# Theory Generic_set_bit

```(*
* Copyright Data61, CSIRO (ABN 41 687 119 230)
*
*)

(* Author: Jeremy Dawson, NICTA *)

section ‹Operation variant for setting and unsetting bits›

theory Generic_set_bit
imports
"HOL-Library.Word"
Most_significant_bit
begin

class set_bit = semiring_bits +
fixes set_bit :: ‹'a ⇒ nat ⇒ bool ⇒ 'a›
assumes bit_set_bit_iff_2n:
‹bit (set_bit a m b) n ⟷
(if m = n then b else bit a n) ∧ 2 ^ n ≠ 0›
begin

lemma bit_set_bit_iff [bit_simps]:
‹bit (set_bit a m b) n ⟷
(if m = n then b else bit a n) ∧ possible_bit TYPE('a) n›

end

lemma set_bit_eq:
‹set_bit a n b = (if b then Bit_Operations.set_bit else unset_bit) n a›
for a :: ‹'a::{semiring_bit_operations, set_bit}›
by (rule bit_eqI) (simp add: bit_simps)

instantiation nat :: set_bit
begin

definition set_bit_nat :: ‹nat ⇒ nat ⇒ bool ⇒ nat›
where ‹set_bit m n b = (if b then Bit_Operations.set_bit else unset_bit) n m› for m n :: nat

instance
by standard (simp add: set_bit_nat_def bit_simps)

end

instantiation int :: set_bit
begin

definition set_bit_int :: ‹int ⇒ nat ⇒ bool ⇒ int›
where ‹set_bit_int i n b = (if b then Bit_Operations.set_bit else Bit_Operations.unset_bit) n i›

instance
by standard (simp add: set_bit_int_def bit_simps)

end

instantiation word :: (len) set_bit
begin

definition set_bit_word :: ‹'a word ⇒ nat ⇒ bool ⇒ 'a word›
where set_bit_unfold: ‹set_bit w n b = (if b then Bit_Operations.set_bit n w else unset_bit n w)›
for w :: ‹'a::len word›

instance
by standard (auto simp add: set_bit_unfold bit_simps dest: bit_imp_le_length)

end

lemma bit_set_bit_word_iff [bit_simps]:
‹bit (set_bit w m b) n ⟷ (if m = n then n < LENGTH('a) ∧ b else bit w n)›
for w :: ‹'a::len word›
by (auto simp add: bit_simps dest: bit_imp_le_length)

context
includes bit_operations_syntax
begin

lemma int_set_bit_0 [simp]: fixes x :: int shows
"set_bit x 0 b = of_bool b + 2 * (x div 2)"

lemma int_set_bit_Suc: fixes x :: int shows
"set_bit x (Suc n) b = of_bool (odd x) + 2 * set_bit (x div 2) n b"
by (simp add: set_bit_eq set_bit_Suc unset_bit_Suc mod2_eq_if)

lemma bin_last_set_bit:
"odd (set_bit x n b :: int) = (if n > 0 then odd x else b)"
by (cases n) (simp_all add: int_set_bit_Suc)

lemma bin_rest_set_bit:
"(set_bit x n b :: int) div 2 = (if n > 0 then set_bit (x div 2) (n - 1) b else x div 2)"
by (cases n) (simp_all add: int_set_bit_Suc)

lemma int_set_bit_numeral: fixes x :: int shows
"set_bit x (numeral w) b = of_bool (odd x) + 2 * set_bit (x div 2) (pred_numeral w) b"

lemmas int_set_bit_numerals [simp] =
int_set_bit_numeral[where x="numeral w'"]
int_set_bit_numeral[where x="- numeral w'"]
int_set_bit_numeral[where x="Numeral1"]
int_set_bit_numeral[where x="1"]
int_set_bit_numeral[where x="0"]
int_set_bit_Suc[where x="numeral w'"]
int_set_bit_Suc[where x="- numeral w'"]
int_set_bit_Suc[where x="Numeral1"]
int_set_bit_Suc[where x="1"]
int_set_bit_Suc[where x="0"]
for w'

lemma fixes i :: int
