# Theory More_Arithmetic

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

section ‹Arithmetic lemmas›

theory More_Arithmetic
imports Main "HOL-Library.Type_Length" "HOL-Library.Bit_Operations"
begin

declare iszero_0 [intro]

declare min.absorb1 [simp] min.absorb2 [simp]

lemma n_less_equal_power_2 [simp]:
"n < 2 ^ n"
by (fact less_exp)

lemma min_pm [simp]: "min a b + (a - b) = a"
for a b :: nat
by arith

lemma min_pm1 [simp]: "a - b + min a b = a"
for a b :: nat
by arith

lemma rev_min_pm [simp]: "min b a + (a - b) = a"
for a b :: nat
by arith

lemma rev_min_pm1 [simp]: "a - b + min b a = a"
for a b :: nat
by arith

lemma min_minus [simp]: "min m (m - k) = m - k"
for m k :: nat
by arith

lemma min_minus' [simp]: "min (m - k) m = m - k"
for m k :: nat
by arith

lemma nat_less_power_trans:
fixes n :: nat
assumes nv: "n < 2 ^ (m - k)"
and     kv: "k ≤ m"
shows "2 ^ k * n < 2 ^ m"
proof (rule order_less_le_trans)
show "2 ^ k * n < 2 ^ k * 2 ^ (m - k)"
by (rule mult_less_mono2 [OF nv zero_less_power]) simp
show "(2::nat) ^ k * 2 ^ (m - k) ≤ 2 ^ m" using nv kv
qed

lemma nat_le_power_trans:
fixes n :: nat
shows "⟦n ≤ 2 ^ (m - k); k ≤ m⟧ ⟹ 2 ^ k * n ≤ 2 ^ m"

fixes x :: nat
assumes yv: "y < 2 ^ n"
and     xv: "x < 2 ^ m"
and     mn: "sz = m + n"
shows   "x * 2 ^ n + y < 2 ^ sz"
proof (subst mn)
from yv obtain qy where "y + qy = 2 ^ n" and "0 < qy"

have "x * 2 ^ n + y < x * 2 ^ n + 2 ^ n" by simp fact+
also have "… = (x + 1) * 2 ^ n" by simp
also have "… ≤ 2 ^ (m + n)" using xv
by (subst power_add) (rule mult_le_mono1, simp)
finally show "x * 2 ^ n + y < 2 ^ (m + n)" .
qed

lemma nat_power_less_diff:
assumes lt: "(2::nat) ^ n * q < 2 ^ m"
shows "q < 2 ^ (m - n)"
using lt
proof (induct n arbitrary: m)
case 0
then show ?case by simp
next
case (Suc n)

have ih: "⋀m. 2 ^ n * q < 2 ^ m ⟹ q < 2 ^ (m - n)"
and prem: "2 ^ Suc n * q < 2 ^ m" by fact+

show ?case
proof (cases m)
case 0
then show ?thesis using Suc by simp
next
case (Suc m')
then show ?thesis using prem
qed
qed

lemma power_2_mult_step_le:
"⟦n' ≤ n; 2 ^ n' * k' < 2 ^ n * k⟧ ⟹ 2 ^ n' * (k' + 1) ≤ 2 ^ n * (k::nat)"
apply (cases "n'=n", simp)
apply (metis Suc_leI le_refl mult_Suc_right mult_le_mono semiring_normalization_rules(7))
apply (drule (1) le_neq_trans)
apply clarsimp
apply (subgoal_tac "∃m. n = n' + m")
prefer 2
apply (metis Suc_leI mult.assoc mult_Suc_right nat_mult_le_cancel_disj)
done

lemma nat_mult_power_less_eq:
"b > 0 ⟹ (a * b ^ n < (b :: nat) ^ m) = (a < b ^ (m - n))"
using mult_less_cancel2[where m = a and k = "b ^ n" and n="b ^ (m - n)"]
mult_less_cancel2[where m="a * b ^ (n - m)" and k="b ^ m" and n=1]
apply (simp add: max_def split: if_split_asm)
done

lemma diff_diff_less:
"(i < m - (m - (n :: nat))) = (i < m ∧ i < n)"
by auto

lemma small_powers_of_2:
‹x < 2 ^ (x - 1)› if ‹x ≥ 3› for x :: nat
proof -
define m where ‹m = x - 3›
with that have ‹x = m + 3›
by simp
moreover have ‹m + 3 < 4 * 2 ^ m›
by (induction m) simp_all
ultimately show ?thesis
by simp
qed

end


# Theory More_Divides

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

section ‹Lemmas on division›

theory More_Divides
imports
"HOL-Library.Word"
begin

declare div_eq_dividend_iff [simp]

lemma int_div_same_is_1 [simp]:
‹a div b = a ⟷ b = 1› if ‹0 < a› for a b :: int
using that by (metis div_by_1 abs_ge_zero abs_of_pos int_div_less_self neq_iff
nonneg1_imp_zdiv_pos_iff zabs_less_one_iff)

lemma int_div_minus_is_minus1 [simp]:
‹a div b = - a ⟷ b = - 1› if ‹0 > a› for a b :: int
using that by (metis div_minus_right equation_minus_iff int_div_same_is_1 neg_0_less_iff_less)

lemma nat_div_eq_Suc_0_iff: "n div m = Suc 0 ⟷ m ≤ n ∧ n < 2 * m"
apply auto
using div_greater_zero_iff apply fastforce
apply (metis One_nat_def div_greater_zero_iff dividend_less_div_times mult.right_neutral mult_Suc mult_numeral_1 numeral_2_eq_2 zero_less_numeral)
done

lemma diff_mod_le:
‹a - a mod b ≤ d - b› if ‹a < d› ‹b dvd d› for a b d :: nat
using that
apply(subst minus_mod_eq_mult_div)
apply(clarsimp simp: dvd_def)
apply(cases ‹b = 0›)
apply simp
apply(subgoal_tac "a div b ≤ k - 1")
prefer 2
apply(subgoal_tac "a div b < k")
apply(subgoal_tac "b * (a div b) < b * ((b * k) div b)")
apply clarsimp
apply(subst div_mult_self1_is_m)
apply arith
apply(rule le_less_trans)
apply simp
apply(subst mult.commute)
apply(rule div_times_less_eq_dividend)
apply assumption
apply clarsimp
apply(subgoal_tac "b * (a div b) ≤ b * (k - 1)")
apply(erule le_trans)
apply simp
done

lemma one_mod_exp_eq_one [simp]:
"1 mod (2 * 2 ^ n) = (1::int)"
using power_gt1 [of 2 n] by (auto intro: mod_pos_pos_trivial)

lemma int_mod_lem: "0 < n ⟹ 0 ≤ b ∧ b < n ⟷ b mod n = b"
for b n :: int
apply safe
apply (erule (1) mod_pos_pos_trivial)
apply (erule_tac [!] subst)
apply auto
done

lemma int_mod_ge': "b < 0 ⟹ 0 < n ⟹ b + n ≤ b mod n"
for b n :: int

lemma int_mod_le': "0 ≤ b - n ⟹ b mod n ≤ b - n"
for b n :: int
by (metis minus_mod_self2 zmod_le_nonneg_dividend)

lemma emep1: "even n ⟹ even d ⟹ 0 ≤ d ⟹ (n + 1) mod d = (n mod d) + 1"
for n d :: int

lemma m1mod2k: "- 1 mod 2 ^ n = (2 ^ n - 1 :: int)"
by (rule zmod_minus1) simp

lemma sb_inc_lem: "a + 2^k < 0 ⟹ a + 2^k + 2^(Suc k) ≤ (a + 2^k) mod 2^(Suc k)"
for a :: int
using int_mod_ge' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"]
by simp

lemma sb_inc_lem': "a < - (2^k) ⟹ a + 2^k + 2^(Suc k) ≤ (a + 2^k) mod 2^(Suc k)"
for a :: int
by (rule sb_inc_lem) simp

lemma sb_dec_lem: "0 ≤ - (2 ^ k) + a ⟹ (a + 2 ^ k) mod (2 * 2 ^ k) ≤ - (2 ^ k) + a"
for a :: int
using int_mod_le'[where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"] by simp

lemma sb_dec_lem': "2 ^ k ≤ a ⟹ (a + 2 ^ k) mod (2 * 2 ^ k) ≤ - (2 ^ k) + a"
for a :: int
by (rule sb_dec_lem) simp

lemma mod_2_neq_1_eq_eq_0: "k mod 2 ≠ 1 ⟷ k mod 2 = 0"
for k :: int
by (fact not_mod_2_eq_1_eq_0)

lemma z1pmod2: "(2 * b + 1) mod 2 = (1::int)"
for b :: int
by arith

lemma p1mod22k': "(1 + 2 * b) mod (2 * 2 ^ n) = 1 + 2 * (b mod 2 ^ n)"
for b :: int
by (rule pos_zmod_mult_2) simp

lemma p1mod22k: "(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + 1"
for b :: int

lemma pos_mod_sign2:
‹0 ≤ a mod 2› for a :: int
by simp

lemma pos_mod_bound2:
‹a mod 2 < 2› for a :: int
by simp

lemma nmod2: "n mod 2 = 0 ∨ n mod 2 = 1"
for n :: int
by arith

lemma eme1p:
"even n ⟹ even d ⟹ 0 ≤ d ⟹ (1 + n) mod d = 1 + n mod d" for n d :: int
using emep1 [of n d] by (simp add: ac_simps)

lemma m1mod22k:
‹- 1 mod (2 * 2 ^ n) = 2 * 2 ^ n - (1::int)›

lemma z1pdiv2: "(2 * b + 1) div 2 = b"
for b :: int
by arith

lemma zdiv_le_dividend:
‹0 ≤ a ⟹ 0 < b ⟹ a div b ≤ a› for a b :: int
by (metis div_by_1 int_one_le_iff_zero_less zdiv_mono2 zero_less_one)

lemma axxmod2: "(1 + x + x) mod 2 = 1 ∧ (0 + x + x) mod 2 = 0"
for x :: int
by arith

lemma axxdiv2: "(1 + x + x) div 2 = x ∧ (0 + x + x) div 2 = x"
for x :: int
by arith

lemmas rdmods =
mod_minus_eq [symmetric]
mod_diff_left_eq [symmetric]
mod_diff_right_eq [symmetric]
mod_mult_right_eq [symmetric]
mod_mult_left_eq [symmetric]

lemma mod_plus_right: "(a + x) mod m = (b + x) mod m ⟷ a mod m = b mod m"
for a b m x :: nat
by (induct x) (simp_all add: mod_Suc, arith)

lemma nat_minus_mod: "(n - n mod m) mod m = 0"
for m n :: nat
by (induct n) (simp_all add: mod_Suc)

lemmas nat_minus_mod_plus_right =
trans [OF nat_minus_mod mod_0 [symmetric],
THEN mod_plus_right [THEN iffD2], simplified]

mod_mult_eq mod_diff_eq
mod_minus_eq

lemmas push_mods = push_mods' [THEN eq_reflection]
lemmas pull_mods = push_mods [symmetric] rdmods [THEN eq_reflection]

lemma nat_mod_eq: "b < n ⟹ a mod n = b mod n ⟹ a mod n = b"
for a b n :: nat
by (induct a) auto

lemmas nat_mod_eq' = refl [THEN [2] nat_mod_eq]

lemma nat_mod_lem: "0 < n ⟹ b < n ⟷ b mod n = b"
for b n :: nat
apply safe
apply (erule nat_mod_eq')
apply (erule subst)
apply (erule mod_less_divisor)
done

lemma mod_nat_add: "x < z ⟹ y < z ⟹ (x + y) mod z = (if x + y < z then x + y else x + y - z)"
for x y z :: nat
apply (rule nat_mod_eq)
apply auto
apply (rule trans)
apply (rule le_mod_geq)
apply simp
apply (rule nat_mod_eq')
apply arith
done

lemma mod_nat_sub: "x < z ⟹ (x - y) mod z = x - y"
for x y :: nat
by (rule nat_mod_eq') arith

lemma int_mod_eq: "0 ≤ b ⟹ b < n ⟹ a mod n = b mod n ⟹ a mod n = b"
for a b n :: int
by (metis mod_pos_pos_trivial)

lemma zmde:
‹b * (a div b) = a - a mod b› for a b :: ‹'a::{group_add,semiring_modulo}›
using mult_div_mod_eq [of b a] by (simp add: eq_diff_eq)

(* already have this for naturals, div_mult_self1/2, but not for ints *)
lemma zdiv_mult_self: "m ≠ 0 ⟹ (a + m * n) div m = a div m + n"
for a m n :: int
by simp

lemma mod_power_lem: "a > 1 ⟹ a ^ n mod a ^ m = (if m ≤ n then 0 else a ^ n)"
for a :: int

lemma nonneg_mod_div: "0 ≤ a ⟹ 0 ≤ b ⟹ 0 ≤ (a mod b) ∧ 0 ≤ a div b"
for a b :: int
by (cases "b = 0") (auto intro: pos_imp_zdiv_nonneg_iff [THEN iffD2])

lemma mod_exp_less_eq_exp:
‹a mod 2 ^ n < 2 ^ n› for a :: int
by (rule pos_mod_bound) simp

lemma div_mult_le:
‹a div b * b ≤ a› for a b :: nat
by (fact div_times_less_eq_dividend)

lemma power_sub:
fixes a :: nat
assumes lt: "n ≤ m"
and     av: "0 < a"
shows "a ^ (m - n) = a ^ m div a ^ n"
proof (subst nat_mult_eq_cancel1 [symmetric])
show "(0::nat) < a ^ n" using av by simp
next
from lt obtain q where mv: "n + q = m"

have "a ^ n * (a ^ m div a ^ n) = a ^ m"
proof (subst mult.commute)
have "a ^ m = (a ^ m div a ^ n) * a ^ n + a ^ m mod a ^ n"
by (rule  div_mult_mod_eq [symmetric])

moreover have "a ^ m mod a ^ n = 0"
by (subst mod_eq_0_iff_dvd, subst dvd_def, rule exI [where x = "a ^ q"],
(subst power_add [symmetric] mv)+, rule refl)

ultimately show "(a ^ m div a ^ n) * a ^ n = a ^ m" by simp
qed

then show "a ^ n * a ^ (m - n) = a ^ n * (a ^ m div a ^ n)" using lt
qed

lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
apply (cut_tac m = q and n = c in mod_less_divisor)
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
apply (erule_tac P = "%x. lhs < rhs x" for lhs rhs in ssubst)
done

lemma less_two_pow_divD:
"⟦ (x :: nat) < 2 ^ n div 2 ^ m ⟧
⟹ n ≥ m ∧ (x < 2 ^ (n - m))"
apply (rule context_conjI)
apply (rule ccontr)
done

lemma less_two_pow_divI:
"⟦ (x :: nat) < 2 ^ (n - m); m ≤ n ⟧ ⟹ x < 2 ^ n div 2 ^ m"

lemmas m2pths = pos_mod_sign mod_exp_less_eq_exp

lemmas int_mod_eq' = mod_pos_pos_trivial (* FIXME delete *)

lemmas int_mod_le = zmod_le_nonneg_dividend (* FIXME: delete *)

lemma power_mod_div:
fixes x :: "nat"
shows "x mod 2 ^ n div 2 ^ m = x div 2 ^ m mod 2 ^ (n - m)" (is "?LHS = ?RHS")
proof (cases "n ≤ m")
case True
then have "?LHS = 0"
apply -
apply (rule div_less)
apply (rule order_less_le_trans [OF mod_less_divisor]; simp)
done
also have "… = ?RHS" using True
by simp
finally show ?thesis .
next
case False
then have lt: "m < n" by simp
then obtain q where nv: "n = m + q" and "0 < q"

then have "x mod 2 ^ n = 2 ^ m * (x div 2 ^ m mod 2 ^ q) + x mod 2 ^ m"

