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diff --git a/theories/Numbers/Natural/Abstract/NBits.v b/theories/Numbers/Natural/Abstract/NBits.v new file mode 100644 index 00000000..c66f003e --- /dev/null +++ b/theories/Numbers/Natural/Abstract/NBits.v @@ -0,0 +1,1463 @@ +(************************************************************************) +(* 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 *) +(************************************************************************) + +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 caracterisations 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 caracterisation 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 caracterisation + 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 [Psize] and [Psize_pos]). +*) + +(** 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. + (* main *) + 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. + (* main *) + 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. + (* main *) + 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. + (* induction over some n such that [a<2^n] and [b<2^n] *) + 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. + (*base*) + 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. + (*step*) + 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. + (* - add *) + 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. + (* - carry *) + 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. + (* - carry0 *) + 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. |