Library Coq.Numbers.Natural.Abstract.NBits


Require Import Bool NAxioms NSub NPow NDiv NParity NLog.

Derived properties of bitwise operations

Module Type NBitsProp
 (Import A : NAxiomsSig')
 (Import B : NSubProp A)
 (Import C : NParityProp A B)
 (Import D : NPowProp A B C)
 (Import E : NDivProp A B)
 (Import F : NLog2Prop A B C D).

Include BoolEqualityFacts A.

Ltac order_nz := try apply pow_nonzero; order'.
Hint Rewrite div_0_l mod_0_l div_1_r mod_1_r : nz.

Some properties of power and division

Lemma pow_sub_r : forall a b c, a~=0 -> c<=b -> a^(b-c) == a^b / a^c.
Proof.
 intros a b c Ha H.
 apply div_unique with 0.
 generalize (pow_nonzero a c Ha) (le_0_l (a^c)); order'.
 nzsimpl. now rewrite <- pow_add_r, add_comm, sub_add.
Qed.

Lemma pow_div_l : forall a b c, b~=0 -> a mod b == 0 ->
 (a/b)^c == a^c / b^c.
Proof.
 intros a b c Hb H.
 apply div_unique with 0.
 generalize (pow_nonzero b c Hb) (le_0_l (b^c)); order'.
 nzsimpl. rewrite <- pow_mul_l. f_equiv. now apply div_exact.
Qed.

An injection from bits true and false to numbers 1 and 0. We declare it as a (local) coercion for shorter statements.

Definition b2n (b:bool) := if b then 1 else 0.
Local Coercion b2n : bool >-> t.

Instance b2n_proper : Proper (Logic.eq ==> eq) b2n.
Proof. solve_proper. Qed.

Lemma exists_div2 a : exists a' (b:bool), a == 2*a' + b.
Proof.
 elim (Even_or_Odd a); [intros (a',H)| intros (a',H)].
 exists a'. exists false. now nzsimpl.
 exists a'. exists true. now simpl.
Qed.

We can compact testbit_odd_0 testbit_even_0 testbit_even_succ testbit_odd_succ in only two lemmas.

Lemma testbit_0_r a (b:bool) : testbit (2*a+b) 0 = b.
Proof.
 destruct b; simpl; rewrite ?add_0_r.
 apply testbit_odd_0.
 apply testbit_even_0.
Qed.

Lemma testbit_succ_r a (b:bool) n :
 testbit (2*a+b) (succ n) = testbit a n.
Proof.
 destruct b; simpl; rewrite ?add_0_r.
 apply testbit_odd_succ, le_0_l.
 apply testbit_even_succ, le_0_l.
Qed.

Alternative characterisations of testbit
This concise equation could have been taken as specification for testbit in the interface, but it would have been hard to implement with little initial knowledge about div and mod

Lemma testbit_spec' a n : a.[n] == (a / 2^n) mod 2.
Proof.
 revert a. induct n.
 intros a. nzsimpl.
 destruct (exists_div2 a) as (a' & b & H). rewrite H at 1.
 rewrite testbit_0_r. apply mod_unique with a'; trivial.
 destruct b; order'.
 intros n IH a.
 destruct (exists_div2 a) as (a' & b & H). rewrite H at 1.
 rewrite testbit_succ_r, IH. f_equiv.
 rewrite pow_succ_r', <- div_div by order_nz. f_equiv.
 apply div_unique with b; trivial.
 destruct b; order'.
Qed.

This characterisation that uses only basic operations and power was initially taken as specification for testbit. We describe a as having a low part and a high part, with the corresponding bit in the middle. This characterisation is moderatly complex to implement, but also moderately usable...

Lemma testbit_spec a n :
  exists l h, 0<=l<2^n /\ a == l + (a.[n] + 2*h)*2^n.
Proof.
 exists (a mod 2^n). exists (a / 2^n / 2). split.
 split; [apply le_0_l | apply mod_upper_bound; order_nz].
 rewrite add_comm, mul_comm, (add_comm a.[n]).
 rewrite (div_mod a (2^n)) at 1 by order_nz. do 2 f_equiv.
 rewrite testbit_spec'. apply div_mod. order'.
Qed.

Lemma testbit_true : forall a n,
 a.[n] = true <-> (a / 2^n) mod 2 == 1.
Proof.
 intros a n.
 rewrite <- testbit_spec'; destruct a.[n]; split; simpl; now try order'.
Qed.

Lemma testbit_false : forall a n,
 a.[n] = false <-> (a / 2^n) mod 2 == 0.
Proof.
 intros a n.
 rewrite <- testbit_spec'; destruct a.[n]; split; simpl; now try order'.
Qed.

Lemma testbit_eqb : forall a n,
 a.[n] = eqb ((a / 2^n) mod 2) 1.
Proof.
 intros a n.
 apply eq_true_iff_eq. now rewrite testbit_true, eqb_eq.
Qed.

Results about the injection b2n

Lemma b2n_inj : forall (a0 b0:bool), a0 == b0 -> a0 = b0.
Proof.
 intros [|] [|]; simpl; trivial; order'.
Qed.

Lemma add_b2n_double_div2 : forall (a0:bool) a, (a0+2*a)/2 == a.
Proof.
 intros a0 a. rewrite mul_comm, div_add by order'.
 now rewrite div_small, add_0_l by (destruct a0; order').
Qed.

Lemma add_b2n_double_bit0 : forall (a0:bool) a, (a0+2*a).[0] = a0.
Proof.
 intros a0 a. apply b2n_inj.
 rewrite testbit_spec'. nzsimpl. rewrite mul_comm, mod_add by order'.
 now rewrite mod_small by (destruct a0; order').
Qed.

Lemma b2n_div2 : forall (a0:bool), a0/2 == 0.
Proof.
 intros a0. rewrite <- (add_b2n_double_div2 a0 0). now nzsimpl.
Qed.

Lemma b2n_bit0 : forall (a0:bool), a0.[0] = a0.
Proof.
 intros a0. rewrite <- (add_b2n_double_bit0 a0 0) at 2. now nzsimpl.
Qed.

The specification of testbit by low and high parts is complete

Lemma testbit_unique : forall a n (a0:bool) l h,
 l<2^n -> a == l + (a0 + 2*h)*2^n -> a.[n] = a0.
Proof.
 intros a n a0 l h Hl EQ.
 apply b2n_inj. rewrite testbit_spec' by trivial.
 symmetry. apply mod_unique with h. destruct a0; simpl; order'.
 symmetry. apply div_unique with l; trivial.
 now rewrite add_comm, (add_comm _ a0), mul_comm.
Qed.

All bits of number 0 are 0

Lemma bits_0 : forall n, 0.[n] = false.
Proof.
 intros n. apply testbit_false. nzsimpl; order_nz.
Qed.

Various ways to refer to the lowest bit of a number

Lemma bit0_odd : forall a, a.[0] = odd a.
Proof.
 intros. symmetry.
 destruct (exists_div2 a) as (a' & b & EQ).
 rewrite EQ, testbit_0_r, add_comm, odd_add_mul_2.
 destruct b; simpl; apply odd_1 || apply odd_0.
Qed.