then have "?LHS = x div 2 ^ m mod 2 ^ q"

also have "… = ?RHS" using nv
by simp

finally show ?thesis .
qed

lemma mod_mod_power:
fixes k :: nat
shows "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ (min m n)"
proof (cases "m ≤ n")
case True

then have "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ m"
apply -
apply (subst mod_less [where n = "2 ^ n"])
apply (rule order_less_le_trans [OF mod_less_divisor])
apply simp+
done
also have "… = k mod  2 ^ (min m n)" using True by simp
finally show ?thesis .
next
case False
then have "n < m" by simp
then obtain d where md: "m = n + d"
then have "k mod 2 ^ m = 2 ^ n * (k div 2 ^ n mod 2 ^ d) + k mod 2 ^ n"
then have "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ n"
then show ?thesis using False
by simp
qed

lemma mod_div_equality_div_eq:
"a div b * b = (a - (a mod b) :: int)"

lemma zmod_helper:
"n mod m = k ⟹ ((n :: int) + a) mod m = (k + a) mod m"

lemma int_div_sub_1:
"⟦ m ≥ 1 ⟧ ⟹ (n - (1 :: int)) div m = (if m dvd n then (n div m) - 1 else n div m)"
apply (subgoal_tac "m = 0 ∨ (n - (1 :: int)) div m = (if m dvd n then (n div m) - 1 else n div m)")
apply fastforce
apply (subst mult_cancel_right[symmetric])
apply (simp only: left_diff_distrib split: if_split)
apply (simp only: mod_div_equality_div_eq)
apply (clarsimp simp: field_simps)
apply (clarsimp simp: dvd_eq_mod_eq_0)
apply (cases "m = 1")
apply simp
apply (subst mod_diff_eq[symmetric], simp add: zmod_minus1)
apply clarsimp
apply (rule mod_pos_pos_trivial)
apply simp
apply simp
apply (cases "(n - 1) mod m = m - 1")
apply (drule zmod_helper[where a=1])
apply simp
apply (subgoal_tac "1 + (n - 1) mod m ≤ m")
apply simp
apply simp
done

lemma power_minus_is_div:
"b ≤ a ⟹ (2 :: nat) ^ (a - b) = 2 ^ a div 2 ^ b"
apply (induct a arbitrary: b)
apply simp
apply (erule le_SucE)
apply simp
done

lemma two_pow_div_gt_le:
"v < 2 ^ n div (2 ^ m :: nat) ⟹ m ≤ n"
by (clarsimp dest!: less_two_pow_divD)

lemma td_gal_lt:
‹0 < c ⟹ a < b * c ⟷ a div c < b›
for a b c :: nat
apply (auto dest: less_mult_imp_div_less)
apply (metis div_le_mono div_mult_self_is_m leD leI)
done

lemma td_gal:
‹0 < c ⟹ b * c ≤ a  ⟷ b ≤ a div c›
for a b c :: nat
by (meson not_le td_gal_lt)

end


# Theory More_Word

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

section ‹Lemmas on words›

theory More_Word
imports
"HOL-Library.Word"
More_Arithmetic
More_Divides
begin

lemma unat_power_lower [simp]:
"unat ((2::'a::len word) ^ n) = 2 ^ n" if "n < LENGTH('a::len)"
using that by transfer simp

lemma unat_p2: "n < LENGTH('a :: len) ⟹ unat (2 ^ n :: 'a word) = 2 ^ n"
by (fact unat_power_lower)

lemma word_div_lt_eq_0:
"x < y ⟹ x div y = 0" for x :: "'a :: len word"
by transfer simp

lemma word_div_eq_1_iff: "n div m = 1 ⟷ n ≥ m ∧ unat n < 2 * unat (m :: 'a :: len word)"
apply (simp only: word_arith_nat_defs word_le_nat_alt word_of_nat_eq_iff flip: nat_div_eq_Suc_0_iff)
apply (simp flip: unat_div unsigned_take_bit_eq)
done

lemma shiftl_power:
"(shiftl1 ^^ x) (y::'a::len word) = 2 ^ x * y"
apply (induct x)
apply simp
done

lemma AND_twice [simp]:
"(w AND m) AND m = w AND m"
by (fact and.right_idem)

"w AND m = z ⟹ w AND m' = z' ⟹ w AND (m OR m') = (z OR z')"
for w m m' z z' :: ‹'a::len word›

lemma p2_gt_0:
"(0 < (2 ^ n :: 'a :: len word)) = (n < LENGTH('a))"
by (simp add : word_gt_0 not_le)

lemma uint_2p_alt:
‹n < LENGTH('a::len) ⟹ uint ((2::'a::len word) ^ n) = 2 ^ n›
using p2_gt_0 [of n, where ?'a = 'a] by (simp add: uint_2p)

lemma p2_eq_0:
‹(2::'a::len word) ^ n = 0 ⟷ LENGTH('a::len) ≤ n›
by (fact exp_eq_zero_iff)

lemma p2len:
‹(2 :: 'a word) ^ LENGTH('a::len) = 0›
by simp

"w AND NOT (mask n) = (w div 2^n) * 2^n"
for w :: ‹'a::len word›
by (rule bit_word_eqI)
(auto simp add: bit_simps simp flip: push_bit_eq_mult drop_bit_eq_div)

"n < size w ⟹ w AND NOT (mask n) = ((w div (2 ^ n)) * (2 ^ n))"
for w :: ‹'a::len word›

"w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
for w :: ‹'a::len word›
by (rule bit_word_eqI)
(auto simp add: bit_simps word_size simp flip: push_bit_eq_mult drop_bit_eq_div)

"0 < n ⟹ w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
for w :: ‹'a::len word›

lemma mask_2pm1: "mask n = 2 ^ n - (1 :: 'a::len word)"

"x + 2 ^ n - 1 = x + mask n"
for x :: ‹'a::len word›

lemma word_and_mask_le_2pm1: "w AND mask n ≤ 2 ^ n - 1"
for w :: ‹'a::len word›

lemma is_aligned_AND_less_0:
"u AND mask n = 0 ⟹ v < 2^n ⟹ u AND v = 0"
for u v :: ‹'a::len word›
done

lemma le_shiftr1:
‹shiftr1 u ≤ shiftr1 v› if ‹u ≤ v›
using that proof transfer
fix k l :: int
assume ‹take_bit LENGTH('a) k ≤ take_bit LENGTH('a) l›
then have ‹take_bit LENGTH('a) (drop_bit 1 (take_bit LENGTH('a) k))
≤ take_bit LENGTH('a) (drop_bit 1 (take_bit LENGTH('a) l))›
done
then show ‹take_bit LENGTH('a) (take_bit LENGTH('a) k div 2)
≤ take_bit LENGTH('a) (take_bit LENGTH('a) l div 2)›
qed

for w :: ‹'a::len word›
apply (cases ‹n ≥ LENGTH('a)›; transfer)
apply auto
apply (metis take_bit_int_less_exp)
apply (metis min_def nat_less_le take_bit_int_eq_self_iff take_bit_take_bit)
done

‹w ≤ mask n ⟷ take_bit n w = w›
for w :: ‹'a::len word›

lemma NOT_eq:
"NOT (x :: 'a :: len word) = - x - 1"
apply (cut_tac x = "x" in word_add_not)
apply (drule eq_diff_eq [THEN iffD2])
by simp

lemma NOT_mask: "NOT (mask n :: 'a::len word) = - (2 ^ n)"

lemma le_m1_iff_lt: "(x > (0 :: 'a :: len word)) = ((y ≤ x - 1) = (y < x))"
by uint_arith

lemma gt0_iff_gem1:
‹0 < x ⟷ x - 1 < x›
for x :: ‹'a::len word›

lemma power_2_ge_iff:
‹2 ^ n - (1 :: 'a::len word) < 2 ^ n ⟷ n < LENGTH('a)›
using gt0_iff_gem1 p2_gt_0 by blast

"n < len_of TYPE ('a) = (((w :: 'a :: len word) ≤ mask n) = (w < 2 ^ n))"
unfolding mask_2pm1 by (rule trans [OF p2_gt_0 [THEN sym] le_m1_iff_lt])

‹n < LENGTH('a) ⟹ mask n < (2 :: 'a::len word) ^ n›

lemma word_unat_power:
"(2 :: 'a :: len word) ^ n = of_nat (2 ^ n)"
by simp

lemma of_nat_mono_maybe:
assumes xlt: "x < 2 ^ len_of TYPE ('a)"
shows   "y < x ⟹ of_nat y < (of_nat x :: 'a :: len word)"
apply (subst word_less_nat_alt)
apply (subst unat_of_nat)+
apply (subst mod_less)
apply (erule order_less_trans [OF _ xlt])
apply (subst mod_less [OF xlt])
apply assumption
done

lemma word_and_max_word:
fixes a::"'a::len word"
shows "x = max_word ⟹ a AND x = a"
by simp

‹x AND mask LENGTH('a) = x› for x :: ‹'a::len word›
proof (rule bit_eqI)
fix n
assume ‹2 ^ n ≠ (0 :: 'a word)›
then have ‹n < LENGTH('a)›
by simp
then show ‹bit (x AND Bit_Operations.mask LENGTH('a)) n ⟷ bit x n›
qed

lemma of_int_uint:
"of_int (uint x) = x"
by (fact word_of_int_uint)

corollary word_plus_and_or_coroll:
"x AND y = 0 ⟹ x + y = x OR y"
for x y :: ‹'a::len word›
using word_plus_and_or[where x=x and y=y]
by simp

corollary word_plus_and_or_coroll2:
"(x AND w) + (x AND NOT w) = x"
for x w :: ‹'a::len word›
apply (simp flip: bit.conj_disj_distrib)
done

lemma word_plus_mono_left:
fixes x :: "'a :: len word"
shows "⟦y ≤ z; x ≤ x + z⟧ ⟹ y + x ≤ z + x"
by unat_arith

lemma less_Suc_unat_less_bound:
"n < Suc (unat (x :: 'a :: len word)) ⟹ n < 2 ^ LENGTH('a)"
by (auto elim!: order_less_le_trans intro: Suc_leI)

lemma up_ucast_inj:
"⟦ ucast x = (ucast y::'b::len word); LENGTH('a) ≤ len_of TYPE ('b) ⟧ ⟹ x = (y::'a::len word)"
by transfer (simp add: min_def split: if_splits)

lemmas ucast_up_inj = up_ucast_inj

lemma up_ucast_inj_eq:
"LENGTH('a) ≤ len_of TYPE ('b) ⟹ (ucast x = (ucast y::'b::len word)) = (x = (y::'a::len word))"
by (fastforce dest: up_ucast_inj)

lemma no_plus_overflow_neg:
"(x :: 'a :: len word) < -y ⟹ x ≤ x + y"

lemma ucast_ucast_eq:
"⟦ ucast x = (ucast (ucast y::'a word)::'c::len word); LENGTH('a) ≤ LENGTH('b);
LENGTH('b) ≤ LENGTH('c) ⟧ ⟹
x = ucast y" for x :: "'a::len word" and y :: "'b::len word"
apply transfer
apply (cases ‹LENGTH('c) = LENGTH('a)›)
done

lemma ucast_0_I:
"x = 0 ⟹ ucast x = 0"
by simp

fixes x :: "'a :: len word"
assumes yv: "y < 2 ^ n"
and     xv: "x < 2 ^ m"
and     mnv: "sz < LENGTH('a :: len)"
and    xv': "x < 2 ^ (LENGTH('a :: len) - n)"
and     mn: "sz = m + n"
shows   "x * 2 ^ n + y < 2 ^ sz"
proof (subst mn)
from mnv mn have nv: "n < LENGTH('a)" and mv: "m < LENGTH('a)"  by auto

have uy: "unat y < 2 ^ n"
by (rule order_less_le_trans [OF unat_mono [OF yv] order_eq_refl],
rule unat_power_lower[OF nv])

have ux: "unat x < 2 ^ m"
by (rule order_less_le_trans [OF unat_mono [OF xv] order_eq_refl],
rule unat_power_lower[OF mv])

then show "x * 2 ^ n + y < 2 ^ (m + n)" using ux uy nv mnv xv'
apply (subst word_less_nat_alt)
apply (subst unat_word_ariths)+
apply (subst mod_less)
apply simp
apply (subst mult.commute)
apply (rule nat_less_power_trans [OF _ order_less_imp_le [OF nv]])
apply (rule order_less_le_trans [OF unat_mono [OF xv']])
apply (cases "n = 0"; simp)
apply (subst unat_power_lower[OF nv])
apply (subst mod_less)
apply (erule order_less_le_trans [OF nat_add_offset_less], assumption)
apply (rule mn)
apply simp
done
qed

lemma word_less_power_trans:
fixes n :: "'a :: len word"
assumes nv: "n < 2 ^ (m - k)"
and     kv: "k ≤ m"
and     mv: "m < len_of TYPE ('a)"
shows "2 ^ k * n < 2 ^ m"
using nv kv mv
apply -
apply (subst word_less_nat_alt)
apply (subst unat_word_ariths)
apply (subst mod_less)
apply simp
apply (rule nat_less_power_trans)
apply (erule order_less_trans [OF unat_mono])
apply simp
apply simp
apply simp
apply (rule nat_less_power_trans)
apply (subst unat_power_lower[where 'a = 'a, symmetric])
apply simp
apply (erule unat_mono)
apply simp
done

lemma  word_less_power_trans2:
fixes n :: "'a::len word"
shows "⟦n < 2 ^ (m - k); k ≤ m; m < LENGTH('a)⟧ ⟹ n * 2 ^ k < 2 ^ m"
by (subst field_simps, rule word_less_power_trans)

lemma Suc_unat_diff_1:
fixes x :: "'a :: len word"
assumes lt: "1 ≤ x"
shows "Suc (unat (x - 1)) = unat x"
proof -
have "0 < unat x"
by (rule order_less_le_trans [where y = 1], simp, subst unat_1 [symmetric],
rule iffD1 [OF word_le_nat_alt lt])

then show ?thesis
by ((subst unat_sub [OF lt])+, simp only:  unat_1)
qed

lemma word_eq_unatI:
‹v = w› if ‹unat v = unat w›
using that by transfer (simp add: nat_eq_iff)

lemma word_div_sub:
fixes x :: "'a :: len word"
assumes yx: "y ≤ x"
and     y0: "0 < y"
shows "(x - y) div y = x div y - 1"
apply (rule word_eq_unatI)
apply (subst unat_div)
apply (subst unat_sub [OF yx])
apply (subst unat_sub)
apply (subst word_le_nat_alt)
apply (subst unat_div)
apply (subst le_div_geq)
apply (rule order_le_less_trans [OF _ unat_mono [OF y0]])
apply simp
apply (subst word_le_nat_alt [symmetric], rule yx)
apply simp
apply (subst unat_div)
apply (subst le_div_geq [OF _ iffD1 [OF word_le_nat_alt yx]])
apply (rule order_le_less_trans [OF _ unat_mono [OF y0]])
apply simp
apply simp
done

lemma word_mult_less_mono1:
fixes i :: "'a :: len word"
assumes ij: "i < j"
and    knz: "0 < k"
and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows  "i * k < j * k"
proof -
from ij ujk knz have jk: "unat i * unat k < 2 ^ len_of TYPE ('a)"
by (auto intro: order_less_subst2 simp: word_less_nat_alt elim: mult_less_mono1)

then show ?thesis using ujk knz ij
by (auto simp: word_less_nat_alt iffD1 [OF unat_mult_lem])
qed

lemma word_mult_less_dest:
fixes i :: "'a :: len word"
assumes ij: "i * k < j * k"
and    uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows  "i < j"
using uik ujk ij
by (auto simp: word_less_nat_alt iffD1 [OF unat_mult_lem] elim: mult_less_mono1)