shows int_set_bit_True_conv_OR [code]: "Generic_set_bit.set_bit i n True = i OR push_bit n 1"
and int_set_bit_False_conv_NAND [code]: "Generic_set_bit.set_bit i n False = i AND NOT (push_bit n 1)"
and int_set_bit_conv_ops: "Generic_set_bit.set_bit i n b = (if b then i OR (push_bit n 1) else i AND NOT (push_bit n 1))"

lemma msb_set_bit [simp]:
"msb (set_bit (x :: int) n b) ⟷ msb x"

lemmas msb_bin_sc = msb_set_bit

abbreviation (input) bin_sc :: ‹nat ⇒ bool ⇒ int ⇒ int›
where ‹bin_sc n b i ≡ set_bit i n b›

lemma bin_sc_eq:
‹bin_sc n False = unset_bit n›
‹bin_sc n True = Bit_Operations.set_bit n›

lemma bin_sc_0 [simp]:
"bin_sc 0 b w = of_bool b + 2 * (λk::int. k div 2) w"

lemma bin_sc_Suc [simp]:
"bin_sc (Suc n) b w = of_bool (odd w) + 2 * bin_sc n b (w div 2)"
by (simp add: set_bit_eq set_bit_Suc unset_bit_Suc mod2_eq_if)

lemma bin_nth_sc [bit_simps]: "bit (bin_sc n b w) n ⟷ b"

lemma bin_sc_sc_same [simp]: "bin_sc n c (bin_sc n b w) = bin_sc n c w"
by (rule bit_eqI) (simp add: bit_simps)

lemma bin_sc_sc_diff: "m ≠ n ⟹ bin_sc m c (bin_sc n b w) = bin_sc n b (bin_sc m c w)"
by (rule bit_eqI) (simp add: bit_simps)

lemma bin_nth_sc_gen: "(bit :: int ⇒ nat ⇒ bool) (bin_sc n b w) m = (if m = n then b else (bit :: int ⇒ nat ⇒ bool) w m)"

lemma bin_sc_nth [simp]: "bin_sc n ((bit :: int ⇒ nat ⇒ bool) w n) w = w"
by (rule bit_eqI) (simp add: bin_nth_sc_gen)

lemma bin_sc_bintr [simp]:
"(take_bit :: nat ⇒ int ⇒ int) m (bin_sc n x ((take_bit :: nat ⇒ int ⇒ int) m w)) = (take_bit :: nat ⇒ int ⇒ int) m (bin_sc n x w)"
apply (rule bit_eqI)
apply (cases x)
apply (auto simp add: bit_simps bin_sc_eq)
done

lemma bin_clr_le: "bin_sc n False w ≤ w"

lemma bin_set_ge: "bin_sc n True w ≥ w"

lemma bintr_bin_clr_le: "(take_bit :: nat ⇒ int ⇒ int) n (bin_sc m False w) ≤ (take_bit :: nat ⇒ int ⇒ int) n w"
by (simp add: set_bit_int_def take_bit_unset_bit_eq unset_bit_less_eq)

lemma bintr_bin_set_ge: "(take_bit :: nat ⇒ int ⇒ int) n (bin_sc m True w) ≥ (take_bit :: nat ⇒ int ⇒ int) n w"
by (simp add: set_bit_int_def take_bit_set_bit_eq set_bit_greater_eq)

lemma bin_sc_FP [simp]: "bin_sc n False 0 = 0"
by (induct n) auto

lemma bin_sc_TM [simp]: "bin_sc n True (- 1) = - 1"
by (induct n) auto

lemmas bin_sc_simps = bin_sc_0 bin_sc_Suc bin_sc_TM bin_sc_FP

lemma bin_sc_minus: "0 < n ⟹ bin_sc (Suc (n - 1)) b w = bin_sc n b w"
by auto

lemmas bin_sc_Suc_minus =
trans [OF bin_sc_minus [symmetric] bin_sc_Suc]

lemma bin_sc_numeral [simp]:
"bin_sc (numeral k) b w =
of_bool (odd w) + 2 * bin_sc (pred_numeral k) b (w div 2)"

lemmas bin_sc_minus_simps =
bin_sc_simps (2,3,4) [THEN [2] trans, OF bin_sc_minus [THEN sym]]

lemma bin_sc_pos:
"0 ≤ i ⟹ 0 ≤ bin_sc n b i"

lemma bin_clr_conv_NAND:
"bin_sc n False i = i AND NOT (push_bit n 1)"
by (rule bit_eqI) (auto simp add: bin_sc_eq bit_simps)

lemma bin_set_conv_OR:
"bin_sc n True i = i OR (push_bit n 1)"
by (rule bit_eqI) (auto simp add: bin_sc_eq bit_simps)