Lemma bit0_eqb : forall a, a.[0] = eqb (a mod 2) 1.
Proof.
 intros a. rewrite testbit_eqb. now nzsimpl.
Qed.

Lemma bit0_mod : forall a, a.[0] == a mod 2.
Proof.
 intros a. rewrite testbit_spec'. now nzsimpl.
Qed.

Hence testing a bit is equivalent to shifting and testing parity

Lemma testbit_odd : forall a n, a.[n] = odd (a>>n).
Proof.
 intros. now rewrite <- bit0_odd, shiftr_spec, add_0_l.
Qed.

log2 gives the highest nonzero bit

Lemma bit_log2 : forall a, a~=0 -> a.[log2 a] = true.
Proof.
 intros a Ha.
 assert (Ha' : 0 < a) by (generalize (le_0_l a); order).
 destruct (log2_spec_alt a Ha') as (r & EQ & (_,Hr)).
 rewrite EQ at 1.
 rewrite testbit_true, add_comm.
 rewrite <- (mul_1_l (2^log2 a)) at 1.
 rewrite div_add by order_nz.
 rewrite div_small by trivial.
 rewrite add_0_l. apply mod_small. order'.
Qed.

Lemma bits_above_log2 : forall a n, log2 a < n ->
 a.[n] = false.
Proof.
 intros a n H.
 rewrite testbit_false.
 rewrite div_small. nzsimpl; order'.
 apply log2_lt_cancel. rewrite log2_pow2; trivial using le_0_l.
Qed.

Hence the number of bits of a is 1+log2 a (see Pos.size_nat and Pos.size).
Testing bits after division or multiplication by a power of two

Lemma div2_bits : forall a n, (a/2).[n] = a.[S n].
Proof.
 intros. apply eq_true_iff_eq.
 rewrite 2 testbit_true.
 rewrite pow_succ_r by apply le_0_l.
 now rewrite div_div by order_nz.
Qed.

Lemma div_pow2_bits : forall a n m, (a/2^n).[m] = a.[m+n].
Proof.
 intros a n. revert a. induct n.
 intros a m. now nzsimpl.
 intros n IH a m. nzsimpl; try apply le_0_l.
 rewrite <- div_div by order_nz.
 now rewrite IH, div2_bits.
Qed.

Lemma double_bits_succ : forall a n, (2*a).[S n] = a.[n].
Proof.
 intros. rewrite <- div2_bits. now rewrite mul_comm, div_mul by order'.
Qed.

Lemma mul_pow2_bits_add : forall a n m, (a*2^n).[m+n] = a.[m].
Proof.
 intros. rewrite <- div_pow2_bits. now rewrite div_mul by order_nz.
Qed.

Lemma mul_pow2_bits_high : forall a n m, n<=m -> (a*2^n).[m] = a.[m-n].
Proof.
 intros.
 rewrite <- (sub_add n m) at 1 by order'.
 now rewrite mul_pow2_bits_add.
Qed.

Lemma mul_pow2_bits_low : forall a n m, m<n -> (a*2^n).[m] = false.
Proof.
 intros. apply testbit_false.
 rewrite <- (sub_add m n) by order'. rewrite pow_add_r, mul_assoc.
 rewrite div_mul by order_nz.
 rewrite <- (succ_pred (n-m)). rewrite pow_succ_r.
 now rewrite (mul_comm 2), mul_assoc, mod_mul by order'.
 apply lt_le_pred.
 apply sub_gt in H. generalize (le_0_l (n-m)); order.
 now apply sub_gt.
Qed.

Selecting the low part of a number can be done by a modulo

Lemma mod_pow2_bits_high : forall a n m, n<=m ->
 (a mod 2^n).[m] = false.
Proof.
 intros a n m H.
 destruct (eq_0_gt_0_cases (a mod 2^n)) as [EQ|LT].
 now rewrite EQ, bits_0.
 apply bits_above_log2.
 apply lt_le_trans with n; trivial.
 apply log2_lt_pow2; trivial.
 apply mod_upper_bound; order_nz.
Qed.

Lemma mod_pow2_bits_low : forall a n m, m<n ->
 (a mod 2^n).[m] = a.[m].
Proof.
 intros a n m H.
 rewrite testbit_eqb.
 rewrite <- (mod_add _ (2^(P (n-m))*(a/2^n))) by order'.
 rewrite <- div_add by order_nz.
 rewrite (mul_comm _ 2), mul_assoc, <- pow_succ_r', succ_pred
   by now apply sub_gt.
 rewrite mul_comm, mul_assoc, <- pow_add_r, (add_comm m), sub_add
   by order.
 rewrite add_comm, <- div_mod by order_nz.
 symmetry. apply testbit_eqb.
Qed.

We now prove that having the same bits implies equality. For that we use a notion of equality over functional streams of bits.

Definition eqf (f g:t -> bool) := forall n:t, f n = g n.

Instance eqf_equiv : Equivalence eqf.
Proof.
 split; congruence.
Qed.

Local Infix "===" := eqf (at level 70, no associativity).

Instance testbit_eqf : Proper (eq==>eqf) testbit.
Proof.
 intros a a' Ha n. now rewrite Ha.
Qed.

Only zero corresponds to the always-false stream.

Lemma bits_inj_0 :
 forall a, (forall n, a.[n] = false) -> a == 0.
Proof.
 intros a H. destruct (eq_decidable a 0) as [EQ|NEQ]; trivial.
 apply bit_log2 in NEQ. now rewrite H in NEQ.
Qed.

If two numbers produce the same stream of bits, they are equal.

Lemma bits_inj : forall a b, testbit a === testbit b -> a == b.
Proof.
 intros a. pattern a.
 apply strong_right_induction with 0;[solve_proper|clear a|apply le_0_l].
 intros a _ IH b H.
 destruct (eq_0_gt_0_cases a) as [EQ|LT].
 rewrite EQ in H |- *. symmetry. apply bits_inj_0.
 intros n. now rewrite <- H, bits_0.
 rewrite (div_mod a 2), (div_mod b 2) by order'.
 f_equiv; [ | now rewrite <- 2 bit0_mod, H].
 f_equiv.
 apply IH; trivial using le_0_l.
 apply div_lt; order'.
 intro n. rewrite 2 div2_bits. apply H.
Qed.

Lemma bits_inj_iff : forall a b, testbit a === testbit b <-> a == b.
Proof.
 split. apply bits_inj. intros EQ; now rewrite EQ.
Qed.

Hint Rewrite lxor_spec lor_spec land_spec ldiff_spec bits_0 : bitwise.

Ltac bitwise := apply bits_inj; intros ?m; autorewrite with bitwise.

