# Theory Bit_Operations

```(*  Author:  Florian Haftmann, TUM
*)

section ‹Bit operations in suitable algebraic structures›

theory Bit_Operations
imports Presburger Groups_List
begin

subsection ‹Abstract bit structures›

class semiring_bits = semiring_parity + semiring_modulo_trivial +
assumes bit_induct [case_names stable rec]:
‹(⋀a. a div 2 = a ⟹ P a)
⟹ (⋀a b. P a ⟹ (of_bool b + 2 * a) div 2 = a ⟹ P (of_bool b + 2 * a))
⟹ P a›
assumes bits_mod_div_trivial [simp]: ‹a mod b div b = 0›
and half_div_exp_eq: ‹a div 2 div 2 ^ n = a div 2 ^ Suc n›
and even_double_div_exp_iff: ‹2 ^ Suc n ≠ 0 ⟹ even (2 * a div 2 ^ Suc n) ⟷ even (a div 2 ^ n)›
fixes bit :: ‹'a ⇒ nat ⇒ bool›
assumes bit_iff_odd: ‹bit a n ⟷ odd (a div 2 ^ n)›
begin

text ‹
Having \<^const>‹bit› as definitional class operation
takes into account that specific instances can be implemented
differently wrt. code generation.
›

lemma half_1 [simp]:
‹1 div 2 = 0›
using even_half_succ_eq [of 0] by simp

lemma div_exp_eq_funpow_half:
‹a div 2 ^ n = ((λa. a div 2) ^^ n) a›
proof -
have ‹((λa. a div 2) ^^ n) = (λa. a div 2 ^ n)›
by (induction n)
(simp_all del: funpow.simps power.simps add: power_0 funpow_Suc_right half_div_exp_eq)
then show ?thesis
by simp
qed

lemma div_exp_eq:
‹a div 2 ^ m div 2 ^ n = a div 2 ^ (m + n)›

lemma bit_0:
‹bit a 0 ⟷ odd a›

lemma bit_Suc:
‹bit a (Suc n) ⟷ bit (a div 2) n›
using div_exp_eq [of a 1 n] by (simp add: bit_iff_odd)

lemma bit_rec:
‹bit a n ⟷ (if n = 0 then odd a else bit (a div 2) (n - 1))›
by (cases n) (simp_all add: bit_Suc bit_0)

context
fixes a
assumes stable: ‹a div 2 = a›
begin

‹a + a mod 2 = 0›
proof -
have ‹a div 2 * 2 + a mod 2 = a›
by (fact div_mult_mod_eq)
then have ‹a * 2 + a mod 2 = a›
then show ?thesis
qed

lemma stable_imp_bit_iff_odd:
‹bit a n ⟷ odd a›
by (induction n) (simp_all add: stable bit_Suc bit_0)

end

lemma bit_iff_odd_imp_stable:
‹a div 2 = a› if ‹⋀n. bit a n ⟷ odd a›
using that proof (induction a rule: bit_induct)
case (stable a)
then show ?case
by simp
next
case (rec a b)
from rec.prems [of 1] have [simp]: ‹b = odd a›
by (simp add: rec.hyps bit_Suc bit_0)
from rec.hyps have hyp: ‹(of_bool (odd a) + 2 * a) div 2 = a›
by simp
have ‹bit a n ⟷ odd a› for n
using rec.prems [of ‹Suc n›] by (simp add: hyp bit_Suc)
then have ‹a div 2 = a›
by (rule rec.IH)
then have ‹of_bool (odd a) + 2 * a = 2 * (a div 2) + of_bool (odd a)›
also have ‹… = a›
using mult_div_mod_eq [of 2 a]
finally show ?case
using ‹a div 2 = a› by (simp add: hyp)
qed

lemma even_succ_div_exp [simp]:
‹(1 + a) div 2 ^ n = a div 2 ^ n› if ‹even a› and ‹n > 0›
proof (cases n)
case 0
with that show ?thesis
by simp
next
case (Suc n)
with ‹even a› have ‹(1 + a) div 2 ^ Suc n = a div 2 ^ Suc n›
proof (induction n)
case 0
then show ?case
by simp
next
case (Suc n)
then show ?case
using div_exp_eq [of _ 1 ‹Suc n›, symmetric]
by simp
qed
with Suc show ?thesis
by simp
qed

lemma even_succ_mod_exp [simp]:
‹(1 + a) mod 2 ^ n = 1 + (a mod 2 ^ n)› if ‹even a› and ‹n > 0›
using div_mult_mod_eq [of ‹1 + a› ‹2 ^ n›] div_mult_mod_eq [of a ‹2 ^ n›] that

named_theorems bit_simps ‹Simplification rules for \<^const>‹bit››

definition possible_bit :: ‹'a itself ⇒ nat ⇒ bool›
where ‹possible_bit TYPE('a) n ⟷ 2 ^ n ≠ 0›
― ‹This auxiliary avoids non-termination with extensionality.›

lemma possible_bit_0 [simp]:
‹possible_bit TYPE('a) 0›

lemma fold_possible_bit:
‹2 ^ n = 0 ⟷ ¬ possible_bit TYPE('a) n›

lemma bit_imp_possible_bit:
‹possible_bit TYPE('a) n› if ‹bit a n›
by (rule ccontr) (use that in ‹auto simp add: bit_iff_odd possible_bit_def›)

lemma impossible_bit:
‹¬ bit a n› if ‹¬ possible_bit TYPE('a) n›
using that by (blast dest: bit_imp_possible_bit)

lemma possible_bit_less_imp:
‹possible_bit TYPE('a) j› if ‹possible_bit TYPE('a) i› ‹j ≤ i›
using power_add [of 2 j ‹i - j›] that mult_not_zero [of ‹2 ^ j› ‹2 ^ (i - j)›]

lemma possible_bit_min [simp]:
‹possible_bit TYPE('a) (min i j) ⟷ possible_bit TYPE('a) i ∨ possible_bit TYPE('a) j›
by (auto simp add: min_def elim: possible_bit_less_imp)

lemma bit_eqI:
‹a = b› if ‹⋀n. possible_bit TYPE('a) n ⟹ bit a n ⟷ bit b n›
proof -
have ‹bit a n ⟷ bit b n› for n
proof (cases ‹possible_bit TYPE('a) n›)
case False
then show ?thesis
next
case True
then show ?thesis
by (rule that)
qed
then show ?thesis proof (induction a arbitrary: b rule: bit_induct)
case (stable a)
from stable(2) [of 0] have **: ‹even b ⟷ even a›
have ‹b div 2 = b›
proof (rule bit_iff_odd_imp_stable)
fix n
from stable have *: ‹bit b n ⟷ bit a n›
by simp
also have ‹bit a n ⟷ odd a›
using stable by (simp add: stable_imp_bit_iff_odd)
finally show ‹bit b n ⟷ odd b›
qed
from ** have ‹a mod 2 = b mod 2›
then have ‹a mod 2 + (a + b) = b mod 2 + (a + b)›
by simp
then have ‹a + a mod 2 + b = b + b mod 2 + a›
with ‹a div 2 = a› ‹b div 2 = b› show ?case
next
case (rec a p)
from rec.prems [of 0] have [simp]: ‹p = odd b›
from rec.hyps have ‹bit a n ⟷ bit (b div 2) n› for n
using rec.prems [of ‹Suc n›] by (simp add: bit_Suc)
then have ‹a = b div 2›
by (rule rec.IH)
then have ‹2 * a = 2 * (b div 2)›
by simp
then have ‹b mod 2 + 2 * a = b mod 2 + 2 * (b div 2)›
by simp
also have ‹… = b›
by (fact mod_mult_div_eq)
finally show ?case
qed
qed

lemma bit_eq_rec:
‹a = b ⟷ (even a ⟷ even b) ∧ a div 2 = b div 2› (is ‹?P = ?Q›)
proof
assume ?P
then show ?Q
by simp
next
assume ?Q