lemma word_set_bit_def:
‹set_bit a n x = word_of_int (bin_sc n x (uint a))›
apply (rule bit_word_eqI)
apply (cases x)
done

lemma set_bit_word_of_int:
"set_bit (word_of_int x) n b = word_of_int (bin_sc n b x)"
by (rule word_eqI) (auto simp add: bit_simps)

lemma word_set_numeral [simp]:
"set_bit (numeral bin::'a::len word) n b =
word_of_int (bin_sc n b (numeral bin))"
unfolding word_numeral_alt by (rule set_bit_word_of_int)

lemma word_set_neg_numeral [simp]:
"set_bit (- numeral bin::'a::len word) n b =
word_of_int (bin_sc n b (- numeral bin))"
unfolding word_neg_numeral_alt by (rule set_bit_word_of_int)

lemma word_set_bit_0 [simp]: "set_bit 0 n b = word_of_int (bin_sc n b 0)"
unfolding word_0_wi by (rule set_bit_word_of_int)

lemma word_set_bit_1 [simp]: "set_bit 1 n b = word_of_int (bin_sc n b 1)"
unfolding word_1_wi by (rule set_bit_word_of_int)

lemma setBit_no: "Bit_Operations.set_bit n (numeral bin) = word_of_int (bin_sc n True (numeral bin))"
by (rule bit_word_eqI) (simp add: bit_simps)

lemma clearBit_no:
"unset_bit n (numeral bin) = word_of_int (bin_sc n False (numeral bin))"
by (rule bit_word_eqI) (simp add: bit_simps)

end

lemma test_bit_set_gen:
"bit (set_bit w n x) m ⟷ (if m = n then n < size w ∧ x else bit w m)"
for w :: "'a::len word"

lemma test_bit_set:
"bit (set_bit w n x) n ⟷ n < size w ∧ x"
for w :: "'a::len word"
by (auto simp add: bit_simps word_size)

lemma word_set_nth: "set_bit w n (bit w n) = w"
for w :: "'a::len word"
by (rule bit_word_eqI) (simp add: bit_simps)

lemma word_set_set_same [simp]: "set_bit (set_bit w n x) n y = set_bit w n y"
for w :: "'a::len word"
by (rule word_eqI) (simp add : test_bit_set_gen word_size)

lemma word_set_set_diff:
fixes w :: "'a::len word"
assumes "m ≠ n"
shows "set_bit (set_bit w m x) n y = set_bit (set_bit w n y) m x"
by (rule word_eqI) (auto simp: test_bit_set_gen word_size assms)

lemma word_set_nth_iff: "set_bit w n b = w ⟷ bit w n = b ∨ n ≥ size w"
for w :: "'a::len word"
apply (rule iffI)
apply (rule disjCI)
apply (drule word_eqD)
apply (erule sym [THEN trans])
apply (erule disjE)
apply clarsimp
apply (rule word_eqI)
apply (clarsimp simp add : test_bit_set_gen)
apply (rule bit_eqI)
done

lemma word_clr_le: "w ≥ set_bit w n False"
for w :: "'a::len word"
apply transfer
done

lemma word_set_ge: "w ≤ set_bit w n True"
for w :: "'a::len word"
apply transfer
done

lemma set_bit_beyond:
"size x ≤ n ⟹ set_bit x n b = x" for x :: "'a :: len word"

lemma one_bit_shiftl: "set_bit 0 n True = (1 :: 'a :: len word) << n"
apply (rule word_eqI)
apply (auto simp add: word_size bit_simps)
done

lemma one_bit_pow: "set_bit 0 n True = (2 :: 'a :: len word) ^ n"
by (rule word_eqI) (simp add: bit_simps)

instantiation integer :: set_bit
begin

context
includes integer.lifting
begin

lift_definition set_bit_integer :: ‹integer ⇒ nat ⇒ bool ⇒ integer›
is set_bit .

instance
by (standard; transfer) (simp add: bit_simps)

end

end

end
```