The streams of bits that correspond to a natural numbers are exactly the ones that are always 0 after some point

Lemma are_bits : forall (f:t->bool), Proper (eq==>Logic.eq) f ->
 ((exists n, f === testbit n) <->
  (exists k, forall m, k<=m -> f m = false)).
Proof.
 intros f Hf. split.
 intros (a,H).
  exists (S (log2 a)). intros m Hm. apply le_succ_l in Hm.
  rewrite H, bits_above_log2; trivial using lt_succ_diag_r.
 intros (k,Hk).
  revert f Hf Hk. induct k.
  intros f Hf H0.
  exists 0. intros m. rewrite bits_0, H0; trivial. apply le_0_l.
  intros k IH f Hf Hk.
  destruct (IH (fun m => f (S m))) as (n, Hn).
  solve_proper.
  intros m Hm. apply Hk. now rewrite <- succ_le_mono.
  exists (f 0 + 2*n). intros m.
  destruct (zero_or_succ m) as [Hm|(m', Hm)]; rewrite Hm.
  symmetry. apply add_b2n_double_bit0.
  rewrite Hn, <- div2_bits.
  rewrite mul_comm, div_add, b2n_div2, add_0_l; trivial. order'.
Qed.

Properties of shifts

Lemma shiftr_spec' : forall a n m, (a >> n).[m] = a.[m+n].
Proof.
 intros. apply shiftr_spec. apply le_0_l.
Qed.

Lemma shiftl_spec_high' : forall a n m, n<=m -> (a << n).[m] = a.[m-n].
Proof.
 intros. apply shiftl_spec_high; trivial. apply le_0_l.
Qed.

Lemma shiftr_div_pow2 : forall a n, a >> n == a / 2^n.
Proof.
 intros. bitwise. rewrite shiftr_spec'.
 symmetry. apply div_pow2_bits.
Qed.

Lemma shiftl_mul_pow2 : forall a n, a << n == a * 2^n.
Proof.
 intros. bitwise.
 destruct (le_gt_cases n m) as [H|H].
 now rewrite shiftl_spec_high', mul_pow2_bits_high.
 now rewrite shiftl_spec_low, mul_pow2_bits_low.
Qed.

Lemma shiftl_spec_alt : forall a n m, (a << n).[m+n] = a.[m].
Proof.
 intros. now rewrite shiftl_mul_pow2, mul_pow2_bits_add.
Qed.

Instance shiftr_wd : Proper (eq==>eq==>eq) shiftr.
Proof.
 intros a a' Ha b b' Hb. now rewrite 2 shiftr_div_pow2, Ha, Hb.
Qed.

Instance shiftl_wd : Proper (eq==>eq==>eq) shiftl.
Proof.
 intros a a' Ha b b' Hb. now rewrite 2 shiftl_mul_pow2, Ha, Hb.
Qed.

Lemma shiftl_shiftl : forall a n m,
 (a << n) << m == a << (n+m).
Proof.
 intros. now rewrite !shiftl_mul_pow2, pow_add_r, mul_assoc.
Qed.

Lemma shiftr_shiftr : forall a n m,
 (a >> n) >> m == a >> (n+m).
Proof.
 intros.
 now rewrite !shiftr_div_pow2, pow_add_r, div_div by order_nz.
Qed.

Lemma shiftr_shiftl_l : forall a n m, m<=n ->
 (a << n) >> m == a << (n-m).
Proof.
 intros.
 rewrite shiftr_div_pow2, !shiftl_mul_pow2.
 rewrite <- (sub_add m n) at 1 by trivial.
 now rewrite pow_add_r, mul_assoc, div_mul by order_nz.
Qed.

Lemma shiftr_shiftl_r : forall a n m, n<=m ->
 (a << n) >> m == a >> (m-n).
Proof.
 intros.
 rewrite !shiftr_div_pow2, shiftl_mul_pow2.
 rewrite <- (sub_add n m) at 1 by trivial.
 rewrite pow_add_r, (mul_comm (2^(m-n))).
 now rewrite <- div_div, div_mul by order_nz.
Qed.

shifts and constants

Lemma shiftl_1_l : forall n, 1 << n == 2^n.
Proof.
 intros. now rewrite shiftl_mul_pow2, mul_1_l.
Qed.

Lemma shiftl_0_r : forall a, a << 0 == a.
Proof.
 intros. rewrite shiftl_mul_pow2. now nzsimpl.
Qed.

Lemma shiftr_0_r : forall a, a >> 0 == a.
Proof.
 intros. rewrite shiftr_div_pow2. now nzsimpl.
Qed.

Lemma shiftl_0_l : forall n, 0 << n == 0.
Proof.
 intros. rewrite shiftl_mul_pow2. now nzsimpl.
Qed.

Lemma shiftr_0_l : forall n, 0 >> n == 0.
Proof.
 intros. rewrite shiftr_div_pow2. nzsimpl; order_nz.
Qed.

Lemma shiftl_eq_0_iff : forall a n, a << n == 0 <-> a == 0.
Proof.
 intros a n. rewrite shiftl_mul_pow2. rewrite eq_mul_0. split.
 intros [H | H]; trivial. contradict H; order_nz.
 intros H. now left.
Qed.

Lemma shiftr_eq_0_iff : forall a n,
 a >> n == 0 <-> a==0 \/ (0<a /\ log2 a < n).
Proof.
 intros a n.
 rewrite shiftr_div_pow2, div_small_iff by order_nz.
 destruct (eq_0_gt_0_cases a) as [EQ|LT].
 rewrite EQ. split. now left. intros _.
  assert (H : 2~=0) by order'.
  generalize (pow_nonzero 2 n H) (le_0_l (2^n)); order.
 rewrite log2_lt_pow2; trivial.
 split. right; split; trivial. intros [H|[_ H]]; now order.
Qed.

Lemma shiftr_eq_0 : forall a n, log2 a < n -> a >> n == 0.
Proof.
 intros a n H. rewrite shiftr_eq_0_iff.
 destruct (eq_0_gt_0_cases a) as [EQ|LT]. now left. right; now split.
Qed.

Properties of div2.

Lemma div2_div : forall a, div2 a == a/2.
Proof.
 intros. rewrite div2_spec, shiftr_div_pow2. now nzsimpl.
Qed.

Instance div2_wd : Proper (eq==>eq) div2.
Proof.
 intros a a' Ha. now rewrite 2 div2_div, Ha.
Qed.

Lemma div2_odd : forall a, a == 2*(div2 a) + odd a.
Proof.
 intros a. rewrite div2_div, <- bit0_odd, bit0_mod.
 apply div_mod. order'.
Qed.

Properties of lxor and others, directly deduced from properties of xorb and others.

Instance lxor_wd : Proper (eq ==> eq ==> eq) lxor.
Proof.
 intros a a' Ha b b' Hb. bitwise. now rewrite Ha, Hb.
Qed.

Instance land_wd : Proper (eq ==> eq ==> eq) land.
Proof.
 intros a a' Ha b b' Hb. bitwise. now rewrite Ha, Hb.
Qed.

Instance lor_wd : Proper (eq ==> eq ==> eq) lor.
Proof.
 intros a a' Ha b b' Hb. bitwise. now rewrite Ha, Hb.
Qed.

Instance ldiff_wd : Proper (eq ==> eq ==> eq) ldiff.
Proof.
 intros a a' Ha b b' Hb. bitwise. now rewrite Ha, Hb.
Qed.

Lemma lxor_eq : forall a a', lxor a a' == 0 -> a == a'.
Proof.
 intros a a' H. bitwise. apply xorb_eq.
 now rewrite <- lxor_spec, H, bits_0.
Qed.