then have ‹even a ⟷ even b› and ‹a div 2 = b div 2›
by simp_all
show ?P
proof (rule bit_eqI)
fix n
show ‹bit a n ⟷ bit b n›
proof (cases n)
case 0
with ‹even a ⟷ even b› show ?thesis
next
case (Suc n)
moreover from ‹a div 2 = b div 2› have ‹bit (a div 2) n = bit (b div 2) n›
by simp
ultimately show ?thesis
qed
qed
qed

lemma bit_eq_iff:
‹a = b ⟷ (∀n. possible_bit TYPE('a) n ⟶ bit a n ⟷ bit b n)›
by (auto intro: bit_eqI simp add: possible_bit_def)

lemma bit_0_eq [simp]:
‹bit 0 = ⊥›
proof -
have ‹0 div 2 ^ n = 0› for n
unfolding div_exp_eq_funpow_half by (induction n) simp_all
then show ?thesis
qed

lemma bit_double_Suc_iff:
‹bit (2 * a) (Suc n) ⟷ possible_bit TYPE('a) (Suc n) ∧ bit a n›
using even_double_div_exp_iff [of n a]
by (cases ‹possible_bit TYPE('a) (Suc n)›)

lemma bit_double_iff [bit_simps]:
‹bit (2 * a) n ⟷ possible_bit TYPE('a) n ∧ n ≠ 0 ∧ bit a (n - 1)›
by (cases n) (simp_all add: bit_0 bit_double_Suc_iff)

lemma even_bit_succ_iff:
‹bit (1 + a) n ⟷ bit a n ∨ n = 0› if ‹even a›
using that by (cases ‹n = 0›) (simp_all add: bit_iff_odd)

lemma odd_bit_iff_bit_pred:
‹bit a n ⟷ bit (a - 1) n ∨ n = 0› if ‹odd a›
proof -
from ‹odd a› obtain b where ‹a = 2 * b + 1› ..
moreover have ‹bit (2 * b) n ∨ n = 0 ⟷ bit (1 + 2 * b) n›
using even_bit_succ_iff by simp
ultimately show ?thesis by (simp add: ac_simps)
qed

lemma bit_exp_iff [bit_simps]:
‹bit (2 ^ m) n ⟷ possible_bit TYPE('a) n ∧ n = m›
proof (cases ‹possible_bit TYPE('a) n›)
case False
then show ?thesis
next
case True
then show ?thesis
proof (induction n arbitrary: m)
case 0
show ?case
next
case (Suc n)
then have ‹possible_bit TYPE('a) n›
show ?case
proof (cases m)
case 0
then show ?thesis
next
case (Suc m)
with Suc.IH [of m] ‹possible_bit TYPE('a) n› show ?thesis
qed
qed
qed

lemma bit_1_iff [bit_simps]:
‹bit 1 n ⟷ n = 0›
using bit_exp_iff [of 0 n] by auto

lemma bit_2_iff [bit_simps]:
‹bit 2 n ⟷ possible_bit TYPE('a) 1 ∧ n = 1›
using bit_exp_iff [of 1 n] by auto

lemma bit_of_bool_iff [bit_simps]:
‹bit (of_bool b) n ⟷ n = 0 ∧ b›

lemma bit_mod_2_iff [simp]:
‹bit (a mod 2) n ⟷ n = 0 ∧ odd a›

end

lemma nat_bit_induct [case_names zero even odd]:
‹P n› if zero: ‹P 0›
and even: ‹⋀n. P n ⟹ n > 0 ⟹ P (2 * n)›
and odd: ‹⋀n. P n ⟹ P (Suc (2 * n))›
proof (induction n rule: less_induct)
case (less n)
show ‹P n›
proof (cases ‹n = 0›)
case True with zero show ?thesis by simp
next
case False
with less have hyp: ‹P (n div 2)› by simp
show ?thesis
proof (cases ‹even n›)
case True
then have ‹n ≠ 1›
by auto
with ‹n ≠ 0› have ‹n div 2 > 0›
by simp
with ‹even n› hyp even [of ‹n div 2›] show ?thesis
by simp
next
case False
with hyp odd [of ‹n div 2›] show ?thesis
by simp
qed
qed
qed

instantiation nat :: semiring_bits
begin

definition bit_nat :: ‹nat ⇒ nat ⇒ bool›
where ‹bit_nat m n ⟷ odd (m div 2 ^ n)›

instance
proof
show ‹P n› if stable: ‹⋀n. n div 2 = n ⟹ P n›
and rec: ‹⋀n b. P n ⟹ (of_bool b + 2 * n) div 2 = n ⟹ P (of_bool b + 2 * n)›
for P and n :: nat
proof (induction n rule: nat_bit_induct)
case zero
from stable [of 0] show ?case
by simp
next
case (even n)
with rec [of n False] show ?case
by simp
next
case (odd n)
with rec [of n True] show ?case
by simp
qed
qed (auto simp add: div_mult2_eq bit_nat_def)

end

lemma possible_bit_nat [simp]:
‹possible_bit TYPE(nat) n›

lemma bit_Suc_0_iff [bit_simps]:
‹bit (Suc 0) n ⟷ n = 0›
using bit_1_iff [of n, where ?'a = nat] by simp

lemma not_bit_Suc_0_Suc [simp]:
‹¬ bit (Suc 0) (Suc n)›

lemma not_bit_Suc_0_numeral [simp]:
‹¬ bit (Suc 0) (numeral n)›

context semiring_bits
begin

lemma bit_of_nat_iff [bit_simps]:
‹bit (of_nat m) n ⟷ possible_bit TYPE('a) n ∧ bit m n›
proof (cases ‹possible_bit TYPE('a) n›)
case False
then show ?thesis
next
case True
then have ‹bit (of_nat m) n ⟷ bit m n›
proof (induction m arbitrary: n rule: nat_bit_induct)
case zero
then show ?case
by simp
next
case (even m)
then show ?case
by (cases n)
(auto simp add: bit_double_iff Bit_Operations.bit_double_iff possible_bit_def bit_0 dest: mult_not_zero)
next
case (odd m)
then show ?case
by (cases n)
(auto simp add: bit_double_iff even_bit_succ_iff possible_bit_def
Bit_Operations.bit_Suc Bit_Operations.bit_0 dest: mult_not_zero)
qed
with True show ?thesis
by simp
qed

end

lemma int_bit_induct [case_names zero minus even odd]:
‹P k› if zero_int: ‹P 0›
and minus_int: ‹P (- 1)›
and even_int: ‹⋀k. P k ⟹ k ≠ 0 ⟹ P (k * 2)›
and odd_int: ‹⋀k. P k ⟹ k ≠ - 1 ⟹ P (1 + (k * 2))› for k :: int
proof (cases ‹k ≥ 0›)
case True
define n where ‹n = nat k›
with True have ‹k = int n›
by simp
then show ‹P k›
proof (induction n arbitrary: k rule: nat_bit_induct)
case zero
then show ?case
next
case (even n)
have ‹P (int n * 2)›
by (rule even_int) (use even in simp_all)
with even show ?case
next
case (odd n)
have ‹P (1 + (int n * 2))›
by (rule odd_int) (use odd in simp_all)
with odd show ?case
qed
next
case False
define n where ‹n = nat (- k - 1)›
with False have ‹k = - int n - 1›
by simp
then show ‹P k›
proof (induction n arbitrary: k rule: nat_bit_induct)
case zero
then show ?case
next
case (even n)
have ‹P (1 + (- int (Suc n) * 2))›
by (rule odd_int) (use even in ‹simp_all add: algebra_simps›)
also have ‹… = - int (2 * n) - 1›
finally show ?case
using even.prems by simp
next
case (odd n)
have ‹P (- int (Suc n) * 2)›
by (rule even_int) (use odd in ‹simp_all add: algebra_simps›)