lemma word_mult_less_cancel:
fixes k :: "'a :: len word"
assumes knz: "0 < k"
and    uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows "(i * k < j * k) = (i < j)"
by (rule iffI [OF word_mult_less_dest [OF _ uik ujk] word_mult_less_mono1 [OF _ knz ujk]])

lemma Suc_div_unat_helper:
assumes szv: "sz < LENGTH('a :: len)"
and   usszv: "us ≤ sz"
shows "2 ^ (sz - us) = Suc (unat (((2::'a :: len word) ^ sz - 1) div 2 ^ us))"
proof -
note usv = order_le_less_trans [OF usszv szv]

from usszv obtain q where qv: "sz = us + q" by (auto simp: le_iff_add)

have "Suc (unat (((2:: 'a word) ^ sz - 1) div 2 ^ us)) =
(2 ^ us + unat ((2:: 'a word) ^ sz - 1)) div 2 ^ us"
apply (subst unat_div unat_power_lower[OF usv])+
done

also have "… = ((2 ^ us - 1) + 2 ^ sz) div 2 ^ us" using szv

also have "… = 2 ^ q + ((2 ^ us - 1) div 2 ^ us)"
apply (subst qv)
apply (subst div_mult_self2; simp)
done

also have "… = 2 ^ (sz - us)" using qv by simp

finally show ?thesis ..
qed

lemma enum_word_nth_eq:
‹(Enum.enum :: 'a::len word list) ! n = word_of_nat n›
if ‹n < 2 ^ LENGTH('a)›
for n
using that by (simp add: enum_word_def)

lemma length_enum_word_eq:
‹length (Enum.enum :: 'a::len word list) = 2 ^ LENGTH('a)›

lemma unat_lt2p [iff]:
‹unat x < 2 ^ LENGTH('a)› for x :: ‹'a::len word›
by transfer simp

lemma of_nat_unat [simp]:
"of_nat ∘ unat = id"
by (rule ext, simp)

lemma Suc_unat_minus_one [simp]:
"x ≠ 0 ⟹ Suc (unat (x - 1)) = unat x"
by (metis Suc_diff_1 unat_gt_0 unat_minus_one)

fixes i :: "'a :: len word"
assumes le: "i + k ≤ j + k"
and    uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows  "i ≤ j"
using uik ujk le

fixes i :: "'a :: len word"
assumes ij: "i ≤ j"
and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows  "i + k ≤ j + k"
proof -
from ij ujk have jk: "unat i + unat k < 2 ^ len_of TYPE ('a)"
by (auto elim: order_le_less_subst2 simp: word_le_nat_alt elim: add_le_mono1)

then show ?thesis using ujk ij
by (auto simp: word_le_nat_alt iffD1 [OF unat_add_lem])
qed

fixes i :: "'a :: len word"
shows "⟦i ≤ j; unat j + unat k < 2 ^ LENGTH('a)⟧ ⟹ k + i ≤ k + j"
by (subst field_simps, subst field_simps, erule (1) word_add_le_mono1)

fixes i :: "'a :: len word"
assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and     ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows  "(i + k ≤ j + k) = (i ≤ j)"
proof
assume "i ≤ j"
show "i + k ≤ j + k" by (rule word_add_le_mono1) fact+
next
assume "i + k ≤ j + k"
show "i ≤ j" by (rule word_add_le_dest) fact+
qed

fixes i :: "'a :: len word"
assumes ij: "i < j"
and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows  "i + k < j + k"
proof -
from ij ujk have jk: "unat i + unat k < 2 ^ len_of TYPE ('a)"
by (auto elim: order_le_less_subst2 simp: word_less_nat_alt elim: add_less_mono1)

then show ?thesis using ujk ij
by (auto simp: word_less_nat_alt iffD1 [OF unat_add_lem])
qed

fixes i :: "'a :: len word"
assumes le: "i + k < j + k"
and    uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows  "i < j"
using uik ujk le

fixes i :: "'a :: len word"
assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
and     ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
shows  "(i + k < j + k) = (i < j)"
proof
assume "i < j"
show "i + k < j + k" by (rule word_add_less_mono1) fact+
next
assume "i + k < j + k"
show "i < j" by (rule word_add_less_dest) fact+
qed

lemma word_mult_less_iff:
fixes i :: "'a :: len word"
assumes knz: "0 < k"
and     uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and     ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows  "(i * k < j * k) = (i < j)"
using assms by (rule word_mult_less_cancel)

lemma word_le_imp_diff_le:
fixes n :: "'a::len word"
shows "⟦k ≤ n; n ≤ m⟧ ⟹ n - k ≤ m"
by (auto simp: unat_sub word_le_nat_alt)

lemma word_less_imp_diff_less:
fixes n :: "'a::len word"
shows "⟦k ≤ n; n < m⟧ ⟹ n - k < m"
by (clarsimp simp: unat_sub word_less_nat_alt
intro!: less_imp_diff_less)

lemma word_mult_le_mono1:
fixes i :: "'a :: len word"
assumes ij: "i ≤ j"
and    knz: "0 < k"
and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows  "i * k ≤ j * k"
proof -
from ij ujk knz have jk: "unat i * unat k < 2 ^ len_of TYPE ('a)"
by (auto elim: order_le_less_subst2 simp: word_le_nat_alt elim: mult_le_mono1)

then show ?thesis using ujk knz ij
by (auto simp: word_le_nat_alt iffD1 [OF unat_mult_lem])
qed

lemma word_mult_le_iff:
fixes i :: "'a :: len word"
assumes knz: "0 < k"
and     uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
and     ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
shows  "(i * k ≤ j * k) = (i ≤ j)"
proof
assume "i ≤ j"
show "i * k ≤ j * k" by (rule word_mult_le_mono1) fact+
next
assume p: "i * k ≤ j * k"

have "0 < unat k" using knz by (simp add: word_less_nat_alt)
then show "i ≤ j" using p
by (clarsimp simp: word_le_nat_alt iffD1 [OF unat_mult_lem uik]
iffD1 [OF unat_mult_lem ujk])
qed

lemma word_diff_less:
fixes n :: "'a :: len word"
shows "⟦0 < n; 0 < m; n ≤ m⟧ ⟹ m - n < m"
apply (subst word_less_nat_alt)
apply (subst unat_sub)
apply assumption
apply (rule diff_less)
done

fixes x :: "'a :: len word"
shows "⟦ p + w ≤ x; p ≤ p + w ⟧ ⟹ p ≤ x"
by unat_arith

lemma word_random:
fixes x :: "'a :: len word"
shows "⟦ p ≤ p + x'; x ≤ x' ⟧ ⟹ p ≤ p + x"
by unat_arith

lemma word_sub_mono:
"⟦ a ≤ c; d ≤ b; a - b ≤ a; c - d ≤ c ⟧
⟹ (a - b) ≤ (c - d :: 'a :: len word)"
by unat_arith

lemma power_not_zero:
"n < LENGTH('a::len) ⟹ (2 :: 'a word) ^ n ≠ 0"
by (metis p2_gt_0 word_neq_0_conv)

lemma word_gt_a_gt_0:
"a < n ⟹ (0 :: 'a::len word) < n"
apply (case_tac "n = 0")
apply clarsimp
apply (clarsimp simp: word_neq_0_conv)
done

lemma word_power_less_1 [simp]:
"sz < LENGTH('a::len) ⟹ (2::'a word) ^ sz - 1 < 2 ^ sz"
apply (subst unat_minus_one)
apply simp_all
done

lemma word_sub_1_le:
"x ≠ 0 ⟹ x - 1 ≤ (x :: ('a :: len) word)"
apply (subst no_ulen_sub)
apply simp
apply (cases "uint x = 0")
apply (insert uint_ge_0[where x=x])
apply arith
done

lemma push_bit_word_eq_nonzero:
‹push_bit n w ≠ 0› if ‹w < 2 ^ m› ‹m + n < LENGTH('a)› ‹w ≠ 0›
for w :: ‹'a::len word›
using that
apply (simp only: word_neq_0_conv word_less_nat_alt
mod_0 unat_word_ariths
unat_power_lower word_le_nat_alt)
apply (metis add_diff_cancel_right' gr0I gr_implies_not0 less_or_eq_imp_le min_def push_bit_eq_0_iff take_bit_nat_eq_self_iff take_bit_push_bit take_bit_take_bit unsigned_push_bit_eq)
done

lemma unat_less_power:
fixes k :: "'a::len word"
assumes szv: "sz < LENGTH('a)"
and     kv:  "k < 2 ^ sz"
shows   "unat k < 2 ^ sz"
using szv unat_mono [OF kv] by simp

lemma unat_mult_power_lem:
assumes kv: "k < 2 ^ (LENGTH('a::len) - sz)"
shows "unat (2 ^ sz * of_nat k :: (('a::len) word)) = 2 ^ sz * k"
proof (cases ‹sz < LENGTH('a)›)
case True
with assms show ?thesis
by (simp add: unat_word_ariths take_bit_eq_mod mod_simps)
(simp add: take_bit_nat_eq_self_iff nat_less_power_trans flip: take_bit_eq_mod)
next
case False
with assms show ?thesis
by simp
qed

lemma word_plus_mcs_4:
"⟦v + x ≤ w + x; x ≤ v + x⟧ ⟹ v ≤ (w::'a::len word)"
by uint_arith

lemma word_plus_mcs_3:
"⟦v ≤ w; x ≤ w + x⟧ ⟹ v + x ≤ w + (x::'a::len word)"
by unat_arith

lemma word_le_minus_one_leq:
"x < y ⟹ x ≤ y - 1" for x :: "'a :: len word"
by transfer (metis le_less_trans less_irrefl take_bit_decr_eq take_bit_nonnegative zle_diff1_eq)

lemma word_less_sub_le[simp]:
fixes x :: "'a :: len word"
assumes nv: "n < LENGTH('a)"
shows "(x ≤ 2 ^ n - 1) = (x < 2 ^ n)"
using le_less_trans word_le_minus_one_leq nv power_2_ge_iff by blast

lemma unat_of_nat_len:
"x < 2 ^ LENGTH('a) ⟹ unat (of_nat x :: 'a::len word) = x"

lemma unat_of_nat_eq:
"x < 2 ^ LENGTH('a) ⟹ unat (of_nat x ::'a::len word) = x"
by (rule unat_of_nat_len)

lemma unat_eq_of_nat:
"n < 2 ^ LENGTH('a) ⟹ (unat (x :: 'a::len word) = n) = (x = of_nat n)"
by transfer
(auto simp add: take_bit_of_nat nat_eq_iff take_bit_nat_eq_self_iff intro: sym)

lemma alignUp_div_helper:
fixes a :: "'a::len word"
assumes kv: "k < 2 ^ (LENGTH('a) - n)"
and     xk: "x = 2 ^ n * of_nat k"
and    le: "a ≤ x"
and    sz: "n < LENGTH('a)"
and   anz: "a mod 2 ^ n ≠ 0"
shows "a div 2 ^ n < of_nat k"
proof -
have kn: "unat (of_nat k :: 'a word) * unat ((2::'a word) ^ n) < 2 ^ LENGTH('a)"
using xk kv sz
apply (subst unat_of_nat_eq)
apply (erule order_less_le_trans)
apply simp
apply (subst unat_power_lower, simp)
apply (subst mult.commute)
apply (rule nat_less_power_trans)
apply simp
apply simp
done

have "unat a div 2 ^ n * 2 ^ n ≠ unat a"
proof -
have "unat a = unat a div 2 ^ n * 2 ^ n + unat a mod 2 ^ n"
also have "… ≠ unat a div 2 ^ n * 2 ^ n" using sz anz
finally show ?thesis ..
qed

then have "a div 2 ^ n * 2 ^ n < a" using sz anz
apply (subst word_less_nat_alt)
apply (subst unat_word_ariths)
apply (subst unat_div)
apply simp
apply (rule order_le_less_trans [OF mod_less_eq_dividend])
apply (erule order_le_neq_trans [OF div_mult_le])
done

also from xk le have "… ≤ of_nat k * 2 ^ n" by (simp add: field_simps)
finally show ?thesis using sz kv
apply -
apply (erule word_mult_less_dest [OF _ _ kn])
apply (rule order_le_less_trans [OF div_mult_le])
apply (rule unat_lt2p)
done
qed

"(x AND NOT (mask n)) = x - (x AND (mask n))"
for x :: ‹'a::len word›

"p - (p AND mask n) = (p AND NOT (mask n))"
"p - (p AND NOT (mask n)) = (p AND mask n)"
for p :: ‹'a::len word›

lemma take_bit_word_eq_self_iff:
‹take_bit n w = w ⟷ n ≥ LENGTH('a) ∨ w < 2 ^ n›
for w :: ‹'a::len word›
using take_bit_int_eq_self_iff [of n ‹take_bit LENGTH('a) (uint w)›]
by (transfer fixing: n) auto

lemma word_power_increasing:
assumes x: "2 ^ x < (2 ^ y::'a::len word)" "x < LENGTH('a::len)" "y < LENGTH('a::len)"
shows "x < y" using x
using assms by transfer simp

for x :: ‹'a::len word›

lemma plus_one_helper[elim!]:
"x < n + (1 :: 'a :: len word) ⟹ x ≤ n"
apply (simp add: word_less_nat_alt word_le_nat_alt field_simps)
apply (case_tac "1 + n = 0")
apply simp_all
apply (subst(asm) unatSuc, assumption)
apply arith
done

lemma plus_one_helper2:
"⟦ x ≤ n; n + 1 ≠ 0 ⟧ ⟹ x < n + (1 :: 'a :: len word)"
by (simp add: word_less_nat_alt word_le_nat_alt field_simps
unatSuc)

lemma less_x_plus_1:
fixes x :: "'a :: len word" shows
"x ≠ max_word ⟹ (y < (x + 1)) = (y < x ∨ y = x)"
apply (rule iffI)
apply (rule disjCI)
apply (drule plus_one_helper)
apply simp
apply (subgoal_tac "x < x + 1")
apply (erule disjE, simp_all)
apply (rule plus_one_helper2 [OF order_refl])
apply (rule notI, drule max_word_wrap)
apply simp
done

lemma word_Suc_leq:
fixes k::"'a::len word" shows "k ≠ max_word ⟹ x < k + 1 ⟷ x ≤ k"
using less_x_plus_1 word_le_less_eq by auto

lemma word_Suc_le:
fixes k::"'a::len word" shows "x ≠ max_word ⟹ x + 1 ≤ k ⟷ x < k"
by (meson not_less word_Suc_leq)

lemma word_lessThan_Suc_atMost:
‹{..< k + 1} = {..k}› if ‹k ≠ - 1› for k :: ‹'a::len word›
using that by (simp add: lessThan_def atMost_def word_Suc_leq)

lemma word_atLeastLessThan_Suc_atLeastAtMost:
‹{l ..< u + 1} = {l..u}› if ‹u ≠ - 1› for l :: ‹'a::len word›
using that by (simp add: atLeastAtMost_def atLeastLessThan_def word_lessThan_Suc_atMost)

lemma word_atLeastAtMost_Suc_greaterThanAtMost:
‹{m<..u} = {m + 1..u}› if ‹m ≠ - 1› for m :: ‹'a::len word›
using that by (simp add: greaterThanAtMost_def greaterThan_def atLeastAtMost_def atLeast_def word_Suc_le)

lemma word_atLeastLessThan_Suc_atLeastAtMost_union:
fixes l::"'a::len word"
assumes "m ≠ max_word" and "l ≤ m" and "m ≤ u"
shows "{l..m} ∪ {m+1..u} = {l..u}"
proof -
from ivl_disj_un_two(8)[OF assms(2) assms(3)] have "{l..u} = {l..m} ∪ {m<..u}" by blast
with assms show ?thesis by(simp add: word_atLeastAtMost_Suc_greaterThanAtMost)
qed

