbit_Zpos, Ztestbit_of_N'; trivial.
rewrite Zabs_N_plus; try easy. simpl Zabs_N.
exact (N.shiftr_spec' (Npos a) (Npos n) (Zabs_N m)).
now apply Zplus_le_0_compat.
(* a < 0 *)
rewrite <- (iter_pos_swap_gen _ _ _ Pdiv2_up) by trivial.
rewrite 2 Ztestbit_Zneg; trivial. f_equal.
rewrite Zabs_N_plus; try easy. simpl Zabs_N.
rewrite (iter_pos_swap_gen _ _ _ _ Ndiv2) by exact Ppred_div2_up.
exact (N.shiftr_spec' (Ppred_N a) (Npos n) (Zabs_N m)).
now apply Zplus_le_0_compat.
Qed.
Lemma Zshiftl_spec_low : forall a n m, m<n ->
Ztestbit (Zshiftl a n) m = false.
Proof.
intros a [ |n|n] [ |m|m] H; try easy; simpl Zshiftl.
rewrite iter_nat_of_P.
destruct (nat_of_P_is_S n) as (n' & ->).
simpl. now destruct (iter_nat n' _ (Zmult 2) a).
destruct a as [ |a|a].
(* a = 0 *)
replace (iter_pos n _ (Zmult 2) 0) with 0
by (apply iter_pos_invariant; intros; subst; trivial).
apply Ztestbit_0_l.
(* a > 0 *)
rewrite <- (iter_pos_swap_gen _ _ _ xO) by trivial.
rewrite Ztestbit_Zpos by easy.
exact (N.shiftl_spec_low (Npos a) (Npos n) (Npos m) H).
(* a < 0 *)
rewrite <- (iter_pos_swap_gen _ _ _ xO) by trivial.
rewrite Ztestbit_Zneg by easy.
now rewrite (Ppred_Pshiftl_low a (Npos n)).
Qed.
Lemma Zshiftl_spec_high : forall a n m, 0<=m -> n<=m ->
Ztestbit (Zshiftl a n) m = Ztestbit a (m-n).
Proof.
intros a n m Hm H.
destruct n as [ |n|n]. simpl. now rewrite Zminus_0_r.
(* n > 0 *)
destruct m as [ |m|m]; try (now destruct H).
assert (0 <= Zpos m - Zpos n) by (now apply Zle_minus_le_0).
assert (EQ : Zabs_N (Zpos m - Zpos n) = (Npos m - Npos n)%N).
apply Z_of_N_eq_rev. rewrite Z_of_N_abs, Zabs_eq by trivial.
now rewrite Z_of_N_minus, !Z_of_N_pos, Zmax_r.
destruct a; unfold Zshiftl.
(* ... a = 0 *)
replace (iter_pos n _ (Zmult 2) 0) with 0
by (apply iter_pos_invariant; intros; subst; trivial).
now rewrite 2 Ztestbit_0_l.
(* ... a > 0 *)
rewrite <- (iter_pos_swap_gen _ _ _ xO) by trivial.
rewrite 2 Ztestbit_Zpos, EQ by easy.
exact (N.shiftl_spec_high' (Npos p) (Npos n) (Npos m) H).
(* ... a < 0 *)
rewrite <- (iter_pos_swap_gen _ _ _ xO) by trivial.
rewrite 2 Ztestbit_Zneg, EQ by easy. f_equal.
simpl Zabs_N.
rewrite <- N.shiftl_spec_high by easy.
now apply (Ppred_Pshiftl_high p (Npos n)).
(* n < 0 *)
unfold Zminus. simpl.
now apply (Zshiftr_spec_aux a (Zpos n) m).
Qed.
Lemma Zshiftr_spec : forall a n m, 0<=m ->
Ztestbit (Zshiftr a n) m = Ztestbit a (m+n).
Proof.
intros a n m Hm.
destruct (Zle_or_lt 0 n).
now apply Zshiftr_spec_aux.
destruct (Zle_or_lt (-n) m).
unfold Zshiftr.
rewrite (Zshiftl_spec_high a (-n) m); trivial.
unfold Zminus. now rewrite Zopp_involutive.
unfold Zshiftr.
rewrite (Zshiftl_spec_low a (-n) m); trivial.
rewrite Ztestbit_neg_r; trivial.
now apply Zlt_plus_swap.