also have ‹… = - int (Suc (2 * n)) - 1›
finally show ?case
using odd.prems by simp
qed
qed

instantiation int :: semiring_bits
begin

definition bit_int :: ‹int ⇒ nat ⇒ bool›
where ‹bit_int k n ⟷ odd (k div 2 ^ n)›

instance
proof
show ‹P k› if stable: ‹⋀k. k div 2 = k ⟹ P k›
and rec: ‹⋀k b. P k ⟹ (of_bool b + 2 * k) div 2 = k ⟹ P (of_bool b + 2 * k)›
for P and k :: int
proof (induction k rule: int_bit_induct)
case zero
from stable [of 0] show ?case
by simp
next
case minus
from stable [of ‹- 1›] show ?case
by simp
next
case (even k)
with rec [of k False] show ?case
next
case (odd k)
with rec [of k True] show ?case
qed
qed (auto simp add: zdiv_zmult2_eq bit_int_def)

end

lemma possible_bit_int [simp]:
‹possible_bit TYPE(int) n›

lemma bit_nat_iff [bit_simps]:
‹bit (nat k) n ⟷ k ≥ 0 ∧ bit k n›
proof (cases ‹k ≥ 0›)
case True
moreover define m where ‹m = nat k›
ultimately have ‹k = int m›
by simp
then show ?thesis
next
case False
then show ?thesis
by simp
qed

subsection ‹Bit operations›

class semiring_bit_operations = semiring_bits +
fixes "and" :: ‹'a ⇒ 'a ⇒ 'a›  (infixr ‹AND› 64)
and or :: ‹'a ⇒ 'a ⇒ 'a›  (infixr ‹OR› 59)
and xor :: ‹'a ⇒ 'a ⇒ 'a›  (infixr ‹XOR› 59)
and mask :: ‹nat ⇒ 'a›
and set_bit :: ‹nat ⇒ 'a ⇒ 'a›
and unset_bit :: ‹nat ⇒ 'a ⇒ 'a›
and flip_bit :: ‹nat ⇒ 'a ⇒ 'a›
and push_bit :: ‹nat ⇒ 'a ⇒ 'a›
and drop_bit :: ‹nat ⇒ 'a ⇒ 'a›
and take_bit :: ‹nat ⇒ 'a ⇒ 'a›
assumes and_rec: ‹a AND b = of_bool (odd a ∧ odd b) + 2 * ((a div 2) AND (b div 2))›
and or_rec: ‹a OR b = of_bool (odd a ∨ odd b) + 2 * ((a div 2) OR (b div 2))›
and xor_rec: ‹a XOR b = of_bool (odd a ≠ odd b) + 2 * ((a div 2) XOR (b div 2))›
and set_bit_eq_or: ‹set_bit n a = a OR push_bit n 1›
and unset_bit_eq_or_xor: ‹unset_bit n a = (a OR push_bit n 1) XOR push_bit n 1›
and flip_bit_eq_xor: ‹flip_bit n a = a XOR push_bit n 1›
and push_bit_eq_mult: ‹push_bit n a = a * 2 ^ n›
and drop_bit_eq_div: ‹drop_bit n a = a div 2 ^ n›
and take_bit_eq_mod: ‹take_bit n a = a mod 2 ^ n›
begin

text ‹
We want the bitwise operations to bind slightly weaker
than ‹+› and ‹-›.

Logically, \<^const>‹push_bit›,
\<^const>‹drop_bit› and \<^const>‹take_bit› are just aliases; having them
as separate operations makes proofs easier, otherwise proof automation
would fiddle with concrete expressions \<^term>‹2 ^ n› in a way obfuscating the basic
algebraic relationships between those operations.

For the sake of code generation operations
are specified as definitional class operations,
taking into account that specific instances of these can be implemented
differently wrt. code generation.
›

lemma bit_iff_odd_drop_bit:
‹bit a n ⟷ odd (drop_bit n a)›

lemma even_drop_bit_iff_not_bit:
‹even (drop_bit n a) ⟷ ¬ bit a n›

lemma bit_and_iff [bit_simps]:
‹bit (a AND b) n ⟷ bit a n ∧ bit b n›
proof (induction n arbitrary: a b)
case 0
show ?case
by (simp add: bit_0 and_rec [of a b] even_bit_succ_iff)
next
case (Suc n)
from Suc [of ‹a div 2› ‹b div 2›]
show ?case
by (simp add: and_rec [of a b] bit_Suc)
(auto simp flip: bit_Suc simp add: bit_double_iff dest: bit_imp_possible_bit)
qed

lemma bit_or_iff [bit_simps]:
‹bit (a OR b) n ⟷ bit a n ∨ bit b n›
proof (induction n arbitrary: a b)
case 0
show ?case
by (simp add: bit_0 or_rec [of a b] even_bit_succ_iff)
next
case (Suc n)
from Suc [of ‹a div 2› ‹b div 2›]
show ?case
by (simp add: or_rec [of a b] bit_Suc)
(auto simp flip: bit_Suc simp add: bit_double_iff dest: bit_imp_possible_bit)
qed

lemma bit_xor_iff [bit_simps]:
‹bit (a XOR b) n ⟷ bit a n ≠ bit b n›
proof (induction n arbitrary: a b)
case 0
show ?case
by (simp add: bit_0 xor_rec [of a b] even_bit_succ_iff)
next
case (Suc n)
from Suc [of ‹a div 2› ‹b div 2›]
show ?case
by (simp add: xor_rec [of a b] bit_Suc)
(auto simp flip: bit_Suc simp add: bit_double_iff dest: bit_imp_possible_bit)
qed

sublocale "and": semilattice ‹(AND)›
by standard (auto simp add: bit_eq_iff bit_and_iff)

sublocale or: semilattice_neutr ‹(OR)› 0
by standard (auto simp add: bit_eq_iff bit_or_iff)

sublocale xor: comm_monoid ‹(XOR)› 0
by standard (auto simp add: bit_eq_iff bit_xor_iff)

lemma even_and_iff:
‹even (a AND b) ⟷ even a ∨ even b›
using bit_and_iff [of a b 0] by (auto simp add: bit_0)

lemma even_or_iff:
‹even (a OR b) ⟷ even a ∧ even b›
using bit_or_iff [of a b 0] by (auto simp add: bit_0)

lemma even_xor_iff:
‹even (a XOR b) ⟷ (even a ⟷ even b)›
using bit_xor_iff [of a b 0] by (auto simp add: bit_0)

lemma zero_and_eq [simp]:
‹0 AND a = 0›

lemma and_zero_eq [simp]:
‹a AND 0 = 0›

lemma one_and_eq:
‹1 AND a = a mod 2›

lemma and_one_eq:
‹a AND 1 = a mod 2›
using one_and_eq [of a] by (simp add: ac_simps)

lemma one_or_eq:
‹1 OR a = a + of_bool (even a)›

lemma or_one_eq:
‹a OR 1 = a + of_bool (even a)›
using one_or_eq [of a] by (simp add: ac_simps)

lemma one_xor_eq:
‹1 XOR a = a + of_bool (even a) - of_bool (odd a)›
(auto simp add: bit_1_iff odd_bit_iff_bit_pred bit_0 elim: oddE)

lemma xor_one_eq:
‹a XOR 1 = a + of_bool (even a) - of_bool (odd a)›
using one_xor_eq [of a] by (simp add: ac_simps)

lemma xor_self_eq [simp]:
‹a XOR a = 0›
by (rule bit_eqI) (simp add: bit_simps)

‹mask n + 1 = 2 ^ n›
proof (induction n)
case 0
then show ?case
by simp
next
case (Suc n)
from Suc.IH [symmetric] have ‹2 ^ Suc n = 2 * mask n + 2›
also have ‹… = 2 * mask n + 1 + 1›
finally have *: ‹2 ^ Suc n = 2 * mask n + 1 + 1› .