Lemma lxor_nilpotent : forall a, lxor a a == 0.
Proof.
 intros. bitwise. apply xorb_nilpotent.
Qed.

Lemma lxor_eq_0_iff : forall a a', lxor a a' == 0 <-> a == a'.
Proof.
 split. apply lxor_eq. intros EQ; rewrite EQ; apply lxor_nilpotent.
Qed.

Lemma lxor_0_l : forall a, lxor 0 a == a.
Proof.
 intros. bitwise. apply xorb_false_l.
Qed.

Lemma lxor_0_r : forall a, lxor a 0 == a.
Proof.
 intros. bitwise. apply xorb_false_r.
Qed.

Lemma lxor_comm : forall a b, lxor a b == lxor b a.
Proof.
 intros. bitwise. apply xorb_comm.
Qed.

Lemma lxor_assoc :
 forall a b c, lxor (lxor a b) c == lxor a (lxor b c).
Proof.
 intros. bitwise. apply xorb_assoc.
Qed.

Lemma lor_0_l : forall a, lor 0 a == a.
Proof.
 intros. bitwise. trivial.
Qed.

Lemma lor_0_r : forall a, lor a 0 == a.
Proof.
 intros. bitwise. apply orb_false_r.
Qed.

Lemma lor_comm : forall a b, lor a b == lor b a.
Proof.
 intros. bitwise. apply orb_comm.
Qed.

Lemma lor_assoc :
 forall a b c, lor a (lor b c) == lor (lor a b) c.
Proof.
 intros. bitwise. apply orb_assoc.
Qed.

Lemma lor_diag : forall a, lor a a == a.
Proof.
 intros. bitwise. apply orb_diag.
Qed.

Lemma lor_eq_0_l : forall a b, lor a b == 0 -> a == 0.
Proof.
 intros a b H. bitwise.
 apply (orb_false_iff a.[m] b.[m]).
 now rewrite <- lor_spec, H, bits_0.
Qed.

Lemma lor_eq_0_iff : forall a b, lor a b == 0 <-> a == 0 /\ b == 0.
Proof.
 intros a b. split.
 split. now apply lor_eq_0_l in H.
 rewrite lor_comm in H. now apply lor_eq_0_l in H.
 intros (EQ,EQ'). now rewrite EQ, lor_0_l.
Qed.

Lemma land_0_l : forall a, land 0 a == 0.
Proof.
 intros. bitwise. trivial.
Qed.

Lemma land_0_r : forall a, land a 0 == 0.
Proof.
 intros. bitwise. apply andb_false_r.
Qed.

Lemma land_comm : forall a b, land a b == land b a.
Proof.
 intros. bitwise. apply andb_comm.
Qed.

Lemma land_assoc :
 forall a b c, land a (land b c) == land (land a b) c.
Proof.
 intros. bitwise. apply andb_assoc.
Qed.

Lemma land_diag : forall a, land a a == a.
Proof.
 intros. bitwise. apply andb_diag.
Qed.

Lemma ldiff_0_l : forall a, ldiff 0 a == 0.
Proof.
 intros. bitwise. trivial.
Qed.

Lemma ldiff_0_r : forall a, ldiff a 0 == a.
Proof.
 intros. bitwise. now rewrite andb_true_r.
Qed.

Lemma ldiff_diag : forall a, ldiff a a == 0.
Proof.
 intros. bitwise. apply andb_negb_r.
Qed.

Lemma lor_land_distr_l : forall a b c,
 lor (land a b) c == land (lor a c) (lor b c).
Proof.
 intros. bitwise. apply orb_andb_distrib_l.
Qed.

Lemma lor_land_distr_r : forall a b c,
 lor a (land b c) == land (lor a b) (lor a c).
Proof.
 intros. bitwise. apply orb_andb_distrib_r.
Qed.

Lemma land_lor_distr_l : forall a b c,
 land (lor a b) c == lor (land a c) (land b c).
Proof.
 intros. bitwise. apply andb_orb_distrib_l.
Qed.

Lemma land_lor_distr_r : forall a b c,
 land a (lor b c) == lor (land a b) (land a c).
Proof.
 intros. bitwise. apply andb_orb_distrib_r.
Qed.

Lemma ldiff_ldiff_l : forall a b c,
 ldiff (ldiff a b) c == ldiff a (lor b c).
Proof.
 intros. bitwise. now rewrite negb_orb, andb_assoc.
Qed.

Lemma lor_ldiff_and : forall a b,
 lor (ldiff a b) (land a b) == a.
Proof.
 intros. bitwise.
 now rewrite <- andb_orb_distrib_r, orb_comm, orb_negb_r, andb_true_r.
Qed.

Lemma land_ldiff : forall a b,
 land (ldiff a b) b == 0.
Proof.
 intros. bitwise.
 now rewrite <-andb_assoc, (andb_comm (negb _)), andb_negb_r, andb_false_r.
Qed.

Properties of setbit and clearbit

Definition setbit a n := lor a (1<<n).
Definition clearbit a n := ldiff a (1<<n).

Lemma setbit_spec' : forall a n, setbit a n == lor a (2^n).
Proof.
 intros. unfold setbit. now rewrite shiftl_1_l.
Qed.

Lemma clearbit_spec' : forall a n, clearbit a n == ldiff a (2^n).
Proof.
 intros. unfold clearbit. now rewrite shiftl_1_l.
Qed.

Instance setbit_wd : Proper (eq==>eq==>eq) setbit.
Proof. unfold setbit. solve_proper. Qed.

Instance clearbit_wd : Proper (eq==>eq==>eq) clearbit.
Proof. unfold clearbit. solve_proper. Qed.

Lemma pow2_bits_true : forall n, (2^n).[n] = true.
Proof.
 intros. rewrite <- (mul_1_l (2^n)). rewrite <- (add_0_l n) at 2.
 now rewrite mul_pow2_bits_add, bit0_odd, odd_1.
Qed.

Lemma pow2_bits_false : forall n m, n~=m -> (2^n).[m] = false.
Proof.
 intros.
 rewrite <- (mul_1_l (2^n)).
 destruct (le_gt_cases n m).
 rewrite mul_pow2_bits_high; trivial.
 rewrite <- (succ_pred (m-n)) by (apply sub_gt; order).
 now rewrite <- div2_bits, div_small, bits_0 by order'.
 rewrite mul_pow2_bits_low; trivial.
Qed.

Lemma pow2_bits_eqb : forall n m, (2^n).[m] = eqb n m.
Proof.
 intros. apply eq_true_iff_eq. rewrite eqb_eq. split.
 destruct (eq_decidable n m) as [H|H]. trivial.
 now rewrite (pow2_bits_false _ _ H).
 intros EQ. rewrite EQ. apply pow2_bits_true.
Qed.

Lemma setbit_eqb : forall a n m,
 (setbit a n).[m] = eqb n m || a.[m].
Proof.
 intros. now rewrite setbit_spec', lor_spec, pow2_bits_eqb, orb_comm.
Qed.