Qed.
(** Correctness proofs for bitwise operations *)
Lemma Zor_spec : forall a b n,
Ztestbit (Zor a b) n = Ztestbit a n || Ztestbit b n.
Proof.
intros a b n.
destruct (Zle_or_lt 0 n) as [Hn|Hn]; [|now rewrite !Ztestbit_neg_r].
destruct a as [ |a|a], b as [ |b|b];
rewrite ?Ztestbit_0_l, ?orb_false_r; trivial; unfold Zor;
rewrite ?Ztestbit_Zpos, ?Ztestbit_Zneg, ?Ppred_Nsucc
by trivial.
now rewrite <- N.lor_spec.
now rewrite N.ldiff_spec, negb_andb, negb_involutive, orb_comm.
now rewrite N.ldiff_spec, negb_andb, negb_involutive.
now rewrite N.land_spec, negb_andb.
Qed.
Lemma Zand_spec : forall a b n,
Ztestbit (Zand a b) n = Ztestbit a n && Ztestbit b n.
Proof.
intros a b n.
destruct (Zle_or_lt 0 n) as [Hn|Hn]; [|now rewrite !Ztestbit_neg_r].
destruct a as [ |a|a], b as [ |b|b];
rewrite ?Ztestbit_0_l, ?andb_false_r; trivial; unfold Zand;
rewrite ?Ztestbit_Zpos, ?Ztestbit_Zneg, ?Ztestbit_of_N', ?Ppred_Nsucc
by trivial.
now rewrite <- N.land_spec.
now rewrite N.ldiff_spec.
now rewrite N.ldiff_spec, andb_comm.
now rewrite N.lor_spec, negb_orb.
Qed.
Lemma Zdiff_spec : forall a b n,
Ztestbit (Zdiff a b) n = Ztestbit a n && negb (Ztestbit b n).
Proof.
intros a b n.
destruct (Zle_or_lt 0 n) as [Hn|Hn]; [|now rewrite !Ztestbit_neg_r].
destruct a as [ |a|a], b as [ |b|b];
rewrite ?Ztestbit_0_l, ?andb_true_r; trivial; unfold Zdiff;
rewrite ?Ztestbit_Zpos, ?Ztestbit_Zneg, ?Ztestbit_of_N', ?Ppred_Nsucc
by trivial.
now rewrite <- N.ldiff_spec.
now rewrite N.land_spec, negb_involutive.
now rewrite N.lor_spec, negb_orb.
now rewrite N.ldiff_spec, negb_involutive, andb_comm.
Qed.
Lemma Zxor_spec : forall a b n,
Ztestbit (Zxor a b) n = xorb (Ztestbit a n) (Ztestbit b n).
Proof.
intros a b n.
destruct (Zle_or_lt 0 n) as [Hn|Hn]; [|now rewrite !Ztestbit_neg_r].
destruct a as [ |a|a], b as [ |b|b];
rewrite ?Ztestbit_0_l, ?xorb_false_l, ?xorb_false_r; trivial; unfold Zxor;
rewrite ?Ztestbit_Zpos, ?Ztestbit_Zneg, ?Ztestbit_of_N', ?Ppred_Nsucc
by trivial.
now rewrite <- N.lxor_spec.
now rewrite N.lxor_spec, negb_xorb_r.
now rewrite N.lxor_spec, negb_xorb_l.
now rewrite N.lxor_spec, xorb_negb_negb.
Qed.
(** An additionnal proof concerning [Pshiftl_nat], used in BigN *)
Lemma Pshiftl_nat_Zpower : forall n p,
Zpos (Pos.shiftl_nat p n) = Zpos p * 2 ^ Z_of_nat n.
Proof.
intros.
rewrite Zmult_comm.
induction n. simpl; auto.
transitivity (2 * (2 ^ Z_of_nat n * Zpos p)).
rewrite <- IHn. auto.
rewrite Zmult_assoc.
rewrite inj_S.
rewrite <- Zpower_succ_r; auto with zarith.
apply Zle_0_nat.
Qed.
|