then show ?case
qed

proof -
also have ‹… = (1 OR 2 * mask n) + 1›
by (simp add: one_or_eq mult_2_right algebra_simps)
finally show ?thesis
by simp
qed

‹bit (mask m) n ⟷ possible_bit TYPE('a) n ∧ n < m›
proof (cases ‹possible_bit TYPE('a) n›)
case False
then show ?thesis
next
case True
then have ‹bit (mask m) n ⟷ n < m›
proof (induction m arbitrary: n)
case 0
then show ?case
next
case (Suc m)
show ?case
proof (cases n)
case 0
then show ?thesis
next
case (Suc n)
with Suc.prems have ‹possible_bit TYPE('a) n›
using possible_bit_less_imp by auto
with Suc.IH [of n] have ‹bit (mask m) n ⟷ n < m› .
with Suc.prems show ?thesis
qed
qed
with True show ?thesis
by simp
qed

‹even (mask n) ⟷ n = 0›

by (auto simp add: bit_eq_iff bit_simps)

lemma push_bit_0_id [simp]:
‹push_bit 0 = id›

lemma push_bit_Suc [simp]:
‹push_bit (Suc n) a = push_bit n (a * 2)›

lemma push_bit_double:
‹push_bit n (a * 2) = push_bit n a * 2›

lemma bit_push_bit_iff [bit_simps]:
‹bit (push_bit m a) n ⟷ m ≤ n ∧ possible_bit TYPE('a) n ∧ bit a (n - m)›
proof (induction n arbitrary: m)
case 0
then show ?case
by (auto simp add: bit_0 push_bit_eq_mult)
next
case (Suc n)
show ?case
proof (cases m)
case 0
then show ?thesis
next
case (Suc m)
with Suc.prems Suc.IH [of m] show ?thesis
apply (simp add: bit_simps mult.commute [of _ 2])
done
qed
qed

lemma funpow_double_eq_push_bit:
‹times 2 ^^ n = push_bit n›
by (induction n) (simp_all add: fun_eq_iff push_bit_double ac_simps)

lemma div_push_bit_of_1_eq_drop_bit:
‹a div push_bit n 1 = drop_bit n a›

lemma bits_ident:
‹push_bit n (drop_bit n a) + take_bit n a = a›
using div_mult_mod_eq by (simp add: push_bit_eq_mult take_bit_eq_mod drop_bit_eq_div)

lemma push_bit_push_bit [simp]:
‹push_bit m (push_bit n a) = push_bit (m + n) a›

lemma push_bit_of_0 [simp]:
‹push_bit n 0 = 0›

lemma push_bit_of_1 [simp]:
‹push_bit n 1 = 2 ^ n›

‹push_bit n (a + b) = push_bit n a + push_bit n b›

lemma push_bit_numeral [simp]:
‹push_bit (numeral l) (numeral k) = push_bit (pred_numeral l) (numeral (Num.Bit0 k))›

lemma bit_drop_bit_eq [bit_simps]:
‹bit (drop_bit n a) = bit a ∘ (+) n›
by rule (simp add: drop_bit_eq_div bit_iff_odd div_exp_eq)

lemma disjunctive_xor_eq_or:
‹a XOR b = a OR b› if ‹a AND b = 0›
using that by (auto simp add: bit_eq_iff bit_simps)

‹a + b = a OR b› if ‹a AND b = 0›
proof (rule bit_eqI)
fix n
assume ‹possible_bit TYPE('a) n›
moreover from that have ‹⋀n. ¬ bit (a AND b) n›
by simp
then have ‹⋀n. ¬ bit a n ∨ ¬ bit b n›
ultimately show ‹bit (a + b) n ⟷ bit (a OR b) n›
proof (induction n arbitrary: a b)
case 0
from "0"(2)[of 0] show ?case
by (auto simp add: even_or_iff bit_0)
next
case (Suc n)
from Suc.prems(2) [of 0] have even: ‹even a ∨ even b›
have bit: ‹¬ bit (a div 2) n ∨ ¬ bit (b div 2) n› for n
using Suc.prems(2) [of ‹Suc n›] by (simp add: bit_Suc)
from Suc.prems have ‹possible_bit TYPE('a) n›
using possible_bit_less_imp by force
with ‹⋀n. ¬ bit (a div 2) n ∨ ¬ bit (b div 2) n› Suc.IH [of ‹a div 2› ‹b div 2›]
have IH: ‹bit (a div 2 + b div 2) n ⟷ bit (a div 2 OR b div 2) n›
have ‹a + b = (a div 2 * 2 + a mod 2) + (b div 2 * 2 + b mod 2)›
using div_mult_mod_eq [of a 2] div_mult_mod_eq [of b 2] by simp
also have ‹… = of_bool (odd a ∨ odd b) + 2 * (a div 2 + b div 2)›
using even by (auto simp add: algebra_simps mod2_eq_if)
finally have ‹bit ((a + b) div 2) n ⟷ bit (a div 2 + b div 2) n›
using ‹possible_bit TYPE('a) (Suc n)› by simp (simp_all flip: bit_Suc add: bit_double_iff possible_bit_def)
also have ‹… ⟷ bit (a div 2 OR b div 2) n›
by (rule IH)
finally show ?case
by (simp add: bit_simps flip: bit_Suc)
qed
qed

‹a + b = a XOR b› if ‹a AND b = 0›

lemma take_bit_0 [simp]:
"take_bit 0 a = 0"

lemma bit_take_bit_iff [bit_simps]:
‹bit (take_bit m a) n ⟷ n < m ∧ bit a n›
proof -
have ‹push_bit m (drop_bit m a) AND take_bit m a = 0› (is ‹?lhs = _›)
proof (rule bit_eqI)
fix n
show ‹bit ?lhs n ⟷ bit 0 n›
proof (cases ‹m ≤ n›)
case False
then show ?thesis
next
case True
moreover define q where ‹q = n - m›
ultimately have ‹n = m + q› by simp
moreover have ‹¬ bit (take_bit m a) (m + q)›
