Theory Word_64
section "Words of Length 64"
theory Word_64
imports
Word_Lemmas
Word_Names
Word_Syntax
Rsplit
More_Word_Operations
begin
context
includes bit_operations_syntax
begin
lemma len64: "len_of (x :: 64 itself) = 64" by simp
lemma ucast_8_64_inj:
"inj (ucast :: 8 word ⇒ 64 word)"
by (rule down_ucast_inj) (clarsimp simp: is_down_def target_size source_size)
lemmas unat_power_lower64' = unat_power_lower[where 'a=64]
lemmas word64_less_sub_le' = word_less_sub_le[where 'a = 64]
lemmas word64_power_less_1' = word_power_less_1[where 'a = 64]
lemmas unat_of_nat64' = unat_of_nat_eq[where 'a=64]
lemmas unat_mask_word64' = unat_mask[where 'a=64]
lemmas word64_minus_one_le' = word_minus_one_le[where 'a=64]
lemmas word64_minus_one_le = word64_minus_one_le'[simplified]
lemma less_4_cases:
"(x::word64) < 4 ⟹ x=0 ∨ x=1 ∨ x=2 ∨ x=3"
apply clarsimp
apply (drule word_less_cases, erule disjE, simp, simp)+
done
lemma ucast_le_ucast_8_64:
"(ucast x ≤ (ucast y :: word64)) = (x ≤ (y :: 8 word))"
by (simp add: ucast_le_ucast)
lemma eq_2_64_0:
"(2 ^ 64 :: word64) = 0"
by simp
lemmas mask_64_max_word = max_word_mask [symmetric, where 'a=64, simplified]
lemma of_nat64_n_less_equal_power_2:
"n < 64 ⟹ ((of_nat n)::64 word) < 2 ^ n"
by (rule of_nat_n_less_equal_power_2, clarsimp simp: word_size)
lemma unat_ucast_10_64 :
fixes x :: "10 word"
shows "unat (ucast x :: word64) = unat x"
by transfer simp
lemma word64_bounds:
"- (2 ^ (size (x :: word64) - 1)) = (-9223372036854775808 :: int)"
"((2 ^ (size (x :: word64) - 1)) - 1) = (9223372036854775807 :: int)"
"- (2 ^ (size (y :: 64 signed word) - 1)) = (-9223372036854775808 :: int)"
"((2 ^ (size (y :: 64 signed word) - 1)) - 1) = (9223372036854775807 :: int)"
by (simp_all add: word_size)
lemmas signed_arith_ineq_checks_to_eq_word64'
= signed_arith_ineq_checks_to_eq[where 'a=64]
signed_arith_ineq_checks_to_eq[where 'a="64 signed"]
lemmas signed_arith_ineq_checks_to_eq_word64
= signed_arith_ineq_checks_to_eq_word64' [unfolded word64_bounds]
lemmas signed_mult_eq_checks64_to_64'
= signed_mult_eq_checks_double_size[where 'a=64 and 'b=64]
signed_mult_eq_checks_double_size[where 'a="64 signed" and 'b=64]
lemmas signed_mult_eq_checks64_to_64 = signed_mult_eq_checks64_to_64'[simplified]
lemmas sdiv_word64_max' = sdiv_word_max [where 'a=64] sdiv_word_max [where 'a="64 signed"]
lemmas sdiv_word64_max = sdiv_word64_max'[simplified word_size, simplified]
lemmas sdiv_word64_min' = sdiv_word_min [where 'a=64] sdiv_word_min [where 'a="64 signed"]
lemmas sdiv_word64_min = sdiv_word64_min' [simplified word_size, simplified]
lemmas sint64_of_int_eq' = sint_of_int_eq [where 'a=64]
lemmas sint64_of_int_eq = sint64_of_int_eq' [simplified]
lemma ucast_of_nats [simp]:
"(ucast (of_nat x :: word64) :: sword64) = (of_nat x)"
"(ucast (of_nat x :: word64) :: 16 sword) = (of_nat x)"
"(ucast (of_nat x :: word64) :: 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_64'
= power_strict_increasing_iff[where b="2 :: nat" and y=31]
lemmas signed_shift_guard_simpler_64 = signed_shift_guard_simpler_64'[simplified]
lemma word64_31_less:
"31 < len_of TYPE (64 signed)" "31 > (0 :: nat)"
"31 < len_of TYPE (64)" "31 > (0 :: nat)"
by auto
lemmas signed_shift_guard_to_word_64
= signed_shift_guard_to_word[OF word64_31_less(1-2)]
signed_shift_guard_to_word[OF word64_31_less(3-4)]
lemma mask_step_down_64:
‹∃x. mask x = b› if ‹b && 1 = 1›
and ‹∃x. x < 64 ∧ mask x = b >> 1› for b :: ‹64word›
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 < 64 ∧ mask x = b >> 1› obtain x where ‹x < 64› ‹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_64:
"⟦i ≥ 0; i ≤ 2 ^ 63⟧ ⟹ (unat ((of_int i)::sword64)) = nat i"
by (simp add: unsigned_of_int nat_take_bit_eq take_bit_nat_eq_self)
lemmas word_ctz_not_minus_1_64 = word_ctz_not_minus_1[where 'a=64, simplified]
lemma word64_and_max_simp:
‹x AND 0xFFFFFFFFFFFFFFFF = x› for x :: ‹64 word›
using word_and_full_mask_simp [of x]
by (simp add: numeral_eq_Suc mask_Suc_exp)
end
end