Theory Word_Lib.Word_32
section "Words of Length 32"
theory Word_32
imports
Word_Lemmas
Word_Syntax
Word_Names
Rsplit
More_Word_Operations
Bitwise
begin
context
includes bit_operations_syntax
begin
type_synonym word32 = "32 word"
lemma len32: "len_of (x :: 32 itself) = 32" by simp
type_synonym sword32 = "32 sword"
lemma ucast_8_32_inj:
"inj (ucast :: 8 word ⇒ 32 word)"
by (rule down_ucast_inj) (clarsimp simp: is_down_def target_size source_size)
lemmas unat_power_lower32' = unat_power_lower[where 'a=32]
lemmas word32_less_sub_le' = word_less_sub_le[where 'a = 32]
lemmas word32_power_less_1' = word_power_less_1[where 'a = 32]
lemmas unat_of_nat32' = unat_of_nat_eq[where 'a=32]
lemmas unat_mask_word32' = unat_mask[where 'a=32]
lemmas word32_minus_one_le' = word_minus_one_le[where 'a=32]
lemmas word32_minus_one_le = word32_minus_one_le'[simplified]
lemma unat_ucast_8_32:
fixes x :: "8 word"
shows "unat (ucast x :: word32) = unat x"
by transfer simp
lemma ucast_le_ucast_8_32:
"(ucast x ≤ (ucast y :: word32)) = (x ≤ (y :: 8 word))"
by (simp add: ucast_le_ucast)
lemma eq_2_32_0:
"(2 ^ 32 :: word32) = 0"
by simp
lemmas mask_32_max_word = max_word_mask [symmetric, where 'a=32, simplified]
lemma of_nat32_n_less_equal_power_2:
"n < 32 ⟹ ((of_nat n)::32 word) < 2 ^ n"
by (rule of_nat_n_less_equal_power_2, clarsimp simp: word_size)
lemma unat_ucast_10_32 :
fixes x :: "10 word"
shows "unat (ucast x :: word32) = unat x"
by transfer simp
lemma word32_bounds:
"- (2 ^ (size (x :: word32) - 1)) = (-2147483648 :: int)"
"((2 ^ (size (x :: word32) - 1)) - 1) = (2147483647 :: int)"
"- (2 ^ (size (y :: 32 signed word) - 1)) = (-2147483648 :: int)"
"((2 ^ (size (y :: 32 signed word) - 1)) - 1) = (2147483647 :: int)"
by (simp_all add: word_size)
lemmas signed_arith_ineq_checks_to_eq_word32'
= signed_arith_ineq_checks_to_eq[where 'a=32]
signed_arith_ineq_checks_to_eq[where 'a="32 signed"]
lemmas signed_arith_ineq_checks_to_eq_word32
= signed_arith_ineq_checks_to_eq_word32' [unfolded word32_bounds]
lemmas signed_mult_eq_checks32_to_64'
= signed_mult_eq_checks_double_size[where 'a=32 and 'b=64]
signed_mult_eq_checks_double_size[where 'a="32 signed" and 'b=64]
lemmas signed_mult_eq_checks32_to_64 = signed_mult_eq_checks32_to_64'[simplified]
lemmas sdiv_word32_max' = sdiv_word_max [where 'a=32] sdiv_word_max [where 'a="32 signed"]
lemmas sdiv_word32_max = sdiv_word32_max'[simplified word_size, simplified]
lemmas sdiv_word32_min' = sdiv_word_min [where 'a=32] sdiv_word_min [where 'a="32 signed"]
lemmas sdiv_word32_min = sdiv_word32_min' [simplified word_size, simplified]
lemmas sint32_of_int_eq' = sint_of_int_eq [where 'a=32]
lemmas sint32_of_int_eq = sint32_of_int_eq' [simplified]
lemma ucast_of_nats [simp]:
"(ucast (of_nat x :: word32) :: sword32) = (of_nat x)"