by (simp add: take_bit_eq_mod bit_iff_odd flip: div_exp_eq)
ultimately show ?thesis
qed
qed
then have ‹push_bit m (drop_bit m a) XOR take_bit m a = push_bit m (drop_bit m a) + take_bit m a›
also have ‹… = a›
finally have ‹bit (push_bit m (drop_bit m a) XOR take_bit m a) n ⟷ bit a n›
by simp
also have ‹… ⟷ (m ≤ n ∨ n < m) ∧ bit a n›
by auto
also have ‹… ⟷ m ≤ n ∧ bit a n ∨ n < m ∧ bit a n›
by auto
also have ‹m ≤ n ∧ bit a n ⟷ bit (push_bit m (drop_bit m a)) n›
by (auto simp add: bit_simps bit_imp_possible_bit)
finally show ?thesis
qed

lemma take_bit_Suc:
‹take_bit (Suc n) a = take_bit n (a div 2) * 2 + a mod 2› (is ‹?lhs = ?rhs›)
proof (rule bit_eqI)
fix m
assume ‹possible_bit TYPE('a) m›
then show ‹bit ?lhs m ⟷ bit ?rhs m›
apply (cases a rule: parity_cases; cases m)
apply (simp_all add: bit_simps even_bit_succ_iff mult.commute [of _ 2] add.commute [of _ 1] flip: bit_Suc)
done
qed

lemma take_bit_rec:
‹take_bit n a = (if n = 0 then 0 else take_bit (n - 1) (a div 2) * 2 + a mod 2)›
by (cases n) (simp_all add: take_bit_Suc)

lemma take_bit_Suc_0 [simp]:
‹take_bit (Suc 0) a = a mod 2›

lemma take_bit_of_0 [simp]:
‹take_bit n 0 = 0›
by (rule bit_eqI) (simp add: bit_simps)

lemma take_bit_of_1 [simp]:
‹take_bit n 1 = of_bool (n > 0)›
by (cases n) (simp_all add: take_bit_Suc)

lemma drop_bit_of_0 [simp]:
‹drop_bit n 0 = 0›
by (rule bit_eqI) (simp add: bit_simps)

lemma drop_bit_of_1 [simp]:
‹drop_bit n 1 = of_bool (n = 0)›
by (rule bit_eqI) (simp add: bit_simps ac_simps)

lemma drop_bit_0 [simp]:
‹drop_bit 0 = id›

lemma drop_bit_Suc:
‹drop_bit (Suc n) a = drop_bit n (a div 2)›
using div_exp_eq [of a 1] by (simp add: drop_bit_eq_div)

lemma drop_bit_rec:
‹drop_bit n a = (if n = 0 then a else drop_bit (n - 1) (a div 2))›
by (cases n) (simp_all add: drop_bit_Suc)

lemma drop_bit_half:
‹drop_bit n (a div 2) = drop_bit n a div 2›
by (induction n arbitrary: a) (simp_all add: drop_bit_Suc)

lemma drop_bit_of_bool [simp]:
‹drop_bit n (of_bool b) = of_bool (n = 0 ∧ b)›
by (cases n) simp_all

lemma even_take_bit_eq [simp]:
‹even (take_bit n a) ⟷ n = 0 ∨ even a›
by (simp add: take_bit_rec [of n a])

lemma take_bit_take_bit [simp]:
‹take_bit m (take_bit n a) = take_bit (min m n) a›
by (rule bit_eqI) (simp add: bit_simps)

lemma drop_bit_drop_bit [simp]:
‹drop_bit m (drop_bit n a) = drop_bit (m + n) a›

lemma push_bit_take_bit:
‹push_bit m (take_bit n a) = take_bit (m + n) (push_bit m a)›
by (rule bit_eqI) (auto simp add: bit_simps)

lemma take_bit_push_bit:
‹take_bit m (push_bit n a) = push_bit n (take_bit (m - n) a)›
by (rule bit_eqI) (auto simp add: bit_simps)

lemma take_bit_drop_bit:
‹take_bit m (drop_bit n a) = drop_bit n (take_bit (m + n) a)›
by (rule bit_eqI) (auto simp add: bit_simps)

lemma drop_bit_take_bit:
‹drop_bit m (take_bit n a) = take_bit (n - m) (drop_bit m a)›
by (rule bit_eqI) (auto simp add: bit_simps)

lemma even_push_bit_iff [simp]:
‹even (push_bit n a) ⟷ n ≠ 0 ∨ even a›

lemma stable_imp_drop_bit_eq:
‹drop_bit n a = a›
if ‹a div 2 = a›
by (induction n) (simp_all add: that drop_bit_Suc)

lemma stable_imp_take_bit_eq:
‹take_bit n a = (if even a then 0 else mask n)›
if ‹a div 2 = a›
by (rule bit_eqI) (use that in ‹simp add: bit_simps stable_imp_bit_iff_odd›)

lemma exp_dvdE:
assumes ‹2 ^ n dvd a›
obtains b where ‹a = push_bit n b›
proof -
from assms obtain b where ‹a = 2 ^ n * b› ..
then have ‹a = push_bit n b›
with that show thesis .
qed

lemma take_bit_eq_0_iff:
‹take_bit n a = 0 ⟷ 2 ^ n dvd a› (is ‹?P ⟷ ?Q›)
proof
assume ?P
then show ?Q
next
assume ?Q
then obtain b where ‹a = push_bit n b›
by (rule exp_dvdE)
then show ?P
qed

lemma take_bit_tightened:
‹take_bit m a = take_bit m b› if ‹take_bit n a = take_bit n b› and ‹m ≤ n›
proof -
from that have ‹take_bit m (take_bit n a) = take_bit m (take_bit n b)›
by simp
then have ‹take_bit (min m n) a = take_bit (min m n) b›
by simp
with that show ?thesis
qed

lemma take_bit_eq_self_iff_drop_bit_eq_0:
‹take_bit n a = a ⟷ drop_bit n a = 0› (is ‹?P ⟷ ?Q›)
proof
assume ?P
show ?Q
proof (rule bit_eqI)
fix m
from ‹?P› have ‹a = take_bit n a› ..