Lemma setbit_iff : forall a n m,
 (setbit a n).[m] = true <-> n==m \/ a.[m] = true.
Proof.
 intros. now rewrite setbit_eqb, orb_true_iff, eqb_eq.
Qed.

Lemma setbit_eq : forall a n, (setbit a n).[n] = true.
Proof.
 intros. apply setbit_iff. now left.
Qed.

Lemma setbit_neq : forall a n m, n~=m ->
 (setbit a n).[m] = a.[m].
Proof.
 intros a n m H. rewrite setbit_eqb.
 rewrite <- eqb_eq in H. apply not_true_is_false in H. now rewrite H.
Qed.

Lemma clearbit_eqb : forall a n m,
 (clearbit a n).[m] = a.[m] && negb (eqb n m).
Proof.
 intros. now rewrite clearbit_spec', ldiff_spec, pow2_bits_eqb.
Qed.

Lemma clearbit_iff : forall a n m,
 (clearbit a n).[m] = true <-> a.[m] = true /\ n~=m.
Proof.
 intros. rewrite clearbit_eqb, andb_true_iff, <- eqb_eq.
 now rewrite negb_true_iff, not_true_iff_false.
Qed.

Lemma clearbit_eq : forall a n, (clearbit a n).[n] = false.
Proof.
 intros. rewrite clearbit_eqb, (proj2 (eqb_eq _ _) (eq_refl n)).
 apply andb_false_r.
Qed.

Lemma clearbit_neq : forall a n m, n~=m ->
 (clearbit a n).[m] = a.[m].
Proof.
 intros a n m H. rewrite clearbit_eqb.
 rewrite <- eqb_eq in H. apply not_true_is_false in H. rewrite H.
 apply andb_true_r.
Qed.

Shifts of bitwise operations

Lemma shiftl_lxor : forall a b n,
 (lxor a b) << n == lxor (a << n) (b << n).
Proof.
 intros. bitwise.
 destruct (le_gt_cases n m).
 now rewrite !shiftl_spec_high', lxor_spec.
 now rewrite !shiftl_spec_low.
Qed.

Lemma shiftr_lxor : forall a b n,
 (lxor a b) >> n == lxor (a >> n) (b >> n).
Proof.
 intros. bitwise. now rewrite !shiftr_spec', lxor_spec.
Qed.

Lemma shiftl_land : forall a b n,
 (land a b) << n == land (a << n) (b << n).
Proof.
 intros. bitwise.
 destruct (le_gt_cases n m).
 now rewrite !shiftl_spec_high', land_spec.
 now rewrite !shiftl_spec_low.
Qed.

Lemma shiftr_land : forall a b n,
 (land a b) >> n == land (a >> n) (b >> n).
Proof.
 intros. bitwise. now rewrite !shiftr_spec', land_spec.
Qed.

Lemma shiftl_lor : forall a b n,
 (lor a b) << n == lor (a << n) (b << n).
Proof.
 intros. bitwise.
 destruct (le_gt_cases n m).
 now rewrite !shiftl_spec_high', lor_spec.
 now rewrite !shiftl_spec_low.
Qed.

Lemma shiftr_lor : forall a b n,
 (lor a b) >> n == lor (a >> n) (b >> n).
Proof.
 intros. bitwise. now rewrite !shiftr_spec', lor_spec.
Qed.

Lemma shiftl_ldiff : forall a b n,
 (ldiff a b) << n == ldiff (a << n) (b << n).
Proof.
 intros. bitwise.
 destruct (le_gt_cases n m).
 now rewrite !shiftl_spec_high', ldiff_spec.
 now rewrite !shiftl_spec_low.
Qed.

Lemma shiftr_ldiff : forall a b n,
 (ldiff a b) >> n == ldiff (a >> n) (b >> n).
Proof.
 intros. bitwise. now rewrite !shiftr_spec', ldiff_spec.
Qed.

We cannot have a function complementing all bits of a number, otherwise it would have an infinity of bit 1. Nonetheless, we can design a bounded complement

Definition ones n := P (1 << n).

Definition lnot a n := lxor a (ones n).

Instance ones_wd : Proper (eq==>eq) ones.
Proof. unfold ones. solve_proper. Qed.

Instance lnot_wd : Proper (eq==>eq==>eq) lnot.
Proof. unfold lnot. solve_proper. Qed.

Lemma ones_equiv : forall n, ones n == P (2^n).
Proof.
 intros; unfold ones; now rewrite shiftl_1_l.
Qed.

Lemma ones_add : forall n m, ones (m+n) == 2^m * ones n + ones m.
Proof.
 intros n m. rewrite !ones_equiv.
 rewrite <- !sub_1_r, mul_sub_distr_l, mul_1_r, <- pow_add_r.
 rewrite add_sub_assoc, sub_add. reflexivity.
 apply pow_le_mono_r. order'.
 rewrite <- (add_0_r m) at 1. apply add_le_mono_l, le_0_l.
 rewrite <- (pow_0_r 2). apply pow_le_mono_r. order'. apply le_0_l.
Qed.

Lemma ones_div_pow2 : forall n m, m<=n -> ones n / 2^m == ones (n-m).
Proof.
 intros n m H. symmetry. apply div_unique with (ones m).
 rewrite ones_equiv.
 apply le_succ_l. rewrite succ_pred; order_nz.
 rewrite <- (sub_add m n H) at 1. rewrite (add_comm _ m).
 apply ones_add.
Qed.

Lemma ones_mod_pow2 : forall n m, m<=n -> (ones n) mod (2^m) == ones m.
Proof.
 intros n m H. symmetry. apply mod_unique with (ones (n-m)).
 rewrite ones_equiv.
 apply le_succ_l. rewrite succ_pred; order_nz.
 rewrite <- (sub_add m n H) at 1. rewrite (add_comm _ m).
 apply ones_add.
Qed.

Lemma ones_spec_low : forall n m, m<n -> (ones n).[m] = true.
Proof.
 intros. apply testbit_true. rewrite ones_div_pow2 by order.
 rewrite <- (pow_1_r 2). rewrite ones_mod_pow2.
 rewrite ones_equiv. now nzsimpl'.
 apply le_add_le_sub_r. nzsimpl. now apply le_succ_l.
Qed.

Lemma ones_spec_high : forall n m, n<=m -> (ones n).[m] = false.
Proof.
 intros.
 destruct (eq_0_gt_0_cases n) as [EQ|LT]; rewrite ones_equiv.
 now rewrite EQ, pow_0_r, one_succ, pred_succ, bits_0.
 apply bits_above_log2.
 rewrite log2_pred_pow2; trivial. rewrite <-le_succ_l, succ_pred; order.
Qed.

Lemma ones_spec_iff : forall n m, (ones n).[m] = true <-> m<n.
Proof.
 intros. split. intros H.
 apply lt_nge. intro H'. apply ones_spec_high in H'.
 rewrite H in H'; discriminate.
 apply ones_spec_low.
Qed.

Lemma lnot_spec_low : forall a n m, m<n ->
 (lnot a n).[m] = negb a.[m].
Proof.
 intros. unfold lnot. now rewrite lxor_spec, ones_spec_low.
Qed.