lemma max_word_less_eq_iff [simp]:
‹- 1 ≤ w ⟷ w = - 1› for w :: ‹'a::len word›
by (fact word_order.extremum_unique)

lemma word_or_zero:
"(a OR b = 0) = (a = 0 ∧ b = 0)"
for a b :: ‹'a::len word›
by (fact or_eq_0_iff)

lemma word_2p_mult_inc:
assumes x: "2 * 2 ^ n < (2::'a::len word) * 2 ^ m"
assumes suc_n: "Suc n < LENGTH('a::len)"
shows "2^n < (2::'a::len word)^m"
by (smt suc_n le_less_trans lessI nat_less_le nat_mult_less_cancel_disj p2_gt_0
power_Suc power_Suc unat_power_lower word_less_nat_alt x)

lemma power_overflow:
"n ≥ LENGTH('a) ⟹ 2 ^ n = (0 :: 'a::len word)"
by simp

lemmas extra_sle_sless_unfolds [simp] =
word_sle_eq[where a=0 and b=1]
word_sle_eq[where a=0 and b="numeral n"]
word_sle_eq[where a=1 and b=0]
word_sle_eq[where a=1 and b="numeral n"]
word_sle_eq[where a="numeral n" and b=0]
word_sle_eq[where a="numeral n" and b=1]
word_sless_alt[where a=0 and b=1]
word_sless_alt[where a=0 and b="numeral n"]
word_sless_alt[where a=1 and b=0]
word_sless_alt[where a=1 and b="numeral n"]
word_sless_alt[where a="numeral n" and b=0]
word_sless_alt[where a="numeral n" and b=1]
for n

lemma word_sint_1:
"sint (1::'a::len word) = (if LENGTH('a) = 1 then -1 else 1)"
by (fact signed_1)

lemma ucast_of_nat:
"is_down (ucast :: 'a :: len word ⇒ 'b :: len word)
⟹ ucast (of_nat n :: 'a word) = (of_nat n :: 'b word)"
by transfer simp

lemma scast_1':
"(scast (1::'a::len word) :: 'b::len word) =
(word_of_int (signed_take_bit (LENGTH('a::len) - Suc 0) (1::int)))"
by transfer simp

lemma scast_1:
"(scast (1::'a::len word) :: 'b::len word) = (if LENGTH('a) = 1 then -1 else 1)"
by (fact signed_1)

lemma unat_minus_one_word:
"unat (-1 :: 'a :: len word) = 2 ^ LENGTH('a) - 1"
apply transfer
done

lemmas word_diff_ls'' = word_diff_ls [where xa=x and x=x for x]
lemmas word_diff_ls' = word_diff_ls'' [simplified]

lemmas word_l_diffs' = word_l_diffs [where xa=x and x=x for x]
lemmas word_l_diffs = word_l_diffs' [simplified]

lemma two_power_increasing:
"⟦ n ≤ m; m < LENGTH('a) ⟧ ⟹ (2 :: 'a :: len word) ^ n ≤ 2 ^ m"

lemma word_leq_le_minus_one:
"⟦ x ≤ y; x ≠ 0 ⟧ ⟹ x - 1 < (y :: 'a :: len word)"
apply (subst unat_minus_one)
apply assumption
apply (cases "unat x")
apply arith
done

by (rule bit_word_eqI) (auto simp add: bit_simps)

for x :: ‹'a::len word›
by (rule bit_word_eqI) (auto simp add: bit_simps)

for x :: ‹'a::len word›
apply (rule trans[rotated], rule_tac w="mask n" in word_plus_and_or_coroll2)
max_absorb2)
done

lemma word_of_nat_less:
"⟦ n < unat x ⟧ ⟹ of_nat n < x"
apply (erule order_le_less_trans[rotated])
done

"unat (mask n :: 'a :: len word) = 2 ^ (min n (LENGTH('a))) - 1"
apply (subst min.commute)
apply (intro conjI impI)
done

"LENGTH('a) ≤ n ⟹ mask n = (-1::'a::len word)"

"n < LENGTH('a) ⟹ Suc (2 ^ n * k + unat (mask n :: 'a::len word)) = 2 ^ n * (k+1)"

lemma sint_of_nat_ge_zero:
"x < 2 ^ (LENGTH('a) - 1) ⟹ sint (of_nat x :: 'a :: len word) ≥ 0"

lemma int_eq_sint:
"x < 2 ^ (LENGTH('a) - 1) ⟹ sint (of_nat x :: 'a :: len word) = int x"
apply transfer
apply (rule signed_take_bit_int_eq_self)
apply simp_all
apply (metis negative_zle numeral_power_eq_of_nat_cancel_iff)
done

lemma sint_of_nat_le:
"⟦ b < 2 ^ (LENGTH('a) - 1); a ≤ b ⟧
⟹ sint (of_nat a :: 'a :: len word) ≤ sint (of_nat b :: 'a :: len word)"
apply (cases ‹LENGTH('a)›)
apply simp_all
apply transfer
apply (subst signed_take_bit_eq_if_positive)
apply (metis bit_take_bit_iff nat_less_le order_less_le_trans take_bit_nat_eq_self_iff)
apply (subst signed_take_bit_eq_if_positive)
apply (metis bit_take_bit_iff nat_less_le take_bit_nat_eq_self_iff)
apply (simp flip: of_nat_take_bit add: take_bit_nat_eq_self)
done

lemma word_le_not_less:
"((b::'a::len word) ≤ a) = (¬(a < b))"
by fastforce

lemma less_is_non_zero_p1:
fixes a :: "'a :: len word"
shows "a < k ⟹ a + 1 ≠ 0"
apply (erule contrapos_pn)
apply (drule max_word_wrap)
done

"(unat x + unat y < 2 ^ LENGTH('a)) ⟹
(unat (x + y :: 'a :: len word) = unat x + unat y)"

lemma word_less_two_pow_divI:
"⟦ (x :: 'a::len word) < 2 ^ (n - m); m ≤ n; n < LENGTH('a) ⟧ ⟹ x < 2 ^ n div 2 ^ m"
apply (subst unat_word_ariths)
apply (subst mod_less)
apply (rule order_le_less_trans [OF div_le_dividend])
apply (rule unat_lt2p)
done

lemma word_less_two_pow_divD:
"⟦ (x :: 'a::len word) < 2 ^ n div 2 ^ m ⟧
⟹ n ≥ m ∧ (x < 2 ^ (n - m))"
apply (cases "n < LENGTH('a)")
apply (cases "m < LENGTH('a)")
apply (subst(asm) unat_word_ariths)
apply (subst(asm) mod_less)
apply (rule order_le_less_trans [OF div_le_dividend])
apply (rule unat_lt2p)
apply (clarsimp dest!: less_two_pow_divD)
done

lemma of_nat_less_two_pow_div_set:
"⟦ n < LENGTH('a) ⟧ ⟹
{x. x < (2 ^ n div 2 ^ m :: 'a::len word)}
= of_nat  {k. k < 2 ^ n div 2 ^ m}"
apply (safe dest!: word_less_two_pow_divD less_two_pow_divD
intro!: word_less_two_pow_divI)
apply (rule_tac x="unat x" in exI)
apply (subst unat_power_lower[symmetric, where 'a='a])
apply simp
apply (erule unat_mono)
apply (subst word_unat_power)
apply (rule of_nat_mono_maybe)
apply (rule power_strict_increasing)
apply simp
apply simp
apply assumption
done

lemma ucast_less:
"LENGTH('b) < LENGTH('a) ⟹
(ucast (x :: 'b :: len word) :: ('a :: len word)) < 2 ^ LENGTH('b)"
by transfer simp

lemma ucast_range_less:
"LENGTH('a :: len) < LENGTH('b :: len) ⟹
range (ucast :: 'a word ⇒ 'b word) = {x. x < 2 ^ len_of TYPE ('a)}"
apply safe
apply (erule ucast_less)
apply (rule_tac x="ucast x" in exI)
apply (rule bit_word_eqI)
done

lemma word_power_less_diff:
"⟦2 ^ n * q < (2::'a::len word) ^ m; q < 2 ^ (LENGTH('a) - n)⟧ ⟹ q < 2 ^ (m - n)"
apply (case_tac "m ≥ LENGTH('a)")
apply (case_tac "n ≥ LENGTH('a)")
apply (cases "n = 0")
apply simp
apply (subst word_less_nat_alt)
apply (subst unat_power_lower)
apply simp
apply (rule nat_power_less_diff)
apply (subst (asm) iffD1 [OF unat_mult_lem])
apply simp
done

lemma word_less_sub_1:
"x < (y :: 'a :: len word) ⟹ x ≤ y - 1"
by (fact word_le_minus_one_leq)

lemma word_sub_mono2:
"⟦ a + b ≤ c + d; c ≤ a; b ≤ a + b; d ≤ c + d ⟧
⟹ b ≤ (d :: 'a :: len word)"
apply (drule(1) word_sub_mono)
apply simp
apply simp
apply simp
done

lemma word_not_le:
"(¬ x ≤ (y :: 'a :: len word)) = (y < x)"
by fastforce

lemma word_subset_less:
"⟦ {x .. x + r - 1} ⊆ {y .. y + s - 1};
x ≤ x + r - 1; y ≤ y + (s :: 'a :: len word) - 1;
s ≠ 0 ⟧
⟹ r ≤ s"
apply (frule subsetD[where c=x])
apply simp
apply (drule subsetD[where c="x + r - 1"])
apply simp
apply (drule(1) word_sub_mono2)
apply (erule word_le_minus_cancel)
apply (rule ccontr)
done

lemma uint_power_lower:
"n < LENGTH('a) ⟹ uint (2 ^ n :: 'a :: len word) = (2 ^ n :: int)"
by (rule uint_2p_alt)

lemma power_le_mono:
"⟦2 ^ n ≤ (2::'a::len word) ^ m; n < LENGTH('a); m < LENGTH('a)⟧
⟹ n ≤ m"
apply safe
apply (simp only: uint_arith_simps(3))
apply (drule uint_power_lower)+
apply simp
done

lemma two_power_eq:
"⟦n < LENGTH('a); m < LENGTH('a)⟧
⟹ ((2::'a::len word) ^ n = 2 ^ m) = (n = m)"
apply safe
apply (rule order_antisym)
done

lemma unat_less_helper:
"x < of_nat n ⟹ unat x < n"
apply (erule order_less_le_trans)
done

lemma nat_uint_less_helper:
"nat (uint y) = z ⟹ x < y ⟹ nat (uint x) < z"
apply (erule subst)
apply (subst unat_eq_nat_uint [symmetric])
apply (subst unat_eq_nat_uint [symmetric])

lemma of_nat_0:
"⟦of_nat n = (0::'a::len word); n < 2 ^ LENGTH('a)⟧ ⟹ n = 0"

lemma of_nat_inj:
"⟦x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)⟧ ⟹
(of_nat x = (of_nat y :: 'a :: len word)) = (x = y)"
by (metis unat_of_nat_len)

lemma div_to_mult_word_lt:
"⟦ (x :: 'a :: len word) ≤ y div z ⟧ ⟹ x * z ≤ y"
apply (cases "z = 0")
apply simp
apply (rule order_trans)
apply (erule(1) word_mult_le_mono1)
apply (rule order_le_less_trans [OF div_mult_le])
apply simp
apply (rule word_div_mult_le)
done

"(ucast :: 'a :: len word ⇒ 'b :: len word) (ucast x) = x AND mask (len_of TYPE ('a))"
apply transfer
done

lemma ucast_ucast_len:
"⟦ x < 2 ^ LENGTH('b) ⟧ ⟹ ucast (ucast x::'b::len word) = (x::'a::len word)"
done

lemma ucast_ucast_id:
"LENGTH('a) < LENGTH('b) ⟹ ucast (ucast (x::'a::len word)::'b::len word) = x"
by (auto intro: ucast_up_ucast_id simp: is_up_def source_size_def target_size_def word_size)

lemma unat_ucast:
"unat (ucast x :: ('a :: len) word) = unat x mod 2 ^ (LENGTH('a))"
proof -
have ‹2 ^ LENGTH('a) = nat (2 ^ LENGTH('a))›
by simp
moreover have ‹unat (ucast x :: 'a word) = unat x mod nat (2 ^ LENGTH('a))›
by transfer (simp flip: nat_mod_distrib take_bit_eq_mod)
ultimately show ?thesis
by (simp only:)
qed

lemma ucast_less_ucast:
"LENGTH('a) ≤ LENGTH('b) ⟹
(ucast x < ((ucast (y :: 'a::len word)) :: 'b::len word)) = (x < y)"
apply (subst mod_less)
apply(rule less_le_trans[OF unat_lt2p], simp)
apply (subst mod_less)
apply(rule less_le_trans[OF unat_lt2p], simp)
apply simp
done

― ‹This weaker version was previously called @{text ucast_less_ucast}. We retain it to
support existing proofs.›
lemmas ucast_less_ucast_weak = ucast_less_ucast[OF order.strict_implies_order]

lemma unat_Suc2:
fixes n :: "'a :: len word"
shows
"n ≠ -1 ⟹ unat (n + 1) = Suc (unat n)"
apply (subst eq_diff_eq[symmetric], simp add: minus_equation_iff)
done

lemma word_div_1:
"(n :: 'a :: len word) div 1 = n"
by (fact bits_div_by_1)

lemma word_minus_one_le:
"-1 ≤ (x :: 'a :: len word) = (x = -1)"
by (fact word_order.extremum_unique)

lemma up_scast_inj:
"⟦ scast x = (scast y :: 'b :: len word); size x ≤ LENGTH('b) ⟧
⟹ x = y"
apply transfer
apply (cases ‹LENGTH('a)›)
apply simp_all
apply (metis order_refl take_bit_signed_take_bit take_bit_tightened)
done

lemma up_scast_inj_eq:
"LENGTH('a) ≤ len_of TYPE ('b) ⟹
(scast x = (scast y::'b::len word)) = (x = (y::'a::len word))"
by (fastforce dest: up_scast_inj simp: word_size)

fixes x :: "'a :: len word"
shows "x ≤ y ⟹ ∃n. y = x + of_nat n"
by (rule exI [where x = "unat (y - x)"]) simp

lemma word_plus_mcs_4':
fixes x :: "'a :: len word"
shows "⟦x + v ≤ x + w; x ≤ x + v⟧ ⟹ v ≤ w"
apply (rule word_plus_mcs_4)
done

lemma unat_eq_1:
‹unat x = Suc 0 ⟷ x = 1›
by (auto intro!: unsigned_word_eqI [where ?'a = nat])

lemma word_unat_Rep_inject1:
‹unat x = unat 1 ⟷ x = 1›

"(w AND NOT (mask n)) AND NOT (mask m) = w AND NOT (mask (max m n))"
for w :: ‹'a::len word›
by (rule bit_word_eqI) (auto simp add: bit_simps)

lemma word_less_cases:
"x < y ⟹ x = y - 1 ∨ x < y - (1 ::'a::len word)"
apply (drule word_less_sub_1)
apply (drule order_le_imp_less_or_eq)
apply auto
done

"(x AND w = 0) = (x AND NOT w = x)"
for x w :: ‹'a::len word›
using word_plus_and_or_coroll2[where x=x and w=w]
by auto

"(x AND w = x) = (x AND NOT w = 0)"
for x w :: ‹'a::len word›
using word_plus_and_or_coroll2[where x=x and w=w]
by auto

lemma compl_of_1: "NOT 1 = (-2 :: 'a :: len word)"
by (fact not_one)

"(x = y) = (x AND m = y AND m ∧ x AND NOT m = y AND NOT m)"
for x y m :: ‹'a::len word›
apply transfer
apply (auto simp add: bit_simps ac_simps)
done