also have ‹¬ bit (take_bit n a) (n + m)›
unfolding bit_simps
finally show ‹bit (drop_bit n a) m ⟷ bit 0 m›
qed
next
assume ?Q
show ?P
proof (rule bit_eqI)
fix m
from ‹?Q› have ‹¬ bit (drop_bit n a) (m - n)›
by simp
then have ‹ ¬ bit a (n + (m - n))›
then show ‹bit (take_bit n a) m ⟷ bit a m›
by (cases ‹m < n›) (auto simp add: bit_simps)
qed
qed

lemma drop_bit_exp_eq:
‹drop_bit m (2 ^ n) = of_bool (m ≤ n ∧ possible_bit TYPE('a) n) * 2 ^ (n - m)›
by (auto simp add: bit_eq_iff bit_simps)

lemma take_bit_and [simp]:
‹take_bit n (a AND b) = take_bit n a AND take_bit n b›
by (auto simp add: bit_eq_iff bit_simps)

lemma take_bit_or [simp]:
‹take_bit n (a OR b) = take_bit n a OR take_bit n b›
by (auto simp add: bit_eq_iff bit_simps)

lemma take_bit_xor [simp]:
‹take_bit n (a XOR b) = take_bit n a XOR take_bit n b›
by (auto simp add: bit_eq_iff bit_simps)

lemma push_bit_and [simp]:
‹push_bit n (a AND b) = push_bit n a AND push_bit n b›
by (auto simp add: bit_eq_iff bit_simps)

lemma push_bit_or [simp]:
‹push_bit n (a OR b) = push_bit n a OR push_bit n b›
by (auto simp add: bit_eq_iff bit_simps)

lemma push_bit_xor [simp]:
‹push_bit n (a XOR b) = push_bit n a XOR push_bit n b›
by (auto simp add: bit_eq_iff bit_simps)

lemma drop_bit_and [simp]:
‹drop_bit n (a AND b) = drop_bit n a AND drop_bit n b›
by (auto simp add: bit_eq_iff bit_simps)

lemma drop_bit_or [simp]:
‹drop_bit n (a OR b) = drop_bit n a OR drop_bit n b›
by (auto simp add: bit_eq_iff bit_simps)

lemma drop_bit_xor [simp]:
‹drop_bit n (a XOR b) = drop_bit n a XOR drop_bit n b›
by (auto simp add: bit_eq_iff bit_simps)

by (rule bit_eqI) (simp add: bit_simps)

‹take_bit n a = a AND mask n›
by (auto simp add: bit_eq_iff bit_simps)

lemma or_eq_0_iff:
‹a OR b = 0 ⟷ a = 0 ∧ b = 0›
by (auto simp add: bit_eq_iff bit_or_iff)

lemma bit_iff_and_drop_bit_eq_1:
‹bit a n ⟷ drop_bit n a AND 1 = 1›
by (simp add: bit_iff_odd_drop_bit and_one_eq odd_iff_mod_2_eq_one)

lemma bit_iff_and_push_bit_not_eq_0:
‹bit a n ⟷ a AND push_bit n 1 ≠ 0›
by (cases ‹possible_bit TYPE('a) n›) (simp_all add: bit_eq_iff bit_simps impossible_bit)

lemma bit_set_bit_iff [bit_simps]:
‹bit (set_bit m a) n ⟷ bit a n ∨ (m = n ∧ possible_bit TYPE('a) n)›
by (auto simp add: set_bit_eq_or bit_or_iff bit_exp_iff)

lemma even_set_bit_iff:
‹even (set_bit m a) ⟷ even a ∧ m ≠ 0›
using bit_set_bit_iff [of m a 0] by (auto simp add: bit_0)

lemma bit_unset_bit_iff [bit_simps]:
‹bit (unset_bit m a) n ⟷ bit a n ∧ m ≠ n›
by (auto simp add: unset_bit_eq_or_xor bit_simps dest: bit_imp_possible_bit)

lemma even_unset_bit_iff:
‹even (unset_bit m a) ⟷ even a ∨ m = 0›
using bit_unset_bit_iff [of m a 0] by (auto simp add: bit_0)

lemma bit_flip_bit_iff [bit_simps]:
‹bit (flip_bit m a) n ⟷ (m = n ⟷ ¬ bit a n) ∧ possible_bit TYPE('a) n›
by (auto simp add: bit_eq_iff bit_simps flip_bit_eq_xor bit_imp_possible_bit)

lemma even_flip_bit_iff:
‹even (flip_bit m a) ⟷ ¬ (even a ⟷ m = 0)›
using bit_flip_bit_iff [of m a 0] by (auto simp: possible_bit_def  bit_0)

lemma and_exp_eq_0_iff_not_bit:
‹a AND 2 ^ n = 0 ⟷ ¬ bit a n› (is ‹?P ⟷ ?Q›)
using bit_imp_possible_bit[of a n]
by (auto simp add: bit_eq_iff bit_simps)

lemma bit_sum_mult_2_cases:
assumes a: ‹∀j. ¬ bit a (Suc j)›
shows ‹bit (a + 2 * b) n = (if n = 0 then odd a else bit (2 * b) n)›
proof -
from a have ‹n = 0› if ‹bit a n› for n using that
by (cases n) simp_all
then have ‹a = 0 ∨ a = 1›
by (auto simp add: bit_eq_iff bit_1_iff)
then show ?thesis
by (cases n) (auto simp add: bit_0 bit_double_iff even_bit_succ_iff)
qed

lemma set_bit_0 [simp]:
‹set_bit 0 a = 1 + 2 * (a div 2)›
by (auto simp add: bit_eq_iff bit_simps even_bit_succ_iff simp flip: bit_Suc)

lemma set_bit_Suc:
‹set_bit (Suc n) a = a mod 2 + 2 * set_bit n (a div 2)›
by (auto simp add: bit_eq_iff bit_sum_mult_2_cases bit_simps bit_0 simp flip: bit_Suc
elim: possible_bit_less_imp)

lemma unset_bit_0 [simp]:
‹unset_bit 0 a = 2 * (a div 2)›
by (auto simp add: bit_eq_iff bit_simps simp flip: bit_Suc)

lemma unset_bit_Suc:
‹unset_bit (Suc n) a = a mod 2 + 2 * unset_bit n (a div 2)›
by (auto simp add: bit_eq_iff bit_sum_mult_2_cases bit_simps bit_0 simp flip: bit_Suc)

lemma flip_bit_0 [simp]:
‹flip_bit 0 a = of_bool (even a) + 2 * (a div 2)›
by (auto simp add: bit_eq_iff bit_simps even_bit_succ_iff bit_0 simp flip: bit_Suc)

lemma flip_bit_Suc:
‹flip_bit (Suc n) a = a mod 2 + 2 * flip_bit n (a div 2)›
by (auto simp add: bit_eq_iff bit_sum_mult_2_cases bit_simps bit_0 simp flip: bit_Suc
elim: possible_bit_less_imp)

lemma flip_bit_eq_if:
‹flip_bit n a = (if bit a n then unset_bit else set_bit) n a›
by (rule bit_eqI) (auto simp add: bit_set_bit_iff bit_unset_bit_iff bit_flip_bit_iff)

lemma take_bit_set_bit_eq:
‹take_bit n (set_bit m a) = (if n ≤ m then take_bit n a else set_bit m (take_bit n a))›
by (rule bit_eqI) (auto simp add: bit_take_bit_iff bit_set_bit_iff)

lemma take_bit_unset_bit_eq:
‹take_bit n (unset_bit m a) = (if n ≤ m then take_bit n a else unset_bit m (take_bit n a))›
by (rule bit_eqI) (auto simp add: bit_take_bit_iff bit_unset_bit_iff)

lemma take_bit_flip_bit_eq:
‹take_bit n (flip_bit m a) = (if n ≤ m then take_bit n a else flip_bit m (take_bit n a))›
by (rule bit_eqI) (auto simp add: bit_take_bit_iff bit_flip_bit_iff)

lemma bit_1_0 [simp]:
‹bit 1 0›

lemma not_bit_1_Suc [simp]:
‹¬ bit 1 (Suc n)›

lemma push_bit_Suc_numeral [simp]:
‹push_bit (Suc n) (numeral k) = push_bit n (numeral (Num.Bit0 k))›

‹mask n = 0 ⟷ n = 0›

lemma bit_horner_sum_bit_iff [bit_simps]:
‹bit (horner_sum of_bool 2 bs) n ⟷ possible_bit TYPE('a) n ∧ n < length bs ∧ bs ! n›
proof (induction bs arbitrary: n)
case Nil
then show ?case
by simp
next
case (Cons b bs)
show ?case
proof (cases n)
case 0
then show ?thesis
next
case (Suc m)
with bit_rec [of _ n] Cons.prems Cons.IH [of m]
show ?thesis
(auto simp add: possible_bit_less_imp bit_simps simp flip: bit_Suc)
qed
qed

lemma horner_sum_bit_eq_take_bit:
‹horner_sum of_bool 2 (map (bit a) [0..<n]) = take_bit n a›
by (rule bit_eqI) (auto simp add: bit_simps)

lemma take_bit_horner_sum_bit_eq:
‹take_bit n (horner_sum of_bool 2 bs) = horner_sum of_bool 2 (take n bs)›
by (auto simp add: bit_eq_iff bit_take_bit_iff bit_horner_sum_bit_iff)

lemma take_bit_sum:
‹take_bit n a = (∑k = 0..<n. push_bit k (of_bool (bit a k)))›
by (simp flip: horner_sum_bit_eq_take_bit add: horner_sum_eq_sum push_bit_eq_mult)

lemma set_bit_eq:
‹set_bit n a = a + of_bool (¬ bit a n) * 2 ^ n›
proof -
have ‹a AND of_bool (¬ bit a n) * 2 ^ n = 0›
by (auto simp add: bit_eq_iff bit_simps)
then show ?thesis
qed

end

class ring_bit_operations = semiring_bit_operations + ring_parity +
fixes not :: ‹'a ⇒ 'a›  (‹NOT›)
assumes not_eq_complement: ‹NOT a = - a - 1›
begin

text ‹
For the sake of code generation \<^const>‹not› is specified as
definitional class operation.  Note that \<^const>‹not› has no
sensible definition for unlimited but only positive bit strings
(type \<^typ>‹nat›).