Lemma lnot_spec_high : forall a n m, n<=m ->
 (lnot a n).[m] = a.[m].
Proof.
 intros. unfold lnot. now rewrite lxor_spec, ones_spec_high, xorb_false_r.
Qed.

Lemma lnot_involutive : forall a n, lnot (lnot a n) n == a.
Proof.
 intros a n. bitwise.
 destruct (le_gt_cases n m).
 now rewrite 2 lnot_spec_high.
 now rewrite 2 lnot_spec_low, negb_involutive.
Qed.

Lemma lnot_0_l : forall n, lnot 0 n == ones n.
Proof.
 intros. unfold lnot. apply lxor_0_l.
Qed.

Lemma lnot_ones : forall n, lnot (ones n) n == 0.
Proof.
 intros. unfold lnot. apply lxor_nilpotent.
Qed.

Bounded complement and other operations

Lemma lor_ones_low : forall a n, log2 a < n ->
 lor a (ones n) == ones n.
Proof.
 intros a n H. bitwise. destruct (le_gt_cases n m).
 rewrite ones_spec_high, bits_above_log2; trivial.
 now apply lt_le_trans with n.
 now rewrite ones_spec_low, orb_true_r.
Qed.

Lemma land_ones : forall a n, land a (ones n) == a mod 2^n.
Proof.
 intros a n. bitwise. destruct (le_gt_cases n m).
 now rewrite ones_spec_high, mod_pow2_bits_high, andb_false_r.
 now rewrite ones_spec_low, mod_pow2_bits_low, andb_true_r.
Qed.

Lemma land_ones_low : forall a n, log2 a < n ->
 land a (ones n) == a.
Proof.
 intros; rewrite land_ones. apply mod_small.
 apply log2_lt_cancel. rewrite log2_pow2; trivial using le_0_l.
Qed.

Lemma ldiff_ones_r : forall a n,
 ldiff a (ones n) == (a >> n) << n.
Proof.
 intros a n. bitwise. destruct (le_gt_cases n m).
 rewrite ones_spec_high, shiftl_spec_high', shiftr_spec'; trivial.
 rewrite sub_add; trivial. apply andb_true_r.
 now rewrite ones_spec_low, shiftl_spec_low, andb_false_r.
Qed.

Lemma ldiff_ones_r_low : forall a n, log2 a < n ->
 ldiff a (ones n) == 0.
Proof.
 intros a n H. bitwise. destruct (le_gt_cases n m).
 rewrite ones_spec_high, bits_above_log2; trivial.
 now apply lt_le_trans with n.
 now rewrite ones_spec_low, andb_false_r.
Qed.

Lemma ldiff_ones_l_low : forall a n, log2 a < n ->
 ldiff (ones n) a == lnot a n.
Proof.
 intros a n H. bitwise. destruct (le_gt_cases n m).
 rewrite ones_spec_high, lnot_spec_high, bits_above_log2; trivial.
 now apply lt_le_trans with n.
 now rewrite ones_spec_low, lnot_spec_low.
Qed.

Lemma lor_lnot_diag : forall a n,
 lor a (lnot a n) == lor a (ones n).
Proof.
 intros a n. bitwise.
 destruct (le_gt_cases n m).
 rewrite lnot_spec_high, ones_spec_high; trivial. now destruct a.[m].
 rewrite lnot_spec_low, ones_spec_low; trivial. now destruct a.[m].
Qed.

Lemma lor_lnot_diag_low : forall a n, log2 a < n ->
 lor a (lnot a n) == ones n.
Proof.
 intros a n H. now rewrite lor_lnot_diag, lor_ones_low.
Qed.

Lemma land_lnot_diag : forall a n,
 land a (lnot a n) == ldiff a (ones n).
Proof.
 intros a n. bitwise.
 destruct (le_gt_cases n m).
 rewrite lnot_spec_high, ones_spec_high; trivial. now destruct a.[m].
 rewrite lnot_spec_low, ones_spec_low; trivial. now destruct a.[m].
Qed.

Lemma land_lnot_diag_low : forall a n, log2 a < n ->
 land a (lnot a n) == 0.
Proof.
 intros. now rewrite land_lnot_diag, ldiff_ones_r_low.
Qed.

Lemma lnot_lor_low : forall a b n, log2 a < n -> log2 b < n ->
 lnot (lor a b) n == land (lnot a n) (lnot b n).
Proof.
 intros a b n Ha Hb. bitwise. destruct (le_gt_cases n m).
 rewrite !lnot_spec_high, lor_spec, !bits_above_log2; trivial.
 now apply lt_le_trans with n.
 now apply lt_le_trans with n.
 now rewrite !lnot_spec_low, lor_spec, negb_orb.
Qed.

Lemma lnot_land_low : forall a b n, log2 a < n -> log2 b < n ->
 lnot (land a b) n == lor (lnot a n) (lnot b n).
Proof.
 intros a b n Ha Hb. bitwise. destruct (le_gt_cases n m).
 rewrite !lnot_spec_high, land_spec, !bits_above_log2; trivial.
 now apply lt_le_trans with n.
 now apply lt_le_trans with n.
 now rewrite !lnot_spec_low, land_spec, negb_andb.
Qed.

Lemma ldiff_land_low : forall a b n, log2 a < n ->
 ldiff a b == land a (lnot b n).
Proof.
 intros a b n Ha. bitwise. destruct (le_gt_cases n m).
 rewrite (bits_above_log2 a m). trivial.
 now apply lt_le_trans with n.
 rewrite !lnot_spec_low; trivial.
Qed.

Lemma lnot_ldiff_low : forall a b n, log2 a < n -> log2 b < n ->
 lnot (ldiff a b) n == lor (lnot a n) b.
Proof.
 intros a b n Ha Hb. bitwise. destruct (le_gt_cases n m).
 rewrite !lnot_spec_high, ldiff_spec, !bits_above_log2; trivial.
 now apply lt_le_trans with n.
 now apply lt_le_trans with n.
 now rewrite !lnot_spec_low, ldiff_spec, negb_andb, negb_involutive.
Qed.

Lemma lxor_lnot_lnot : forall a b n,
 lxor (lnot a n) (lnot b n) == lxor a b.
Proof.
 intros a b n. bitwise. destruct (le_gt_cases n m).
 rewrite !lnot_spec_high; trivial.
 rewrite !lnot_spec_low, xorb_negb_negb; trivial.
Qed.

Lemma lnot_lxor_l : forall a b n,
 lnot (lxor a b) n == lxor (lnot a n) b.
Proof.
 intros a b n. bitwise. destruct (le_gt_cases n m).
 rewrite !lnot_spec_high, lxor_spec; trivial.
 rewrite !lnot_spec_low, lxor_spec, negb_xorb_l; trivial.
Qed.

Lemma lnot_lxor_r : forall a b n,
 lnot (lxor a b) n == lxor a (lnot b n).
Proof.
 intros a b n. bitwise. destruct (le_gt_cases n m).
 rewrite !lnot_spec_high, lxor_spec; trivial.
 rewrite !lnot_spec_low, lxor_spec, negb_xorb_r; trivial.
Qed.

