# Theory Word_Lib.Most_significant_bit

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

(* Author: Jeremy Dawson, NICTA *)

section ‹Dedicated operation for the most significant bit›

theory Most_significant_bit
imports
"HOL-Library.Word"
Bit_Shifts_Infix_Syntax
More_Word
More_Arithmetic
begin

class msb =
fixes msb :: ‹'a ⇒ bool›

instantiation int :: msb
begin

definition ‹msb x ⟷ x < 0› for x :: int

instance ..

end

lemma msb_bin_rest [simp]: "msb (x div 2) = msb x"
for x :: int

context
includes bit_operations_syntax
begin

lemma int_msb_and [simp]: "msb ((x :: int) AND y) ⟷ msb x ∧ msb y"

lemma int_msb_or [simp]: "msb ((x :: int) OR y) ⟷ msb x ∨ msb y"

lemma int_msb_xor [simp]: "msb ((x :: int) XOR y) ⟷ msb x ≠ msb y"

lemma int_msb_not [simp]: "msb (NOT (x :: int)) ⟷ ¬ msb x"

end

lemma msb_shiftl [simp]: "msb ((x :: int) << n) ⟷ msb x"

lemma msb_shiftr [simp]: "msb ((x :: int) >> r) ⟷ msb x"

lemma msb_0 [simp]: "msb (0 :: int) = False"

lemma msb_1 [simp]: "msb (1 :: int) = False"

lemma msb_numeral [simp]:
"msb (numeral n :: int) = False"
"msb (- numeral n :: int) = True"

instantiation word :: (len) msb
begin

definition msb_word :: ‹'a word ⇒ bool›
where msb_word_iff_bit: ‹msb w ⟷ bit w (LENGTH('a) - Suc 0)› for w :: ‹'a::len word›

instance ..

end

lemma msb_word_eq:
‹msb w ⟷ bit w (LENGTH('a) - 1)› for w :: ‹'a::len word›

lemma word_msb_sint: "msb w ⟷ sint w < 0"

lemma msb_word_iff_sless_0:
‹msb w ⟷ w <s 0›

lemma msb_word_of_int:
"msb (word_of_int x::'a::len word) = bit x (LENGTH('a) - 1)"

lemma word_msb_numeral [simp]:
"msb (numeral w::'a::len word) = bit (numeral w :: int) (LENGTH('a) - 1)"
unfolding word_numeral_alt by (rule msb_word_of_int)

lemma word_msb_neg_numeral [simp]:
"msb (- numeral w::'a::len word) = bit (- numeral w :: int) (LENGTH('a) - 1)"
unfolding word_neg_numeral_alt by (rule msb_word_of_int)

lemma word_msb_0 [simp]: "¬ msb (0::'a::len word)"

lemma word_msb_1 [simp]: "msb (1::'a::len word) ⟷ LENGTH('a) = 1"

lemma word_msb_nth: "msb w = bit (uint w) (LENGTH('a) - 1)"
for w :: "'a::len word"

lemma msb_nth: "msb w = bit w (LENGTH('a) - 1)"
for w :: "'a::len word"
by (fact msb_word_eq)

lemma word_msb_n1 [simp]: "msb (-1::'a::len word)"

lemma msb_shift: "msb w ⟷ w >> LENGTH('a) - 1 ≠ 0"
for w :: "'a::len word"
by (simp add: drop_bit_eq_zero_iff_not_bit_last msb_word_eq shiftr_def)

lemmas word_ops_msb = msb1 [unfolded msb_nth [symmetric, unfolded One_nat_def]]

lemma word_sint_msb_eq: "sint x = uint x - (if msb x then 2 ^ size x else 0)"
apply (cases ‹LENGTH('a)›)
apply transfer
done

lemma word_sle_msb_le: "x <=s y ⟷ (msb y ⟶ msb x) ∧ ((msb x ∧ ¬ msb y) ∨ x ≤ y)"
apply (simp add: word_sle_eq word_sint_msb_eq word_size word_le_def)
apply safe
apply (rule order_trans[OF _ uint_ge_0])
apply (erule notE[OF leD])
apply (rule order_less_le_trans[OF _ uint_ge_0])
apply simp
done

lemma word_sless_msb_less: "x <s y ⟷ (msb y ⟶ msb x) ∧ ((msb x ∧ ¬ msb y) ∨ x < y)"
by (auto simp add: word_sless_eq word_sle_msb_le)

lemma not_msb_from_less:
"(v :: 'a word) < 2 ^ (LENGTH('a :: len) - 1) ⟹ ¬ msb v"
apply (drule word_eqD, drule(1) iffD2)
done

lemma sint_eq_uint:
"¬ msb x ⟹ sint x = uint x"
apply (cases ‹LENGTH('a)›)
apply transfer
done

lemma scast_eq_ucast:
"¬ msb x ⟹ scast x = ucast x"
apply (cases ‹LENGTH('a)›)
apply simp
apply (rule bit_word_eqI)
apply (auto simp add: bit_signed_iff bit_unsigned_iff min_def msb_word_eq)
apply (erule notE)
apply (metis le_less_Suc_eq test_bit_bin)
done

lemma msb_ucast_eq:
"LENGTH('a) = LENGTH('b) ⟹
msb (ucast x :: ('a::len) word) = msb (x :: ('b::len) word)"

lemma msb_big:
‹msb a ⟷ 2 ^ (LENGTH('a) - Suc 0) ≤ a›
for a :: ‹'a::len word›
using bang_is_le [of a ‹LENGTH('a) - Suc 0›]
apply (auto simp add: msb_nth word_le_not_less)
apply (rule ccontr)
apply (erule notE)
apply (rule ccontr)
apply (clarsimp simp: not_less)
apply (subgoal_tac "a = take_bit (LENGTH('a) - Suc 0) a")
apply (cut_tac and_mask_less' [where w=a and n="LENGTH('a) - Suc 0"])
apply auto
apply (rule sym)
done

instantiation integer :: msb
begin

context
includes integer.lifting
begin

lift_definition msb_integer :: ‹integer ⇒ bool› is msb .

instance ..

end

end

end
```