"0xFF = (mask 8 :: 'a::len word)"

"0x1FF = (mask 9 :: 'a::len word)"

lemma ucast_of_nat_small:
"x < 2 ^ LENGTH('a) ⟹ ucast (of_nat x :: 'a :: len word) = (of_nat x :: 'b :: len word)"
apply transfer
apply (auto simp add: take_bit_of_nat min_def not_le)
apply (metis linorder_not_less min_def take_bit_nat_eq_self take_bit_take_bit)
done

lemma word_le_make_less:
fixes x :: "'a :: len word"
shows "y ≠ -1 ⟹ (x ≤ y) = (x < (y + 1))"
apply safe
apply (erule plus_one_helper2)
done

lemmas finite_word = finite [where 'a="'a::len word"]

lemma word_to_1_set:
"{0 ..< (1 :: 'a :: len word)} = {0}"
by fastforce

lemma word_leq_minus_one_le:
fixes x :: "'a::len word"
shows "⟦y ≠ 0; x ≤ y - 1 ⟧ ⟹ x < y"
using le_m1_iff_lt word_neq_0_conv by blast

lemma word_count_from_top:
"n ≠ 0 ⟹ {0 ..< n :: 'a :: len word} = {0 ..< n - 1} ∪ {n - 1}"
apply (rule set_eqI, rule iffI)
apply simp
apply (drule word_le_minus_one_leq)
apply (rule disjCI)
apply simp
apply simp
apply (erule word_leq_minus_one_le)
apply fastforce
done

lemma word_minus_one_le_leq:
"⟦ x - 1 < y ⟧ ⟹ x ≤ (y :: 'a :: len word)"
apply (cases "x = 0")
apply simp
apply (subst(asm) unat_minus_one)
apply (cases "unat x")
apply arith
done

lemma word_div_less:
"m < n ⟹ m div n = 0" for m :: "'a :: len word"

lemma word_must_wrap:
"⟦ x ≤ n - 1; n ≤ x ⟧ ⟹ n = (0 :: 'a :: len word)"
using dual_order.trans sub_wrap word_less_1 by blast

lemma range_subset_card:
"⟦ {a :: 'a :: len word .. b} ⊆ {c .. d}; b ≥ a ⟧ ⟹ d ≥ c ∧ d - c ≥ b - a"
using word_sub_le word_sub_mono by fastforce

lemma less_1_simp:
"n - 1 < m = (n ≤ (m :: 'a :: len word) ∧ n ≠ 0)"
by unat_arith

lemma word_power_mod_div:
fixes x :: "'a::len word"
shows "⟦ n < LENGTH('a); m < LENGTH('a)⟧
⟹ x mod 2 ^ n div 2 ^ m = x div 2 ^ m mod 2 ^ (n - m)"
apply (simp add: word_arith_nat_div unat_mod power_mod_div)
apply (subst unat_arith_simps(3))
apply (subst unat_mod)
apply (subst unat_of_nat)+
done

lemma word_range_minus_1':
fixes a :: "'a :: len word"
shows "a ≠ 0 ⟹ {a - 1<..b} = {a..b}"
by (simp add: greaterThanAtMost_def atLeastAtMost_def greaterThan_def atLeast_def less_1_simp)

lemma word_range_minus_1:
fixes a :: "'a :: len word"
shows "b ≠ 0 ⟹ {a..b - 1} = {a..<b}"
apply (simp add: atLeastLessThan_def atLeastAtMost_def atMost_def lessThan_def)
apply (rule arg_cong [where f = "λx. {a..} ∩ x"])
apply rule
apply clarsimp
apply (erule contrapos_pp)
apply (simp add: linorder_not_less linorder_not_le word_must_wrap)
apply (clarsimp)
apply (drule word_le_minus_one_leq)
apply (auto simp: word_less_sub_1)
done

lemma ucast_nat_def:
"of_nat (unat x) = (ucast :: 'a :: len word ⇒ 'b :: len word) x"
by transfer simp

lemma overflow_plus_one_self:
"(1 + p ≤ p) = (p = (-1 :: 'a :: len word))"
apply rule
apply (rule ccontr)
apply (drule plus_one_helper2)
apply (rule notI)
apply (drule arg_cong[where f="λx. x - 1"])
apply simp
apply simp
done

lemma plus_1_less:
"(x + 1 ≤ (x :: 'a :: len word)) = (x = -1)"
apply (rule iffI)
apply (rule ccontr)
apply (cut_tac plus_one_helper2[where x=x, OF order_refl])
apply simp
apply clarsimp
apply (drule arg_cong[where f="λx. x - 1"])
apply simp
apply simp
done

lemma pos_mult_pos_ge:
"[|x > (0::int); n>=0 |] ==> n * x >= n*1"
apply (simp only: mult_left_mono)
done

lemma word_plus_strict_mono_right:
fixes x :: "'a :: len word"
shows "⟦y < z; x ≤ x + z⟧ ⟹ x + y < x + z"
by unat_arith

lemma word_div_mult:
"0 < c ⟹ a < b * c ⟹ a div c < b" for a b c :: "'a::len word"
by (rule classical)
(use div_to_mult_word_lt [of b a c] in
‹auto simp add: word_less_nat_alt word_le_nat_alt unat_div›)

lemma word_less_power_trans_ofnat:
"⟦n < 2 ^ (m - k); k ≤ m; m < LENGTH('a)⟧
⟹ of_nat n * 2 ^ k < (2::'a::len word) ^ m"
apply (subst mult.commute)
apply (rule word_less_power_trans)
apply (simp_all add: word_less_nat_alt less_le_trans take_bit_eq_mod)
done

lemma word_1_le_power:
"n < LENGTH('a) ⟹ (1 :: 'a :: len word) ≤ 2 ^ n"
by (rule inc_le[where i=0, simplified], erule iffD2[OF p2_gt_0])

lemma unat_1_0:
"1 ≤ (x::'a::len word) = (0 < unat x)"

lemma x_less_2_0_1':
fixes x :: "'a::len word"
shows "⟦LENGTH('a) ≠ 1; x < 2⟧ ⟹ x = 0 ∨ x = 1"
apply (cases ‹2 ≤ LENGTH('a)›)
apply simp_all
apply transfer
apply auto
apply (metis add.commute add.right_neutral even_two_times_div_two mod_div_trivial mod_pos_pos_trivial mult.commute mult_zero_left not_less not_take_bit_negative odd_two_times_div_two_succ)
done

lemma of_nat_power:
shows "⟦ p < 2 ^ x; x < len_of TYPE ('a) ⟧ ⟹ of_nat p < (2 :: 'a :: len word) ^ x"
apply (rule order_less_le_trans)
apply (rule of_nat_mono_maybe)
apply (erule power_strict_increasing)
apply simp
apply assumption
apply (simp add: word_unat_power del: of_nat_power)
done

lemma of_nat_n_less_equal_power_2:
"n < LENGTH('a::len) ⟹ ((of_nat n)::'a word) < 2 ^ n"
apply (induct n)
apply clarsimp
apply clarsimp
apply (metis of_nat_power n_less_equal_power_2 of_nat_Suc power_Suc)
done

fixes w :: "'a::len word"
assumes eqm: "w = w AND mask n"
and      sz: "n < len_of TYPE ('a)"
shows "w < (2::'a word) ^ n"
by (subst eqm, rule and_mask_less' [OF sz])

lemma of_nat_mono_maybe':
fixes Y :: "nat"
assumes xlt: "x < 2 ^ len_of TYPE ('a)"
assumes ylt: "y < 2 ^ len_of TYPE ('a)"
shows   "(y < x) = (of_nat y < (of_nat x :: 'a :: len word))"
apply (subst word_less_nat_alt)
apply (subst unat_of_nat)+
apply (subst mod_less)
apply (rule ylt)
apply (subst mod_less)
apply (rule xlt)
apply simp
done

lemma of_nat_mono_maybe_le:
"⟦x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)⟧ ⟹
(y ≤ x) = ((of_nat y :: 'a :: len word) ≤ of_nat x)"
apply (clarsimp simp: le_less)
apply (rule disj_cong)
apply (rule of_nat_mono_maybe', assumption+)
apply auto
using of_nat_inj apply blast
done

for w :: ‹'a::len word›
by (rule bit_word_eqI) (simp add: bit_simps)

"(w AND NOT (mask n)) + (w AND mask n) = w"
for w :: ‹'a::len word›
apply (rule bit_word_eqI)
done

fixes x :: "'a :: len word"
and     m2: "x AND NOT (mask n) = y AND NOT (mask n)"
shows "x = y"
proof -
have *: ‹x = x AND mask n OR x AND NOT (mask n)› for x :: ‹'a word›
by (rule bit_word_eqI) (auto simp add: bit_simps)
from assms * [of x] * [of y] show ?thesis
by simp
qed

lemma neq_0_no_wrap:
fixes x :: "'a :: len word"
shows "⟦ x ≤ x + y; x ≠ 0 ⟧ ⟹ x + y ≠ 0"
by clarsimp

lemma unatSuc2:
fixes n :: "'a :: len word"
shows "n + 1 ≠ 0 ⟹ unat (n + 1) = Suc (unat n)"

lemma word_of_nat_le:
"n ≤ unat x ⟹ of_nat n ≤ x"
apply (erule order_trans[rotated])
done

lemma word_unat_less_le:
"a ≤ of_nat b ⟹ unat a ≤ b"
by (metis eq_iff le_cases le_unat_uoi word_of_nat_le)

fixes x :: "'a :: len word"
shows "2 < LENGTH('a) ⟹ (0 < x AND 1) = (x AND 1 = 1)"

fixes x :: "'a :: len word"
fixes y :: "'b :: len word"
shows
"LENGTH('b) ≥ LENGTH('a) ⟹
ucast (ucast x + y) = x + ucast y"
apply transfer
apply simp
apply simp
done

lemma lt1_neq0:
fixes x :: "'a :: len word"
shows "(1 ≤ x) = (x ≠ 0)" by unat_arith

lemma word_plus_one_nonzero:
fixes x :: "'a :: len word"
shows "⟦x ≤ x + y; y ≠ 0⟧ ⟹ x + 1 ≠ 0"
apply (subst lt1_neq0 [symmetric])
apply (erule word_random)
done

lemma word_sub_plus_one_nonzero:
fixes n :: "'a :: len word"
shows "⟦n' ≤ n; n' ≠ 0⟧ ⟹ (n - n') + 1 ≠ 0"
apply (subst lt1_neq0 [symmetric])
apply (rule word_random [where x' = n'])
apply simp
apply (erule word_sub_le)
done

lemma word_le_minus_mono_right:
fixes x :: "'a :: len word"
shows "⟦ z ≤ y; y ≤ x; z ≤ x ⟧ ⟹ x - y ≤ x - z"
apply (rule word_sub_mono)
apply simp
apply assumption
apply (erule word_sub_le)
apply (erule word_sub_le)
done

lemma word_0_sle_from_less:
‹0 ≤s x› if ‹x < 2 ^ (LENGTH('a) - 1)› for x :: ‹'a::len word›
using that
apply transfer
apply (cases ‹LENGTH('a)›)
apply simp_all
apply (metis bit_take_bit_iff min_def nat_less_le not_less_eq take_bit_int_eq_self_iff take_bit_take_bit)
done

lemma ucast_sub_ucast:
fixes x :: "'a::len word"
assumes "y ≤ x"
assumes T: "LENGTH('a) ≤ LENGTH('b)"
shows "ucast (x - y) = (ucast x - ucast y :: 'b::len word)"
proof -
from T
have P: "unat x < 2 ^ LENGTH('b)" "unat y < 2 ^ LENGTH('b)"
by (fastforce intro!: less_le_trans[OF unat_lt2p])+
then show ?thesis
by (simp add: unat_arith_simps unat_ucast assms[simplified unat_arith_simps])
qed

lemma word_1_0:
"⟦a + (1::('a::len) word) ≤ b; a < of_nat x⟧ ⟹ a < b"
apply transfer
apply (subst (asm) take_bit_incr_eq)
using take_bit_int_less_exp le_less_trans by blast

lemma unat_of_nat_less:"⟦ a < b; unat b = c ⟧ ⟹ a < of_nat c"
by fastforce

lemma word_le_plus_1: "⟦ (y::('a::len) word) < y + n; a < n ⟧ ⟹ y + a ≤ y + a + 1"
by unat_arith

lemma word_le_plus:"⟦(a::('a::len) word) < a + b; c < b⟧ ⟹ a ≤ a + c"
by (metis order_less_imp_le word_random)

lemma sint_minus1 [simp]: "(sint x = -1) = (x = -1)"
apply (cases ‹LENGTH('a)›)
apply simp_all
apply transfer
apply (simp flip: signed_take_bit_eq_iff_take_bit_eq)
done

lemma sint_0 [simp]: "(sint x = 0) = (x = 0)"
by (fact signed_eq_0_iff)

(* It is not always that case that "sint 1 = 1", because of 1-bit word sizes.
* This lemma produces the different cases. *)
lemma sint_1_cases:
P if ‹⟦ len_of TYPE ('a::len) = 1; (a::'a word) = 0; sint a = 0 ⟧ ⟹ P›
‹⟦ len_of TYPE ('a) = 1; a = 1; sint (1 :: 'a word) = -1 ⟧ ⟹ P›
‹⟦ len_of TYPE ('a) > 1; sint (1 :: 'a word) = 1 ⟧ ⟹ P›
proof (cases ‹LENGTH('a) = 1›)
case True
then have ‹a = 0 ∨ a = 1›
by transfer auto
with True that show ?thesis
by auto
next
case False
with that show ?thesis
qed

lemma sint_int_min:
"sint (- (2 ^ (LENGTH('a) - Suc 0)) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))"
apply (cases ‹LENGTH('a)›)
apply simp_all
apply transfer
done

lemma sint_int_max_plus_1:
"sint (2 ^ (LENGTH('a) - Suc 0) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))"
apply (cases ‹LENGTH('a)›)
apply simp_all
apply (subst word_of_int_2p [symmetric])
apply (subst int_word_sint)
apply simp
done

lemma uint_range':
‹0 ≤ uint x ∧ uint x < 2 ^ LENGTH('a)›
for x :: ‹'a::len word›
by transfer simp

lemma sint_of_int_eq:
"⟦ - (2 ^ (LENGTH('a) - 1)) ≤ x; x < 2 ^ (LENGTH('a) - 1) ⟧ ⟹ sint (of_int x :: ('a::len) word) = x"

lemma of_int_sint:
"of_int (sint a) = a"
by simp

lemma sint_ucast_eq_uint:
"⟦ ¬ is_down (ucast :: ('a::len word ⇒ 'b::len word)) ⟧
⟹ sint ((ucast :: ('a::len word ⇒ 'b::len word)) x) = uint x"
apply transfer
done

lemma word_less_nowrapI':
"(x :: 'a :: len word) ≤ z - k ⟹ k ≤ z ⟹ 0 < k ⟹ x < x + k"
by uint_arith

"mask n + 1 = (2 ^ n :: 'a::len word)"

lemma unat_inj: "inj unat"
by (metis eq_iff injI word_le_nat_alt)

lemma unat_ucast_upcast:
"is_up (ucast :: 'b word ⇒ 'a word)
⟹ unat (ucast x :: ('a::len) word) = unat (x :: ('b::len) word)"
unfolding ucast_eq unat_eq_nat_uint
apply transfer
apply simp
done

lemma ucast_mono:
"⟦ (x :: 'b :: len word) < y; y < 2 ^ LENGTH('a) ⟧
⟹ ucast x < ((ucast y) :: 'a :: len word)"
apply (simp only: flip: ucast_nat_def)
apply (rule of_nat_mono_maybe)
apply (rule unat_less_helper)
apply simp
done

lemma ucast_mono_le:
"⟦x ≤ y; y < 2 ^ LENGTH('b)⟧ ⟹ (ucast (x :: 'a :: len word) :: 'b :: len word) ≤ ucast y"
apply (simp only: flip: ucast_nat_def)
apply (subst of_nat_mono_maybe_le[symmetric])
apply (rule unat_less_helper)
apply simp
apply (rule unat_less_helper)
apply (erule le_less_trans)
done

lemma ucast_mono_le':
"⟦ unat y < 2 ^ LENGTH('b); LENGTH('b::len) < LENGTH('a::len); x ≤ y ⟧
⟹ ucast x ≤ (ucast y :: 'b word)" for x y :: ‹'a::len word›
by (auto simp: word_less_nat_alt intro: ucast_mono_le)