Lemma lxor_lor : forall a b, land a b == 0 ->
 lxor a b == lor a b.
Proof.
 intros a b H. bitwise.
 assert (a.[m] && b.[m] = false)
   by now rewrite <- land_spec, H, bits_0.
 now destruct a.[m], b.[m].
Qed.

Bitwise operations and log2

Lemma log2_bits_unique : forall a n,
 a.[n] = true ->
 (forall m, n<m -> a.[m] = false) ->
 log2 a == n.
Proof.
 intros a n H H'.
 destruct (eq_0_gt_0_cases a) as [Ha|Ha].
 now rewrite Ha, bits_0 in H.
 apply le_antisymm; apply le_ngt; intros LT.
 specialize (H' _ LT). now rewrite bit_log2 in H' by order.
 now rewrite bits_above_log2 in H by order.
Qed.

Lemma log2_shiftr : forall a n, log2 (a >> n) == log2 a - n.
Proof.
 intros a n.
 destruct (eq_0_gt_0_cases a) as [Ha|Ha].
 now rewrite Ha, shiftr_0_l, log2_nonpos, sub_0_l by order.
 destruct (lt_ge_cases (log2 a) n).
 rewrite shiftr_eq_0, log2_nonpos by order.
 symmetry. rewrite sub_0_le; order.
 apply log2_bits_unique.
 now rewrite shiftr_spec', sub_add, bit_log2 by order.
 intros m Hm.
 rewrite shiftr_spec'; trivial. apply bits_above_log2; try order.
 now apply lt_sub_lt_add_r.
Qed.

Lemma log2_shiftl : forall a n, a~=0 -> log2 (a << n) == log2 a + n.
Proof.
 intros a n Ha.
 rewrite shiftl_mul_pow2, add_comm by trivial.
 apply log2_mul_pow2. generalize (le_0_l a); order. apply le_0_l.
Qed.

Lemma log2_lor : forall a b,
 log2 (lor a b) == max (log2 a) (log2 b).
Proof.
 assert (AUX : forall a b, a<=b -> log2 (lor a b) == log2 b).
  intros a b H.
  destruct (eq_0_gt_0_cases a) as [Ha|Ha]. now rewrite Ha, lor_0_l.
  apply log2_bits_unique.
  now rewrite lor_spec, bit_log2, orb_true_r by order.
  intros m Hm. assert (H' := log2_le_mono _ _ H).
  now rewrite lor_spec, 2 bits_above_log2 by order.
 intros a b. destruct (le_ge_cases a b) as [H|H].
 rewrite max_r by now apply log2_le_mono.
 now apply AUX.
 rewrite max_l by now apply log2_le_mono.
 rewrite lor_comm. now apply AUX.
Qed.

Lemma log2_land : forall a b,
 log2 (land a b) <= min (log2 a) (log2 b).
Proof.
 assert (AUX : forall a b, a<=b -> log2 (land a b) <= log2 a).
  intros a b H.
  apply le_ngt. intros H'.
  destruct (eq_decidable (land a b) 0) as [EQ|NEQ].
  rewrite EQ in H'. apply log2_lt_cancel in H'. generalize (le_0_l a); order.
  generalize (bit_log2 (land a b) NEQ).
  now rewrite land_spec, bits_above_log2.
 intros a b.
 destruct (le_ge_cases a b) as [H|H].
 rewrite min_l by now apply log2_le_mono. now apply AUX.
 rewrite min_r by now apply log2_le_mono. rewrite land_comm. now apply AUX.
Qed.

Lemma log2_lxor : forall a b,
 log2 (lxor a b) <= max (log2 a) (log2 b).
Proof.
 assert (AUX : forall a b, a<=b -> log2 (lxor a b) <= log2 b).
  intros a b H.
  apply le_ngt. intros H'.
  destruct (eq_decidable (lxor a b) 0) as [EQ|NEQ].
  rewrite EQ in H'. apply log2_lt_cancel in H'. generalize (le_0_l a); order.
  generalize (bit_log2 (lxor a b) NEQ).
  rewrite lxor_spec, 2 bits_above_log2; try order. discriminate.
  apply le_lt_trans with (log2 b); trivial. now apply log2_le_mono.
 intros a b.
 destruct (le_ge_cases a b) as [H|H].
 rewrite max_r by now apply log2_le_mono. now apply AUX.
 rewrite max_l by now apply log2_le_mono. rewrite lxor_comm. now apply AUX.
Qed.

Bitwise operations and arithmetical operations

Local Notation xor3 a b c := (xorb (xorb a b) c).
Local Notation lxor3 a b c := (lxor (lxor a b) c).

Local Notation nextcarry a b c := ((a&&b) || (c && (a||b))).
Local Notation lnextcarry a b c := (lor (land a b) (land c (lor a b))).

Lemma add_bit0 : forall a b, (a+b).[0] = xorb a.[0] b.[0].
Proof.
 intros. now rewrite !bit0_odd, odd_add.
Qed.

Lemma add3_bit0 : forall a b c,
 (a+b+c).[0] = xor3 a.[0] b.[0] c.[0].
Proof.
 intros. now rewrite !add_bit0.
Qed.

Lemma add3_bits_div2 : forall (a0 b0 c0 : bool),
 (a0 + b0 + c0)/2 == nextcarry a0 b0 c0.
Proof.
 assert (H : 1+1 == 2) by now nzsimpl'.
 intros [|] [|] [|]; simpl; rewrite ?add_0_l, ?add_0_r, ?H;
  (apply div_same; order') || (apply div_small; order') || idtac.
 symmetry. apply div_unique with 1. order'. now nzsimpl'.
Qed.

Lemma add_carry_div2 : forall a b (c0:bool),
 (a + b + c0)/2 == a/2 + b/2 + nextcarry a.[0] b.[0] c0.
Proof.
 intros.
 rewrite <- add3_bits_div2.
 rewrite (add_comm ((a/2)+_)).
 rewrite <- div_add by order'.
 f_equiv.
 rewrite <- !div2_div, mul_comm, mul_add_distr_l.
 rewrite (div2_odd a), <- bit0_odd at 1. fold (b2n a.[0]).
 rewrite (div2_odd b), <- bit0_odd at 1. fold (b2n b.[0]).
 rewrite add_shuffle1.
 rewrite <-(add_assoc _ _ c0). apply add_comm.
Qed.

The main result concerning addition: we express the bits of the sum in term of bits of a and b and of some carry stream which is also recursively determined by another equation.