"(ucast (of_nat x :: word32) :: 16 sword) = (of_nat x)"
"(ucast (of_nat x :: word32) :: 8 sword) = (of_nat x)"
"(ucast (of_nat x :: 16 word) :: 16 sword) = (of_nat x)"
"(ucast (of_nat x :: 16 word) :: 8 sword) = (of_nat x)"
"(ucast (of_nat x :: 8 word) :: 8 sword) = (of_nat x)"
by (simp_all add: of_nat_take_bit take_bit_word_eq_self unsigned_of_nat)
lemmas signed_shift_guard_simpler_32'
= power_strict_increasing_iff[where b="2 :: nat" and y=31]
lemmas signed_shift_guard_simpler_32 = signed_shift_guard_simpler_32'[simplified]
lemma word32_31_less:
"31 < len_of TYPE (32 signed)" "31 > (0 :: nat)"
"31 < len_of TYPE (32)" "31 > (0 :: nat)"
by auto
lemmas signed_shift_guard_to_word_32
= signed_shift_guard_to_word[OF word32_31_less(1-2)]
signed_shift_guard_to_word[OF word32_31_less(3-4)]
lemma has_zero_byte:
"~~ (((((v::word32) && 0x7f7f7f7f) + 0x7f7f7f7f) || v) || 0x7f7f7f7f) ≠ 0
⟹ v && 0xff000000 = 0 ∨ v && 0xff0000 = 0 ∨ v && 0xff00 = 0 ∨ v && 0xff = 0"
by word_bitwise auto
lemma mask_step_down_32:
‹∃x. mask x = b› if ‹b && 1 = 1›
and ‹∃x. x < 32 ∧ mask x = b >> 1› for b :: ‹32word›
proof -
from ‹b && 1 = 1› have ‹odd b›
by (auto simp add: mod_2_eq_odd and_one_eq)
then have ‹b mod 2 = 1›
using odd_iff_mod_2_eq_one by blast
from ‹∃x. x < 32 ∧ mask x = b >> 1› obtain x where ‹x < 32› ‹mask x = b >> 1› by blast
then have ‹mask x = b div 2›
using shiftr1_is_div_2 [of b] by simp
with ‹b mod 2 = 1› have ‹2 * mask x + 1 = 2 * (b div 2) + b mod 2›
by (simp only:)
also have ‹… = b›
by (simp add: mult_div_mod_eq)
finally have ‹2 * mask x + 1 = b› .
moreover have ‹mask (Suc x) = 2 * mask x + (1 :: 'a::len word)›
by (simp add: mask_Suc_rec)
ultimately show ?thesis
by auto
qed
lemma unat_of_int_32:
"⟦i ≥ 0; i ≤2 ^ 31⟧ ⟹ (unat ((of_int i)::sword32)) = nat i"
by (simp add: unsigned_of_int nat_take_bit_eq take_bit_nat_eq_self)
lemmas word_ctz_not_minus_1_32 = word_ctz_not_minus_1[where 'a=32, simplified]
lemma cast_chunk_assemble_id_64[simp]:
"(((ucast ((ucast (x::64 word))::32 word))::64 word) || (((ucast ((ucast (x >> 32))::32 word))::64 word) << 32)) = x"
by (simp add:cast_chunk_assemble_id)
lemma cast_chunk_assemble_id_64'[simp]:
"(((ucast ((scast (x::64 word))::32 word))::64 word) || (((ucast ((scast (x >> 32))::32 word))::64 word) << 32)) = x"
by (simp add:cast_chunk_scast_assemble_id)
lemma cast_down_u64: "(scast::64 word ⇒ 32 word) = (ucast::64 word ⇒ 32 word)"
by (subst down_cast_same[symmetric]; simp add:is_down)+
lemma cast_down_s64: "(scast::64 sword ⇒ 32 word) = (ucast::64 sword ⇒ 32 word)"
by (subst down_cast_same[symmetric]; simp add:is_down)
lemma word32_and_max_simp:
‹x AND 0xFFFFFFFF = x› for x :: ‹32 word›
using word_and_full_mask_simp [of x]
by (simp add: numeral_eq_Suc mask_Suc_exp)
end
end