"((x:: 'a :: len word) AND NOT (mask n)) + (2 ^ n - 1) = x OR mask n"
apply (subst word_plus_and_or_coroll; rule bit_word_eqI)
done

lemma le_step_down_word:"⟦(i::('a::len) word) ≤ n; i = n ⟶ P; i ≤ n - 1 ⟶ P⟧ ⟹ P"
by unat_arith

lemma le_step_down_word_2:
fixes x :: "'a::len word"
shows "⟦x ≤  y; x ≠ y⟧ ⟹ x ≤ y - 1"
by (subst (asm) word_le_less_eq,
clarsimp,

lemma and_and_not[simp]:"(a AND b) AND NOT b = 0"
for a b :: ‹'a::len word›
apply (subst word_bw_assocs(1))
apply clarsimp
done

apply (rule_tac x=1 in exI)
done

lemma not_switch:"NOT a = x ⟹ a = NOT x"
by auto

end


# Theory Signed_Words

(*
* Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
*
*)

section "Signed Words"

theory Signed_Words
imports "HOL-Library.Word"
begin

text ‹Signed words as separate (isomorphic) word length class. Useful for tagging words in C.›

typedef ('a::len0) signed = "UNIV :: 'a set" ..

lemma card_signed [simp]: "CARD (('a::len0) signed) = CARD('a)"
unfolding type_definition.card [OF type_definition_signed]
by simp

instantiation signed :: (len0) len0
begin

definition
len_signed [simp]: "len_of (x::'a::len0 signed itself) = LENGTH('a)"

instance ..

end

instance signed :: (len) len
by (intro_classes, simp)

lemma scast_scast_id [simp]:
"scast (scast x :: ('a::len) signed word) = (x :: 'a word)"
"scast (scast y :: ('a::len) word) = (y :: 'a signed word)"
by (auto simp: is_up scast_up_scast_id)

lemma ucast_scast_id [simp]:
"ucast (scast (x :: 'a::len signed word) :: 'a word) = x"

lemma scast_of_nat [simp]:
"scast (of_nat x :: 'a::len signed word) = (of_nat x :: 'a word)"

lemma scast_ucast_id [simp]:
"scast (ucast (x :: 'a::len word) :: 'a signed word) = x"

lemma scast_eq_scast_id [simp]:
"((scast (a :: 'a::len signed word) :: 'a word) = scast b) = (a = b)"
by (metis ucast_scast_id)

lemma ucast_eq_ucast_id [simp]:
"((ucast (a :: 'a::len word) :: 'a signed word) = ucast b) = (a = b)"
by (metis scast_ucast_id)

lemma scast_ucast_norm [simp]:
"(ucast (a :: 'a::len word) = (b :: 'a signed word)) = (a = scast b)"
"((b :: 'a signed word) = ucast (a :: 'a::len word)) = (a = scast b)"
by (metis scast_ucast_id ucast_scast_id)+

lemma scast_2_power [simp]: "scast ((2 :: 'a::len signed word) ^ x) = ((2 :: 'a word) ^ x)"
by (rule bit_word_eqI) (auto simp add: bit_simps)

lemma ucast_nat_def':
"of_nat (unat x) = (ucast :: 'a :: len word ⇒ ('b :: len) signed word) x"
by (fact of_nat_unat)

lemma zero_sle_ucast_up:
"¬ is_down (ucast :: 'a word ⇒ 'b signed word) ⟹
(0 <=s ((ucast (b::('a::len) word)) :: ('b::len) signed word))"

lemma word_le_ucast_sless:
"⟦ x ≤ y; y ≠ -1; LENGTH('a) < LENGTH('b) ⟧ ⟹
(ucast x :: ('b :: len) signed word) <s ucast (y + 1)"
for x y :: ‹'a::len word›
apply (cases ‹LENGTH('b)›)
apply simp_all
apply transfer
done

lemma zero_sle_ucast:
"(0 <=s ((ucast (b::('a::len) word)) :: ('a::len) signed word))
= (uint b < 2 ^ (LENGTH('a) - 1))"
apply transfer
apply (cases ‹LENGTH('a)›)
done

type_synonym 'a sword = "'a signed word"

end


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

(* Author: Jeremy Dawson, NICTA *)

section ‹Operation variants with traditional syntax›

imports "HOL-Library.Word" More_Word Signed_Words
begin

class semiring_bit_syntax = semiring_bit_shifts
begin

definition test_bit :: ‹'a ⇒ nat ⇒ bool›  (infixl "!!" 100)
where test_bit_eq_bit: ‹test_bit = bit›

definition shiftl :: ‹'a ⇒ nat ⇒ 'a›  (infixl "<<" 55)
where shiftl_eq_push_bit: ‹a << n = push_bit n a›

definition shiftr :: ‹'a ⇒ nat ⇒ 'a›  (infixl ">>" 55)
where shiftr_eq_drop_bit: ‹a >> n = drop_bit n a›

end

instance word :: (len) semiring_bit_syntax ..

context
includes lifting_syntax
begin

lemma test_bit_word_transfer [transfer_rule]:
‹(pcr_word ===> (=)) (λk n. n < LENGTH('a) ∧ bit k n) (test_bit :: 'a::len word ⇒ _)›
by (unfold test_bit_eq_bit) transfer_prover

lemma shiftl_word_transfer [transfer_rule]:
‹(pcr_word ===> (=) ===> pcr_word) (λk n. push_bit n k) shiftl›
by (unfold shiftl_eq_push_bit) transfer_prover

lemma shiftr_word_transfer [transfer_rule]:
‹(pcr_word ===> (=) ===> pcr_word) (λk n. (drop_bit n ∘ take_bit LENGTH('a)) k) (shiftr :: 'a::len word ⇒ _)›
by (unfold shiftr_eq_drop_bit) transfer_prover

end

lemma test_bit_word_eq:
‹test_bit = (bit :: 'a::len word ⇒ _)›
by (fact test_bit_eq_bit)

lemma shiftl_word_eq:
‹w << n = push_bit n w› for w :: ‹'a::len word›
by (fact shiftl_eq_push_bit)

lemma shiftr_word_eq:
‹w >> n = drop_bit n w› for w :: ‹'a::len word›
by (fact shiftr_eq_drop_bit)

lemma test_bit_eq_iff: "test_bit u = test_bit v ⟷ u = v"
for u v :: "'a::len word"
by (simp add: bit_eq_iff test_bit_eq_bit fun_eq_iff)

lemma test_bit_size: "w !! n ⟹ n < size w"
for w :: "'a::len word"
by transfer simp

lemma word_eq_iff: "x = y ⟷ (∀n<LENGTH('a). x !! n = y !! n)" (is ‹?P ⟷ ?Q›)
for x y :: "'a::len word"
by transfer (auto simp add: bit_eq_iff bit_take_bit_iff)

lemma word_eqI: "(⋀n. n < size u ⟶ u !! n = v !! n) ⟹ u = v"
for u :: "'a::len word"

lemma word_eqD: "u = v ⟹ u !! x = v !! x"
for u v :: "'a::len word"
by simp

lemma test_bit_bin': "w !! n ⟷ n < size w ∧ bit (uint w) n"

lemmas test_bit_bin = test_bit_bin' [unfolded word_size]

lemma word_test_bit_def:
‹test_bit a = bit (uint a)›
by transfer (simp add: fun_eq_iff bit_take_bit_iff)

lemmas test_bit_def' = word_test_bit_def [THEN fun_cong]

lemma word_test_bit_transfer [transfer_rule]:
"(rel_fun pcr_word (rel_fun (=) (=)))
(λx n. n < LENGTH('a) ∧ bit x n) (test_bit :: 'a::len word ⇒ _)"
by (simp only: test_bit_eq_bit) transfer_prover

lemma test_bit_wi [simp]:
"(word_of_int x :: 'a::len word) !! n ⟷ n < LENGTH('a) ∧ bit x n"
by transfer simp

lemma word_ops_nth_size:
"n < size x ⟹
(x OR y) !! n = (x !! n | y !! n) ∧
(x AND y) !! n = (x !! n ∧ y !! n) ∧
(x XOR y) !! n = (x !! n ≠ y !! n) ∧
(NOT x) !! n = (¬ x !! n)"
for x :: "'a::len word"
by transfer (simp add: bit_or_iff bit_and_iff bit_xor_iff bit_not_iff)

lemma word_ao_nth:
"(x OR y) !! n = (x !! n | y !! n) ∧
(x AND y) !! n = (x !! n ∧ y !! n)"
for x :: "'a::len word"
by transfer (auto simp add: bit_or_iff bit_and_iff)

lemmas msb0 = len_gt_0 [THEN diff_Suc_less, THEN word_ops_nth_size [unfolded word_size]]
lemmas msb1 = msb0 [where i = 0]

lemma test_bit_numeral [simp]:
"(numeral w :: 'a::len word) !! n ⟷
n < LENGTH('a) ∧ bit (numeral w :: int) n"
by transfer (rule refl)

lemma test_bit_neg_numeral [simp]:
"(- numeral w :: 'a::len word) !! n ⟷
n < LENGTH('a) ∧ bit (- numeral w :: int) n"
by transfer (rule refl)

lemma test_bit_1 [iff]: "(1 :: 'a::len word) !! n ⟷ n = 0"
by transfer (auto simp add: bit_1_iff)

lemma nth_0 [simp]: "¬ (0 :: 'a::len word) !! n"
by transfer simp

lemma nth_minus1 [simp]: "(-1 :: 'a::len word) !! n ⟷ n < LENGTH('a)"
by transfer simp

lemma shiftl1_code [code]:
‹shiftl1 w = push_bit 1 w›

lemma uint_shiftr_eq:
‹uint (w >> n) = uint w div 2 ^ n›
by transfer (simp flip: drop_bit_eq_div add: drop_bit_take_bit min_def le_less less_diff_conv)

lemma shiftr1_code [code]:
‹shiftr1 w = drop_bit 1 w›

lemma shiftl_def:
‹w << n = (shiftl1 ^^ n) w›
proof -
have ‹push_bit n = (((*) 2 ^^ n) :: int ⇒ int)› for n
then show ?thesis
by transfer simp
qed

lemma shiftr_def:
‹w >> n = (shiftr1 ^^ n) w›
proof -
have ‹shiftr1 ^^ n = (drop_bit n :: 'a word ⇒ 'a word)›
apply (induction n)
apply simp
apply (simp only: shiftr1_eq_div_2 [abs_def] drop_bit_eq_div [abs_def] funpow_Suc_right)
apply (use div_exp_eq [of _ 1, where ?'a = ‹'a word›] in simp)
done
then show ?thesis
qed

lemma bit_shiftl_word_iff [bit_simps]:
‹bit (w << m) n ⟷ m ≤ n ∧ n < LENGTH('a) ∧ bit w (n - m)›
for w :: ‹'a::len word›
by (simp add: shiftl_word_eq bit_push_bit_iff not_le)

lemma bit_shiftr_word_iff [bit_simps]:
‹bit (w >> m) n ⟷ bit w (m + n)›
for w :: ‹'a::len word›

lift_definition sshiftr :: ‹'a::len word ⇒ nat ⇒ 'a word›  (infixl ‹>>>› 55)
is ‹λk n. take_bit LENGTH('a) (drop_bit n (signed_take_bit (LENGTH('a) - Suc 0) k))›
by (simp flip: signed_take_bit_decr_length_iff)

lemma sshiftr_eq [code]:
‹w >>> n = signed_drop_bit n w›
by transfer simp

lemma sshiftr_eq_funpow_sshiftr1:
‹w >>> n = (sshiftr1 ^^ n) w›
apply (rule sym)
apply (induction n)
apply simp_all
done

lemma uint_sshiftr_eq:
‹uint (w >>> n) = take_bit LENGTH('a) (sint w div 2 ^  n)›
for w :: ‹'a::len word›
by transfer (simp flip: drop_bit_eq_div)

lemma sshift1_code [code]:
‹sshiftr1 w = signed_drop_bit 1 w›

lemma sshiftr_0 [simp]: "0 >>> n = 0"
by transfer simp

lemma sshiftr_n1 [simp]: "-1 >>> n = -1"
by transfer simp

lemma bit_sshiftr_word_iff [bit_simps]:
‹bit (w >>> m) n ⟷ bit w (if LENGTH('a) - m ≤ n ∧ n < LENGTH('a) then LENGTH('a) - 1 else (m + n))›
for w :: ‹'a::len word›
apply transfer
apply (auto simp add: bit_take_bit_iff bit_drop_bit_eq bit_signed_take_bit_iff min_def not_le simp flip: bit_Suc)
using le_less_Suc_eq apply fastforce
using le_less_Suc_eq apply fastforce
done

lemma nth_sshiftr :
"(w >>> m) !! n =
(n < size w ∧ (if n + m ≥ size w then w !! (size w - 1) else w !! (n + m)))"
apply transfer
apply (auto simp add: bit_take_bit_iff bit_drop_bit_eq bit_signed_take_bit_iff min_def not_le ac_simps)
using le_less_Suc_eq apply fastforce
using le_less_Suc_eq apply fastforce
done

lemma sshiftr_numeral [simp]:
‹(numeral k >>> numeral n :: 'a::len word) =
word_of_int (drop_bit (numeral n) (signed_take_bit (LENGTH('a) - 1) (numeral k)))›
apply (rule word_eqI)
apply (cases ‹LENGTH('a)›)
apply (simp_all add: word_size bit_drop_bit_eq nth_sshiftr bit_signed_take_bit_iff min_def not_le not_less less_Suc_eq_le ac_simps)
done

setup ‹
(\<^term>‹shiftl :: 'a::len word ⇒ _›, "bvshl"),
(\<^term>‹shiftr :: 'a::len word ⇒ _›, "bvlshr"),
(\<^term>‹sshiftr :: 'a::len word ⇒ _›, "bvashr")
])
›

lemma revcast_down_us [OF refl]:
"rc = revcast ⟹ source_size rc = target_size rc + n ⟹ rc w = ucast (w >>> n)"
for w :: "'a::len word"
apply (rule bit_word_eqI)
apply (simp add: bit_revcast_iff bit_ucast_iff bit_sshiftr_word_iff ac_simps)
done

lemma revcast_down_ss [OF refl]:
"rc = revcast ⟹ source_size rc = target_size rc + n ⟹ rc w = scast (w >>> n)"
for w :: "'a::len word"
apply (rule bit_word_eqI)
apply (simp add: bit_revcast_iff bit_word_scast_iff bit_sshiftr_word_iff ac_simps)
done

lemma sshiftr_div_2n: "sint (w >>> n) = sint w div 2 ^ n"
using sint_signed_drop_bit_eq [of n w]

lemmas lsb0 = len_gt_0 [THEN word_ops_nth_size [unfolded word_size]]

lemma nth_sint:
fixes w :: "'a::len word"
defines "l ≡ LENGTH('a)"
shows "bit (sint w) n = (if n < l - 1 then w !! n else w !! (l - 1))"
unfolding sint_uint l_def
by (auto simp: bit_signed_take_bit_iff word_test_bit_def not_less min_def)

lemma test_bit_2p: "(word_of_int (2 ^ n)::'a::len word) !! m ⟷ m = n ∧ m < LENGTH('a)"
by transfer (auto simp add: bit_exp_iff)

lemma nth_w2p: "((2::'a::len word) ^ n) !! m ⟷ m = n ∧ m < LENGTH('a::len)"
by transfer (auto simp add: bit_exp_iff)

lemma bang_is_le: "x !! m ⟹ 2 ^ m ≤ x"
for x :: "'a::len word"
apply (rule xtrans(3))
apply (rule_tac [2] y = "x" in le_word_or2)
apply (rule word_eqI)
apply (auto simp add: word_ao_nth nth_w2p word_size)
done