Lemma add_carry_bits : forall a b (c0:bool), exists c,
 a+b+c0 == lxor3 a b c /\ c/2 == lnextcarry a b c /\ c.[0] = c0.
Proof.
 intros a b c0.
 set (n:=max a b).
 assert (Ha : a<2^n).
  apply lt_le_trans with (2^a). apply pow_gt_lin_r, lt_1_2.
  apply pow_le_mono_r. order'. unfold n.
  destruct (le_ge_cases a b); [rewrite max_r|rewrite max_l]; order'.
 assert (Hb : b<2^n).
  apply lt_le_trans with (2^b). apply pow_gt_lin_r, lt_1_2.
  apply pow_le_mono_r. order'. unfold n.
  destruct (le_ge_cases a b); [rewrite max_r|rewrite max_l]; order'.
 clearbody n.
 revert a b c0 Ha Hb. induct n.
 intros a b c0. rewrite !pow_0_r, !one_succ, !lt_succ_r. intros Ha Hb.
 exists c0.
 setoid_replace a with 0 by (generalize (le_0_l a); order').
 setoid_replace b with 0 by (generalize (le_0_l b); order').
 rewrite !add_0_l, !lxor_0_l, !lor_0_r, !land_0_r, !lor_0_r.
 rewrite b2n_div2, b2n_bit0; now repeat split.
 intros n IH a b c0 Ha Hb.
 set (c1:=nextcarry a.[0] b.[0] c0).
 destruct (IH (a/2) (b/2) c1) as (c & IH1 & IH2 & Hc); clear IH.
 apply div_lt_upper_bound; trivial. order'. now rewrite <- pow_succ_r'.
 apply div_lt_upper_bound; trivial. order'. now rewrite <- pow_succ_r'.
 exists (c0 + 2*c). repeat split.
 bitwise.
 destruct (zero_or_succ m) as [EQ|[m' EQ]]; rewrite EQ; clear EQ.
 now rewrite add_b2n_double_bit0, add3_bit0, b2n_bit0.
 rewrite <- !div2_bits, <- 2 lxor_spec.
 f_equiv.
 rewrite add_b2n_double_div2, <- IH1. apply add_carry_div2.
 rewrite add_b2n_double_div2.
 bitwise.
 destruct (zero_or_succ m) as [EQ|[m' EQ]]; rewrite EQ; clear EQ.
 now rewrite add_b2n_double_bit0.
 rewrite <- !div2_bits, IH2. autorewrite with bitwise.
 now rewrite add_b2n_double_div2.
 apply add_b2n_double_bit0.
Qed.

Particular case : the second bit of an addition

Lemma add_bit1 : forall a b,
 (a+b).[1] = xor3 a.[1] b.[1] (a.[0] && b.[0]).
Proof.
 intros a b.
 destruct (add_carry_bits a b false) as (c & EQ1 & EQ2 & Hc).
 simpl in EQ1; rewrite add_0_r in EQ1. rewrite EQ1.
 autorewrite with bitwise. f_equal.
 rewrite one_succ, <- div2_bits, EQ2.
 autorewrite with bitwise.
 rewrite Hc. simpl. apply orb_false_r.
Qed.

In an addition, there will be no carries iff there is no common bits in the numbers to add

Lemma nocarry_equiv : forall a b c,
 c/2 == lnextcarry a b c -> c.[0] = false ->
 (c == 0 <-> land a b == 0).
Proof.
 intros a b c H H'.
 split. intros EQ; rewrite EQ in *.
 rewrite div_0_l in H by order'.
 symmetry in H. now apply lor_eq_0_l in H.
 intros EQ. rewrite EQ, lor_0_l in H.
 apply bits_inj_0.
 induct n. trivial.
 intros n IH.
 rewrite <- div2_bits, H.
 autorewrite with bitwise.
 now rewrite IH.
Qed.

When there is no common bits, the addition is just a xor

Lemma add_nocarry_lxor : forall a b, land a b == 0 ->
 a+b == lxor a b.
Proof.
 intros a b H.
 destruct (add_carry_bits a b false) as (c & EQ1 & EQ2 & Hc).
 simpl in EQ1; rewrite add_0_r in EQ1. rewrite EQ1.
 apply (nocarry_equiv a b c) in H; trivial.
 rewrite H. now rewrite lxor_0_r.
Qed.

A null ldiff implies being smaller

Lemma ldiff_le : forall a b, ldiff a b == 0 -> a <= b.
Proof.
 cut (forall n a b, a < 2^n -> ldiff a b == 0 -> a <= b).
 intros H a b. apply (H a), pow_gt_lin_r; order'.
 induct n.
 intros a b Ha _. rewrite pow_0_r, one_succ, lt_succ_r in Ha.
 assert (Ha' : a == 0) by (generalize (le_0_l a); order').
 rewrite Ha'. apply le_0_l.
 intros n IH a b Ha H.
 assert (NEQ : 2 ~= 0) by order'.
 rewrite (div_mod a 2 NEQ), (div_mod b 2 NEQ).
 apply add_le_mono.
 apply mul_le_mono_l.
 apply IH.
 apply div_lt_upper_bound; trivial. now rewrite <- pow_succ_r'.
 rewrite <- (pow_1_r 2), <- 2 shiftr_div_pow2.
 now rewrite <- shiftr_ldiff, H, shiftr_div_pow2, pow_1_r, div_0_l.
 rewrite <- 2 bit0_mod.
 apply bits_inj_iff in H. specialize (H 0).
 rewrite ldiff_spec, bits_0 in H.
 destruct a.[0], b.[0]; try discriminate; simpl; order'.
Qed.

Subtraction can be a ldiff when the opposite ldiff is null.

Lemma sub_nocarry_ldiff : forall a b, ldiff b a == 0 ->
 a-b == ldiff a b.
Proof.
 intros a b H.
 apply add_cancel_r with b.
 rewrite sub_add.
 symmetry.
 rewrite add_nocarry_lxor.
 bitwise.
 apply bits_inj_iff in H. specialize (H m).
 rewrite ldiff_spec, bits_0 in H.
 now destruct a.[m], b.[m].
 apply land_ldiff.
 now apply ldiff_le.
Qed.

We can express lnot in term of subtraction

Lemma add_lnot_diag_low : forall a n, log2 a < n ->
 a + lnot a n == ones n.
Proof.
 intros a n H.
 assert (H' := land_lnot_diag_low a n H).
 rewrite add_nocarry_lxor, lxor_lor by trivial.
 now apply lor_lnot_diag_low.
Qed.

Lemma lnot_sub_low : forall a n, log2 a < n ->
 lnot a n == ones n - a.
Proof.
 intros a n H.
 now rewrite <- (add_lnot_diag_low a n H), add_comm, add_sub.
Qed.

Adding numbers with no common bits cannot lead to a much bigger number

Lemma add_nocarry_lt_pow2 : forall a b n, land a b == 0 ->
 a < 2^n -> b < 2^n -> a+b < 2^n.
Proof.
 intros a b n H Ha Hb.
 rewrite add_nocarry_lxor by trivial.
 apply div_small_iff. order_nz.
 rewrite <- shiftr_div_pow2, shiftr_lxor, !shiftr_div_pow2.
 rewrite 2 div_small by trivial.
 apply lxor_0_l.
Qed.

Lemma add_nocarry_mod_lt_pow2 : forall a b n, land a b == 0 ->
 a mod 2^n + b mod 2^n < 2^n.
Proof.
 intros a b n H.
 apply add_nocarry_lt_pow2.
 bitwise.
 destruct (le_gt_cases n m).
 now rewrite mod_pow2_bits_high.
 now rewrite !mod_pow2_bits_low, <- land_spec, H, bits_0.
 apply mod_upper_bound; order_nz.
 apply mod_upper_bound; order_nz.
Qed.

End NBitsProp.