›

lemma bits_minus_1_mod_2_eq [simp]:
‹(- 1) mod 2 = 1›

lemma minus_eq_not_plus_1:
‹- a = NOT a + 1›
using not_eq_complement [of a] by simp

lemma minus_eq_not_minus_1:
‹- a = NOT (a - 1)›
using not_eq_complement [of ‹a - 1›] by simp (simp add: algebra_simps)

lemma not_rec:
‹NOT a = of_bool (even a) + 2 * NOT (a div 2)›
by (simp add: not_eq_complement algebra_simps mod_2_eq_odd flip: minus_mod_eq_mult_div)

lemma decr_eq_not_minus:
‹a - 1 = NOT (- a)›
using not_eq_complement [of ‹- a›] by simp

lemma even_not_iff [simp]:
‹even (NOT a) ⟷ odd a›

lemma bit_not_iff [bit_simps]:
‹bit (NOT a) n ⟷ possible_bit TYPE('a) n ∧ ¬ bit a n›
proof (cases ‹possible_bit TYPE('a) n›)
case False
then show ?thesis
by (auto dest: bit_imp_possible_bit)
next
case True
moreover have ‹bit (NOT a) n ⟷ ¬ bit a n›
using ‹possible_bit TYPE('a) n› proof (induction n arbitrary: a)
case 0
then show ?case
next
case (Suc n)
from Suc.prems Suc.IH [of ‹a div 2›]
show ?case
by (simp add: impossible_bit possible_bit_less_imp not_rec [of a] even_bit_succ_iff bit_double_iff flip: bit_Suc)
qed
ultimately show ?thesis
by simp
qed

lemma bit_not_exp_iff [bit_simps]:
‹bit (NOT (2 ^ m)) n ⟷ possible_bit TYPE('a) n ∧ n ≠ m›
by (auto simp add: bit_not_iff bit_exp_iff)

lemma bit_minus_iff [bit_simps]:
‹bit (- a) n ⟷ possible_bit TYPE('a) n ∧ ¬ bit (a - 1) n›

lemma bit_minus_1_iff [simp]:
‹bit (- 1) n ⟷ possible_bit TYPE('a) n›

lemma bit_minus_exp_iff [bit_simps]:
‹bit (- (2 ^ m)) n ⟷ possible_bit TYPE('a) n ∧ n ≥ m›

lemma bit_minus_2_iff [simp]:
‹bit (- 2) n ⟷ possible_bit TYPE('a) n ∧ n > 0›

lemma bit_decr_iff:
‹bit (a - 1) n ⟷ possible_bit TYPE('a) n ∧ ¬ bit (- a) n›

lemma bit_not_iff_eq:
‹bit (NOT a) n ⟷ 2 ^ n ≠ 0 ∧ ¬ bit a n›

lemma not_one_eq [simp]:
‹NOT 1 = - 2›

sublocale "and": semilattice_neutr ‹(AND)› ‹- 1›
by standard (rule bit_eqI, simp add: bit_and_iff)

sublocale bit: abstract_boolean_algebra ‹(AND)› ‹(OR)› NOT 0 ‹- 1›
by standard (auto simp add: bit_and_iff bit_or_iff bit_not_iff intro: bit_eqI)

sublocale bit: abstract_boolean_algebra_sym_diff ‹(AND)› ‹(OR)› NOT 0 ‹- 1› ‹(XOR)›
apply standard
apply (rule bit_eqI)
done

lemma and_eq_not_not_or:
‹a AND b = NOT (NOT a OR NOT b)›
by simp

lemma or_eq_not_not_and:
‹a OR b = NOT (NOT a AND NOT b)›
by simp

‹NOT (a + b) = NOT a - b›

lemma not_diff_distrib:
‹NOT (a - b) = NOT a + b›
using not_add_distrib [of a ‹- b›] by simp

lemma and_eq_minus_1_iff:
‹a AND b = - 1 ⟷ a = - 1 ∧ b = - 1›
by (auto simp: bit_eq_iff bit_simps)

lemma disjunctive_and_not_eq_xor:
‹a AND NOT b = a XOR b› if ‹NOT a AND b = 0›
using that by (auto simp add: bit_eq_iff bit_simps)

lemma disjunctive_diff_eq_and_not:
‹a - b = a AND NOT b› if ‹NOT a AND b = 0›
proof -
from that have ‹NOT a + b = NOT a OR b›
then have ‹NOT (NOT a + b) = NOT (NOT a OR b)›
by simp
then show ?thesis
qed

lemma disjunctive_diff_eq_xor:
‹a AND NOT b = a XOR b› if ‹NOT a AND b = 0›
using that by (simp add: disjunctive_and_not_eq_xor disjunctive_diff_eq_and_not)

lemma push_bit_minus:
‹push_bit n (- a) = - push_bit n a›

lemma take_bit_not_take_bit:
‹take_bit n (NOT (take_bit n a)) = take_bit n (NOT a)›
by (auto simp add: bit_eq_iff bit_take_bit_iff bit_not_iff)

lemma take_bit_not_iff:
‹take_bit n (NOT a) = take_bit n (NOT b) ⟷ take_bit n a = take_bit n b›
by (auto simp add: bit_eq_iff bit_simps)

‹take_bit n (NOT a) = mask n - take_bit n a›
proof -
have ‹NOT (mask n) AND take_bit n a = 0›
moreover have ‹take_bit n (NOT a) = mask n AND NOT (take_bit n a)›
by (auto simp add: bit_eq_iff bit_simps)
ultimately show ?thesis
qed

‹mask n = take_bit n (- 1)›

‹take_bit n (- 1) = mask n›

‹- (2 ^ n) = NOT (mask n)›

‹push_bit n (- 1) = NOT (mask n)›

‹take_bit m (NOT (mask n)) = 0› if ‹n ≥ m›

lemma unset_bit_eq_and_not:
‹unset_bit n a = a AND NOT (push_bit n 1)›
by (rule bit_eqI) (auto simp add: bit_simps)

lemma push_bit_Suc_minus_numeral [simp]:
‹push_bit (Suc n) (- numeral k) = push_bit n (- numeral (Num.Bit0 k))›
apply (simp only: numeral_Bit0)
apply simp
apply (simp only: numeral_mult mult_2_right numeral_add)
done

lemma push_bit_minus_numeral [simp]:
‹push_bit (numeral l) (- numeral k) = push_bit (pred_numeral l) (- numeral (Num.Bit0 k))›
by (simp only: numeral_eq_Suc push_bit_Suc_minus_numeral)

lemma take_bit_Suc_minus_1_eq:
‹take_bit (Suc n) (- 1) = 2 ^ Suc n - 1›

lemma take_bit_numeral_minus_1_eq:
‹take_bit (numeral k) (- 1) = 2 ^ numeral k - 1›

by (rule bit_eqI) (auto simp add: bit_simps not_less possible_bit_less_imp)

‹push_bit n (take_bit m (drop_bit n a)) = a AND mask (m + n) AND NOT (mask n)›
by (rule bit_eqI) (auto simp add: bit_simps)

lemma push_bit_numeral_minus_1 [simp]:
‹push_bit (numeral n) (- 1) = - (2 ^ numeral n)›

lemma unset_bit_eq:
‹unset_bit n a = a - of_bool (bit a n) * 2 ^ n›
proof -
have ‹NOT a AND of_bool (bit a n) * 2 ^ n = 0›
by (auto simp add: bit_eq_iff bit_simps)
moreover have ‹unset_bit n a = a AND NOT (of_bool (bit a n) * 2 ^ n)›
by (auto simp add: bit_eq_iff bit_simps)
ultimately show ?thesis
qed

end

subsection ‹Common algebraic structure›

class linordered_euclidean_semiring_bit_operations =
linordered_euclidean_semiring + semiring_bit_operations
begin

lemma possible_bit [simp]:
‹possible_bit TYPE('a) n›

lemma take_bit_of_exp [simp]:
‹take_bit m (2 ^ n) = of_bool (n < m) * 2 ^ n›

lemma take_bit_of_2 [simp]:
‹take_bit n 2 = of_bool (2 ≤ n) * 2›
using take_bit_of_exp [of n 1] by simp

lemma push_bit_eq_0_iff [simp]:
‹push_bit n a = 0 ⟷ a = 0›

‹take_bit n (take_bit n a + take_bit n b) = take_bit n (a + b)›

lemma take_bit_of_1_eq_0_iff [simp]:
‹take_bit n 1 = 0 ⟷ n = 0›

lemma drop_bit_Suc_bit0 [simp]:
‹drop_bit (Suc n) (numeral (Num.Bit0 k)) = drop_bit n (numeral k)›

lemma drop_bit_Suc_bit1 [simp]:
‹drop_bit (Suc n) (numeral (Num.Bit1 k)) = drop_bit n (numeral k)›

lemma drop_bit_numeral_bit0 [simp]:
‹drop_bit (numeral l) (numeral (Num.Bit0 k)) = drop_bit (pred_numeral l) (numeral k)›

lemma drop_bit_numeral_bit1 [simp]:
‹drop_bit (numeral l) (numeral (Num.Bit1 k)) = drop_bit (pred_numeral l) (numeral k)›

lemma take_bit_Suc_1 [simp]:
‹take_bit (Suc n) 1 = 1›

lemma take_bit_Suc_bit0:
‹take_bit (Suc n) (numeral (Num.Bit0 k)) = take_bit n (numeral k) * 2›

lemma take_bit_Suc_bit1:
‹take_bit (Suc n) (numeral (Num.Bit1 k)) = take_bit n (numeral k) * 2 + 1›
by (simp add: take_bit_Suc numeral_Bit1_div_2 mod_2_eq_odd)

lemma take_bit_numeral_1 [simp]:
‹take_bit (numeral l) 1 = 1›
by (simp add: take_bit_rec [of ‹numeral l› 1])

lemma take_bit_numeral_bit0:
‹take_bit (numeral l) (numeral (Num.Bit0 k)) = take_bit (pred_numeral l) (numeral k) * 2›

lemma take_bit_numeral_bit1:
‹take_bit (numeral l) (numeral (Num.Bit1 k)) = take_bit (pred_numeral l) (numeral k) * 2 + 1›
by (simp add: take_bit_rec numeral_Bit1_div_2 mod_2_eq_odd)

lemma bit_of_nat_iff_bit [bit_simps]:
‹bit (of_nat m) n ⟷ bit m n›
proof -
have ‹even (m div 2 ^ n) ⟷ even (of_nat (m div 2 ^ n))›
by simp
also have ‹of_nat (m div 2 ^ n) = of_nat m div of_nat (2 ^ n)›
finally show ?thesis
qed

by (rule bit_eqI) (auto simp add: bit_simps possible_bit_def)

lemma bit_push_bit_iff':
‹bit (push_bit m a) n ⟷ m ≤ n ∧ bit a (n - m)›

lemma take_bit_Suc_from_most:
‹take_bit (Suc n) a = 2 ^ n * of_bool (bit a n) + take_bit n a›
using mod_mult2_eq' [of a ‹2 ^ n› 2]
by (simp only: take_bit_eq_mod power_Suc2)

lemma take_bit_nonnegative [simp]:
‹0 ≤ take_bit n a›
using horner_sum_nonnegative by (simp flip: horner_sum_bit_eq_take_bit)

lemma not_take_bit_negative [simp]:
‹¬ take_bit n a < 0›

lemma bit_imp_take_bit_positive:
‹0 < take_bit m a› if ‹n < m› and ‹bit a n›
proof (rule ccontr)
assume ‹¬ 0 < take_bit m a›
then have ‹take_bit m a = 0›
by (auto simp add: not_less intro: order_antisym)
then have ‹bit (take_bit m a) n = bit 0 n›
by simp
with that show False
qed

lemma take_bit_mult:
‹take_bit n (take_bit n a * take_bit n b) = take_bit n (a * b)›

lemma drop_bit_push_bit:
‹drop_bit m (push_bit n a) = drop_bit (m - n) (push_bit (n - m) a)›
by (cases ‹m ≤ n›)
(auto simp add: mult.left_commute [of _ ‹2 ^ n›] mult.commute [of _ ‹2 ^ n›] mult.assoc

end

subsection ‹Instance \<^typ>‹int››

locale fold2_bit_int =
fixes f :: ‹bool ⇒ bool ⇒ bool›
begin

context
begin

```