‹mask n = (1 << n) - (1 :: 'a::len word)›

lemma nth_ucast:
"(ucast w::'a::len word) !! n = (w !! n ∧ n < LENGTH('a))"
by transfer (simp add: bit_take_bit_iff ac_simps)

lemma shiftl_0 [simp]: "(0::'a::len word) << n = 0"
by transfer simp

lemma shiftr_0 [simp]: "(0::'a::len word) >> n = 0"
by transfer simp

lemma nth_shiftl1: "shiftl1 w !! n ⟷ n < size w ∧ n > 0 ∧ w !! (n - 1)"
by transfer (auto simp add: bit_double_iff)

lemma nth_shiftl': "(w << m) !! n ⟷ n < size w ∧ n >= m ∧ w !! (n - m)"
for w :: "'a::len word"
by transfer (auto simp add: bit_push_bit_iff)

lemmas nth_shiftl = nth_shiftl' [unfolded word_size]

lemma nth_shiftr1: "shiftr1 w !! n = w !! Suc n"
by transfer (auto simp add: bit_take_bit_iff simp flip: bit_Suc)

lemma nth_shiftr: "(w >> m) !! n = w !! (n + m)"
for w :: "'a::len word"
apply (unfold shiftr_def)
apply (induct "m" arbitrary: n)
done

lemma nth_sshiftr1: "sshiftr1 w !! n = (if n = size w - 1 then w !! n else w !! Suc n)"
apply transfer
apply (auto simp add: bit_take_bit_iff bit_signed_take_bit_iff min_def simp flip: bit_Suc)
using le_less_Suc_eq apply fastforce
using le_less_Suc_eq apply fastforce
done

lemma shiftr_div_2n: "uint (shiftr w n) = uint w div 2 ^ n"
by (fact uint_shiftr_eq)

lemma shiftl_rev: "shiftl w n = word_reverse (shiftr (word_reverse w) n)"
by (induct n) (auto simp add: shiftl_def shiftr_def shiftl1_rev)

lemma rev_shiftl: "word_reverse w << n = word_reverse (w >> n)"

lemma shiftr_rev: "w >> n = word_reverse (word_reverse w << n)"

lemma rev_shiftr: "word_reverse w >> n = word_reverse (w << n)"

lemma shiftl_numeral [simp]:
‹numeral k << numeral l = (push_bit (numeral l) (numeral k) :: 'a::len word)›
by (fact shiftl_word_eq)

lemma shiftl_zero_size: "size x ≤ n ⟹ x << n = 0"
for x :: "'a::len word"
apply transfer
done

lemma shiftl_t2n: "shiftl w n = 2 ^ n * w"
for w :: "'a::len word"
by (induct n) (auto simp: shiftl_def shiftl1_2t)

lemma shiftr_numeral [simp]:
‹(numeral k >> numeral n :: 'a::len word) = drop_bit (numeral n) (numeral k)›
by (fact shiftr_word_eq)

lemma shiftr_numeral_Suc [simp]:
‹(numeral k >> Suc 0 :: 'a::len word) = drop_bit (Suc 0) (numeral k)›
by (fact shiftr_word_eq)

lemma drop_bit_numeral_bit0_1 [simp]:
‹drop_bit (Suc 0) (numeral k) =
(word_of_int (drop_bit (Suc 0) (take_bit LENGTH('a) (numeral k))) :: 'a::len word)›
by (metis Word_eq_word_of_int drop_bit_word.abs_eq of_int_numeral)

‹(mask n :: 'a::len word) !! i ⟷ i < n ∧ i < size (mask n :: 'a word)›

lemma slice_shiftr: "slice n w = ucast (w >> n)"
apply (rule bit_word_eqI)
apply (cases ‹n ≤ LENGTH('b)›)
apply (auto simp add: bit_slice_iff bit_ucast_iff bit_shiftr_word_iff ac_simps
dest: bit_imp_le_length)
done

lemma nth_slice: "(slice n w :: 'a::len word) !! m = (w !! (m + n) ∧ m < LENGTH('a))"
by (simp add: slice_shiftr nth_ucast nth_shiftr)

lemma revcast_down_uu [OF refl]:
"rc = revcast ⟹ source_size rc = target_size rc + n ⟹ rc w = ucast (w >> n)"
for w :: "'a::len word"
apply (rule bit_word_eqI)
apply (simp add: bit_revcast_iff bit_ucast_iff bit_shiftr_word_iff ac_simps)
done

lemma revcast_down_su [OF refl]:
"rc = revcast ⟹ source_size rc = target_size rc + n ⟹ rc w = scast (w >> n)"
for w :: "'a::len word"
apply (rule bit_word_eqI)
apply (simp add: bit_revcast_iff bit_word_scast_iff bit_shiftr_word_iff ac_simps)
done

lemma cast_down_rev [OF refl]:
"uc = ucast ⟹ source_size uc = target_size uc + n ⟹ uc w = revcast (w << n)"
for w :: "'a::len word"
apply (rule bit_word_eqI)
apply (simp add: bit_revcast_iff bit_word_ucast_iff bit_shiftl_word_iff)
done

lemma revcast_up [OF refl]:
"rc = revcast ⟹ source_size rc + n = target_size rc ⟹
rc w = (ucast w :: 'a::len word) << n"
apply (rule bit_word_eqI)
apply (simp add: bit_revcast_iff bit_word_ucast_iff bit_shiftl_word_iff)
apply auto
done

lemmas rc1 = revcast_up [THEN
revcast_rev_ucast [symmetric, THEN trans, THEN word_rev_gal, symmetric]]
lemmas rc2 = revcast_down_uu [THEN
revcast_rev_ucast [symmetric, THEN trans, THEN word_rev_gal, symmetric]]

lemmas ucast_up =
rc1 [simplified rev_shiftr [symmetric] revcast_ucast [symmetric]]
lemmas ucast_down =
rc2 [simplified rev_shiftr revcast_ucast [symmetric]]

― ‹problem posed by TPHOLs referee:
criterion for overflow of addition of signed integers›

lemma sofl_test:
‹sint x + sint y = sint (x + y) ⟷
(x + y XOR x) AND (x + y XOR y) >> (size x - 1) = 0›
for x y :: ‹'a::len word›
proof -
obtain n where n: ‹LENGTH('a) = Suc n›
by (cases ‹LENGTH('a)›) simp_all
have *: ‹sint x + sint y + 2 ^ Suc n > signed_take_bit n (sint x + sint y) ⟹ sint x + sint y ≥ - (2 ^ n)›
‹signed_take_bit n (sint x + sint y) > sint x + sint y - 2 ^ Suc n ⟹ 2 ^ n > sint x + sint y›
using signed_take_bit_int_greater_eq [of ‹sint x + sint y› n] signed_take_bit_int_less_eq [of n ‹sint x + sint y›]
by (auto intro: ccontr)
have ‹sint x + sint y = sint (x + y) ⟷
(sint (x + y) < 0 ⟷ sint x < 0) ∨
(sint (x + y) < 0 ⟷ sint y < 0)›
using sint_less [of x] sint_greater_eq [of x] sint_less [of y] sint_greater_eq [of y]
signed_take_bit_int_eq_self [of ‹LENGTH('a) - 1› ‹sint x + sint y›]
apply (unfold sint_word_ariths)
apply (subst signed_take_bit_int_eq_self)
prefer 4
apply (subst signed_take_bit_int_eq_self)
prefer 7
apply (subst signed_take_bit_int_eq_self)
prefer 10
apply (subst signed_take_bit_int_eq_self)
apply (auto simp add: signed_take_bit_int_eq_self signed_take_bit_eq_take_bit_minus take_bit_Suc_from_most n not_less intro!: *)
apply (smt (z3) take_bit_nonnegative)
apply (smt (z3) take_bit_int_less_exp)
apply (smt (z3) take_bit_nonnegative)
apply (smt (z3) take_bit_int_less_exp)
done
then show ?thesis
apply (simp only: One_nat_def word_size shiftr_word_eq drop_bit_eq_zero_iff_not_bit_last bit_and_iff bit_xor_iff)
done
qed

lemma shiftr_zero_size: "size x ≤ n ⟹ x >> n = 0"
for x :: "'a :: len word"
by (rule word_eqI) (auto simp add: nth_shiftr dest: test_bit_size)

lemma test_bit_cat [OF refl]:
"wc = word_cat a b ⟹ wc !! n = (n < size wc ∧
(if n < size b then b !! n else a !! (n - size b)))"
apply (simp add: word_size not_less; transfer)
apply (auto simp add: bit_concat_bit_iff bit_take_bit_iff)
done

― ‹keep quantifiers for use in simplification›
lemma test_bit_split':
"word_split c = (a, b) ⟶
(∀n m.
b !! n = (n < size b ∧ c !! n) ∧
a !! m = (m < size a ∧ c !! (m + size b)))"
by (auto simp add: word_split_bin' test_bit_bin bit_unsigned_iff word_size bit_drop_bit_eq ac_simps
dest: bit_imp_le_length)

lemma test_bit_split:
"word_split c = (a, b) ⟹
(∀n::nat. b !! n ⟷ n < size b ∧ c !! n) ∧
(∀m::nat. a !! m ⟷ m < size a ∧ c !! (m + size b))"

lemma test_bit_split_eq:
"word_split c = (a, b) ⟷
((∀n::nat. b !! n = (n < size b ∧ c !! n)) ∧
(∀m::nat. a !! m = (m < size a ∧ c !! (m + size b))))"
apply (rule_tac iffI)
apply (rule_tac conjI)
apply (erule test_bit_split [THEN conjunct1])
apply (erule test_bit_split [THEN conjunct2])
apply (case_tac "word_split c")
apply (frule test_bit_split)
apply (erule trans)
apply (fastforce intro!: word_eqI simp add: word_size)
done

lemma test_bit_rcat:
"sw = size (hd wl) ⟹ rc = word_rcat wl ⟹ rc !! n =
(n < size rc ∧ n div sw < size wl ∧ (rev wl) ! (n div sw) !! (n mod sw))"
for wl :: "'a::len word list"
by (simp add: word_size word_rcat_def foldl_map rev_map bit_horner_sum_uint_exp_iff)

lemmas test_bit_cong = arg_cong [where f = "test_bit", THEN fun_cong]

lemma max_test_bit: "(max_word::'a::len word) !! n ⟷ n < LENGTH('a)"
by (fact nth_minus1)

lemma shiftr_x_0 [iff]: "x >> 0 = x"
for x :: "'a::len word"
by transfer simp

lemma shiftl_x_0 [simp]: "x << 0 = x"
for x :: "'a::len word"

lemma shiftl_1 [simp]: "(1::'a::len word) << n = 2^n"

lemma shiftr_1[simp]: "(1::'a::len word) >> n = (if n = 0 then 1 else 0)"
by (induct n) (auto simp: shiftr_def)

lemma map_nth_0 [simp]: "map ((!!) (0::'a::len word)) xs = replicate (length xs) False"
by (induct xs) auto

lemma word_and_1:
"n AND 1 = (if n !! 0 then 1 else 0)" for n :: "_ word"
by (rule bit_word_eqI) (auto simp add: bit_and_iff test_bit_eq_bit bit_1_iff intro: gr0I)

lemma test_bit_1' [simp]:
"(1 :: 'a :: len word) !! n ⟷ 0 < LENGTH('a) ∧ n = 0"
by simp

lemma shiftl0:
"x << 0 = (x :: 'a :: len word)"
by (fact shiftl_x_0)

lemma word_ops_nth [simp]:
fixes x y :: ‹'a::len word›
shows
word_or_nth:  "(x OR y) !! n = (x !! n ∨ y !! n)" and
word_and_nth: "(x AND y) !! n = (x !! n ∧ y !! n)" and
word_xor_nth: "(x XOR y) !! n = (x !! n ≠ y !! n)"
by ((cases "n < size x",
auto dest: test_bit_size simp: word_ops_nth_size word_size)[1])+

"w AND NOT (mask n) = (w >> n) << n"
for w :: ‹'a::len word›
apply (rule word_eqI)
apply (simp add : word_ops_nth_size word_size)
apply (simp add : nth_shiftr nth_shiftl)
by auto

"w AND mask n = (w << (size w - n)) >> (size w - n)"
for w :: ‹'a::len word›
apply (rule word_eqI)
apply (simp add : word_ops_nth_size word_size)
apply (simp add : nth_shiftr nth_shiftl)
by auto

lemma nth_w2p_same:
"(2^n :: 'a :: len word) !! n = (n < LENGTH('a))"

lemma shiftr_div_2n_w: "n < size w ⟹ w >> n = w div (2^n :: 'a :: len word)"
apply (unfold word_div_def)
apply (metis uint_shiftr_eq word_of_int_uint)
done

lemma le_shiftr:
"u ≤ v ⟹ u >> (n :: nat) ≤ (v :: 'a :: len word) >> n"
apply (unfold shiftr_def)
apply (induct_tac "n")
apply auto
apply (erule le_shiftr1)
done

"n <= m ⟹ mask n >> m = (0 :: 'a::len word)"
apply (rule word_eqI)
done

‹mask m >> m = (0::'a::len word)›

lemma word_leI:
"(⋀n.  ⟦n < size (u::'a::len word); u !! n ⟧ ⟹ (v::'a::len word) !! n) ⟹ u <= v"
apply (rule xtrans(4))
apply (rule word_and_le2)
apply (rule word_eqI)
apply safe
apply assumption
apply (erule_tac [2] asm_rl)
apply (unfold word_size)
by auto

"(w ≤ mask n) = (w >> n = 0)"
for w :: ‹'a::len word›
apply safe
apply (rule word_le_0_iff [THEN iffD1])
apply (rule xtrans(3))
apply (erule_tac [2] le_shiftr)
apply simp
apply (rule word_leI)
apply (rename_tac n')
apply (drule_tac x = "n' - n" in word_eqD)
apply (simp add : nth_shiftr word_size)
apply (case_tac "n <= n'")
by auto

"(w AND mask n = w) = (w >> n = 0)"
for w :: ‹'a::len word›
apply (unfold test_bit_eq_iff [THEN sym])
apply (rule iffI)
apply (rule ext)
apply (rule_tac [2] ext)
apply (auto simp add : word_ao_nth nth_shiftr)
apply (drule arg_cong)
apply (drule iffD2)
apply assumption
prefer 2
apply (simp add : word_size test_bit_bin)
apply transfer
apply (auto simp add: fun_eq_iff bit_simps)
done

by (auto intro!: word_eqI simp: and_not_mask nth_shiftl nth_shiftr word_size)

lemma bang_eq:
fixes x :: "'a::len word"
shows "(x = y) = (∀n. x !! n = y !! n)"
by (subst test_bit_eq_iff[symmetric]) fastforce

lemma shiftl_over_and_dist:
fixes a::"'a::len word"
shows "(a AND b) << c = (a << c) AND (b << c)"
apply(rule word_eqI)
done

lemma shiftr_over_and_dist:
fixes a::"'a::len word"
shows "a AND b >> c = (a >> c) AND (b >> c)"
apply(rule word_eqI)
done

lemma sshiftr_over_and_dist:
fixes a::"'a::len word"
shows "a AND b >>> c = (a >>> c) AND (b >>> c)"
apply(rule word_eqI)
done

lemma shiftl_over_or_dist:
fixes a::"'a::len word"
shows "a OR b << c = (a << c) OR (b << c)"
apply(rule word_eqI)
done

lemma shiftr_over_or_dist:
fixes a::"'a::len word"
shows "a OR b >> c = (a >> c) OR (b >> c)"
apply(rule word_eqI)
done

lemma sshiftr_over_or_dist:
fixes a::"'a::len word"
shows "a OR b >>> c = (a >>> c) OR (b >>> c)"
apply(rule word_eqI)
done

lemmas shift_over_ao_dists =
shiftl_over_or_dist shiftr_over_or_dist
sshiftr_over_or_dist shiftl_over_and_dist
shiftr_over_and_dist sshiftr_over_and_dist

lemma shiftl_shiftl:
fixes a::"'a::len word"
shows "a << b << c = a << (b + c)"
apply(rule word_eqI)
done

lemma shiftr_shiftr:
fixes a::"'a::len word"
shows "a >> b >> c = a >> (b + c)"
apply(rule word_eqI)
done

lemma shiftl_shiftr1:
fixes a::"'a::len word"
shows "c ≤ b ⟹ a << b >> c = a AND (mask (size a - b)) << (b - c)"
apply(rule word_eqI)
apply(auto simp:nth_shiftr nth_shiftl word_size word_ao_nth)
done

lemma shiftl_shiftr2:
fixes a::"'a::len word"
shows "b < c ⟹ a << b >> c = (a >> (c - b)) AND (mask (size a - c))"
apply(rule word_eqI)
apply(auto simp:nth_shiftr nth_shiftl word_size word_ao_nth)
done

lemma shiftr_shiftl1:
fixes a::"'a::len word"
shows "c ≤ b ⟹ a >> b << c = (a >> (b - c)) AND (NOT (mask c))"
apply(rule word_eqI)
apply(auto simp:nth_shiftr nth_shiftl word_size word_ops_nth_size)
done

lemma shiftr_shiftl2:
fixes a::"'a::len word"
shows "b < c ⟹ a >> b << c = (a << (c - b)) AND (NOT (mask c))"
apply(rule word_eqI)
apply(auto simp:nth_shiftr nth_shiftl word_size word_ops_nth_size)
done

lemmas multi_shift_simps =
shiftl_shiftl shiftr_shiftr
shiftl_shiftr1 shiftl_shiftr2
shiftr_shiftl1 shiftr_shiftl2

"n ≤ LENGTH('a) ⟹ (mask n >> m :: ('a :: len) word) = mask (n - m)"
apply (rule word_eqI)
apply arith
done

fixes x :: "'a :: len word"
shows "(x + y) << n = (x << n) + (y << n)"

"(x AND NOT (mask y)) >> y = x >> y"
for x :: ‹'a::len word›
apply (rule bit_eqI)
using bit_imp_le_length apply auto
done

lemma shiftr_div_2n':
"unat (w >> n) = unat w div 2 ^ n"
apply (unfold unat_eq_nat_uint)
apply (subst shiftr_div_2n)
apply (subst nat_div_distrib)
apply simp
done

lemma shiftl_shiftr_id:
assumes nv: "n < LENGTH('a)"
and     xv: "x < 2 ^ (LENGTH('a) - n)"
shows "x << n >> n = (x::'a::len word)"
apply (rule word_eq_unatI)
apply (subst shiftr_div_2n')
apply (cases n)
apply simp
apply (subst iffD1 [OF unat_mult_lem])+
apply (subst unat_power_lower[OF nv])
apply (rule nat_less_power_trans [OF _ order_less_imp_le [OF nv]])
apply (rule order_less_le_trans [OF unat_mono [OF xv] order_eq_refl])
apply (rule unat_power_lower)
apply simp
apply (subst unat_power_lower[OF nv])
apply simp
done

lemma ucast_shiftl_eq_0:
fixes w :: "'a :: len word"
shows "⟦ n ≥ LENGTH('b) ⟧ ⟹ ucast (w << n) = (0 :: 'b :: len word)"

lemma word_shift_nonzero:
"⟦ (x::'a::len word) ≤ 2 ^ m; m + n < LENGTH('a::len); x ≠ 0⟧
⟹ x << n ≠ 0"
apply (simp only: word_neq_0_conv word_less_nat_alt
shiftl_t2n mod_0 unat_word_ariths
unat_power_lower word_le_nat_alt)
apply (subst mod_less)
apply (rule order_le_less_trans)
apply (erule mult_le_mono2)
apply (rule power_strict_increasing)
apply simp
apply simp
apply simp
done

lemma word_shiftr_lt:
fixes w :: "'a::len word"
shows "unat (w >> n) < (2 ^ (LENGTH('a) - n))"
apply (subst shiftr_div_2n')
apply transfer
apply (simp flip: drop_bit_eq_div add: drop_bit_nat_eq drop_bit_take_bit)
done

"(NOT(mask n) :: 'a :: len word) !! m = (n ≤ m ∧ m < LENGTH('a))"
by (metis not_le nth_mask test_bit_bin word_ops_nth_size word_size)

lemma upper_bits_unset_is_l2p:
‹(∀n' ≥ n. n' < LENGTH('a) ⟶ ¬ p !! n') ⟷ (p < 2 ^ n)› (is ‹?P ⟷ ?Q›)
if ‹n < LENGTH('a)›
for p :: "'a :: len word"
proof
assume ?Q
then show ?P
by (meson bang_is_le le_less_trans not_le word_power_increasing)
next
assume ?P
have ‹take_bit n p = p›
proof (rule bit_word_eqI)
fix q
assume ‹q < LENGTH('a)›
show ‹bit (take_bit n p) q ⟷ bit p q›
proof (cases ‹q < n›)
case True
then show ?thesis
next
case False
then have ‹n ≤ q›
by simp
with ‹?P› ‹q < LENGTH('a)› have ‹¬ bit p q›
then show ?thesis
qed
qed
with that show ?Q
using take_bit_word_eq_self_iff [of n p] by auto
qed

lemma less_2p_is_upper_bits_unset:
"p < 2 ^ n ⟷ n < LENGTH('a) ∧ (∀n' ≥ n. n' < LENGTH('a) ⟶ ¬ p !! n')" for p :: "'a :: len word"
by (meson le_less_trans le_mask_iff_lt_2n upper_bits_unset_is_l2p word_zero_le)

lemma test_bit_over:
"n ≥ size (x::'a::len word) ⟹ (x !! n) = False"
by transfer auto

"w ≤ mask n ⟷ (∀i ∈ {n ..< size w}. ¬ w !! i)"
for w :: ‹'a::len word›

lemma test_bit_conj_lt:
"(x !! m ∧ m < LENGTH('a)) = x !! m" for x :: "'a :: len word"
using test_bit_bin by blast

lemma neg_test_bit:
"(NOT x) !! n = (¬ x !! n ∧ n < LENGTH('a))" for x :: "'a::len word"
by (cases "n < LENGTH('a)") (auto simp add: test_bit_over word_ops_nth_size word_size)

lemma shiftr_less_t2n':
"⟦ x AND mask (n + m) = x; m < LENGTH('a) ⟧ ⟹ x >> n < 2 ^ m" for x :: "'a :: len word"
apply transfer
done

lemma shiftr_less_t2n:
"x < 2 ^ (n + m) ⟹ x >> n < 2 ^ m" for x :: "'a :: len word"
apply (rule shiftr_less_t2n')
apply (rule ccontr)
apply (subst (asm) p2_eq_0[symmetric])
done

lemma shiftr_eq_0:
"n ≥ LENGTH('a) ⟹ ((w::'a::len word) >> n) = 0"
apply (cut_tac shiftr_less_t2n'[of w n 0], simp)
apply simp
done

"n+m ≥ LENGTH('a :: len) ⟹ ((w::'a::len word) >> n) AND NOT (mask m) = 0"
by (rule bit_word_eqI) (auto simp add: bit_simps dest: bit_imp_le_length)

lemma shiftl_less_t2n:
fixes x :: "'a :: len word"
shows "⟦ x < (2 ^ (m - n)); m < LENGTH('a) ⟧ ⟹ (x << n) < 2 ^ m"
apply transfer
done

lemma shiftl_less_t2n':
"(x::'a::len word) < 2 ^ m ⟹ m+n < LENGTH('a) ⟹ x << n < 2 ^ (m + n)"
by (rule shiftl_less_t2n) simp_all

lemma nth_w2p_scast [simp]:
"((scast ((2::'a::len signed word) ^ n) :: 'a word) !! m)
⟷ ((((2::'a::len  word) ^ n) :: 'a word) !! m)"
by transfer (auto simp add: bit_simps)

lemma scast_bit_test [simp]:
"scast ((1 :: 'a::len signed word) << n) = (1 :: 'a word) << n"
by (clarsimp simp: word_eq_iff)

lemma signed_shift_guard_to_word:
"⟦ n < len_of TYPE ('a); n > 0 ⟧
⟹ (unat (x :: 'a :: len word) * 2 ^ y < 2 ^ n)
= (x = 0 ∨ x < (1 << n >> y))"
apply (simp only: nat_mult_power_less_eq)
apply (cases "y ≤ n")
apply (simp only: shiftl_shiftr1)
apply (rule order_less_le_trans[rotated], rule power_increasing[where n=1])
apply simp
apply simp
apply simp
apply (simp add: nat_mult_power_less_eq word_less_nat_alt word_size)
apply auto[1]
apply (simp only: shiftl_shiftr2, simp add: unat_eq_0)
done

lemma nth_bounded:
"⟦(x :: 'a :: len word) !! n; x < 2 ^ m; m ≤ len_of TYPE ('a)⟧ ⟹ n < m"
apply (rule ccontr)
apply (auto simp add: not_less test_bit_word_eq)
apply (meson bit_imp_le_length bit_uint_iff less_2p_is_upper_bits_unset test_bit_bin)
done

"(x << n) AND mask n = 0"
for x :: ‹'a::len word›

"(a >> (size a - b)) AND mask b = a >> (size a - b)"
for a :: ‹'a::len word›
using shiftl_shiftr2[where a=a and b=0 and c="size a - b"]
apply (cases "b < size a")
apply simp
p2_eq_0[THEN iffD2])
done

lemma shiftl_shiftr3:
"b ≤ c ⟹ a << b >> c = (a >> c - b) AND mask (size a - c)"
for a :: ‹'a::len word›
apply (cases "b = c")
done

"m ≤ size w ⟹ (w AND mask m) >> n = (w >> n) AND mask (m-n)"
for w :: ‹'a::len word›

"m+n ≤ size w ⟹ (w AND mask m) << n = (w << n) AND mask (m+n)"
for w :: ‹'a::len word›

for x :: ‹'a::len word›

lemma word_and_1_shiftl:
"x AND (1 << n) = (if x !! n then (1 << n) else 0)" for x :: "'a :: len word"
apply (rule bit_word_eqI; transfer)
apply (auto simp add: bit_simps not_le ac_simps)
done

lemmas word_and_1_shiftls'
= word_and_1_shiftl[where n=0]
word_and_1_shiftl[where n=1]
word_and_1_shiftl[where n=2]

lemmas word_and_1_shiftls = word_and_1_shiftls' [simplified]

"x AND (mask n << m) = ((x >> m) AND mask n) << m"
for x :: ‹'a::len word›
apply (rule bit_word_eqI; transfer)
apply (auto simp add: bit_simps not_le ac_simps)
done

lemma shift_times_fold:
"(x :: 'a :: len word) * (2 ^ n) << m = x << (m + n)"

lemma of_bool_nth:
"of_bool (x !! v) = (x >> v) AND 1"
for x :: ‹'a::len word›
by (simp add: test_bit_word_eq shiftr_word_eq bit_eq_iff)
(auto simp add: bit_1_iff bit_and_iff bit_drop_bit_eq intro: ccontr)

"(x >> n) AND mask (size x - n) = x >> n" for x :: "'a :: len word"
apply transfer
done

"m = (size x - n) ⟹ (x >> n) AND mask m = x >> n" for x :: "'a :: len word"

lemma and_eq_0_is_nth:
fixes x :: "'a :: len word"
shows "y = 1 << n ⟹ ((x AND y) = 0) = (¬ (x !! n))"
apply safe
apply (drule_tac u="(x AND (1 << n))" and x=n in word_eqD)
apply (rule bit_word_eqI)
apply (auto simp add: bit_simps test_bit_eq_bit)
done

lemma and_neq_0_is_nth:
‹x AND y ≠ 0 ⟷ x !! n› if ‹y = 2 ^ n› for x y :: ‹'a::len word›
using that apply (simp add: bit_simps not_le)
apply transfer
apply auto
done

lemma nth_is_and_neq_0:
"(x::'a::len word) !! n = (x AND 2 ^ n ≠ 0)"
by (subst and_neq_0_is_nth; rule refl)

lemma word_shift_zero:
"⟦ x << n = 0; x ≤ 2^m; m + n < LENGTH('a)⟧ ⟹ (x::'a::len word) = 0"
apply (rule ccontr)
apply (drule (2) word_shift_nonzero)
apply simp
done

for w :: ‹'a::len word›

end


# Theory Word_EqI

(*
* Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
*
*)

section "Solving Word Equalities"

theory Word_EqI
imports
More_Word
"HOL-Eisbach.Eisbach_Tools"
begin

text ‹
Some word equalities can be solved by considering the problem bitwise for all
@{prop "n < LENGTH('a::len)"}, which is different to running @{text word_bitwise}
and expanding into an explicit list of bits.
›

named_theorems word_eqI_simps

lemmas [word_eqI_simps] =
word_ops_nth_size
word_size
word_or_zero
nth_ucast
nth_w2p nth_shiftl
nth_shiftr
less_2p_is_upper_bits_unset
bang_eq
neg_test_bit
is_up
is_down

lemmas word_eqI_rule = word_eqI [rule_format]

lemma test_bit_lenD:
"x !! n ⟹ n < LENGTH('a) ∧ x !! n" for x :: "'a :: len word"
by (fastforce dest: test_bit_size simp: word_size)

method word_eqI uses simp simp_del split split_del cong flip =
((* reduce conclusion to test_bit: *)
rule word_eqI_rule,
(* make sure we're in clarsimp normal form: *)
(clarsimp simp: simp simp del: simp_del simp flip: flip split: split split del: split_del cong: cong)?,
(* turn x < 2^n assumptions into mask equations: *)
(* expand and distribute test_bit everywhere: *)
(clarsimp simp: word_eqI_simps simp simp del: simp_del simp flip: flip
split: split split del: split_del cong: cong)?,
((drule test_bit_lenD)+)?,
(* try to make progress (can't use +, would loop): *)
(clarsimp simp: word_eqI_simps simp simp del: simp_del simp flip: flip
split: split split del: split_del cong: cong)?,
(* helps sometimes, rarely: *)
(simp add: simp test_bit_conj_lt del: simp_del flip: flip split: split split del: split_del cong: cong)?)

method word_eqI_solve uses simp simp_del split split_del cong flip =
solves ‹word_eqI simp: simp simp_del: simp_del split: split split_del: split_del
cong: cong simp flip: flip;
(fastforce dest: test_bit_size simp: word_eqI_simps simp flip: flip
simp: simp simp del: simp_del split: split split del: split_del cong: cong)?›

end


# Theory Bit_Comprehension

(*
* Copyright Brian Huffman, PSU; Jeremy Dawson and Gerwin Klein, NICTA
*
*)

section ‹Comprehension syntax for bit expressions›

theory Bit_Comprehension
imports "HOL-Library.Word"
begin

class bit_comprehension = ring_bit_operations +
fixes set_bits :: ‹(nat ⇒ bool) ⇒ 'a›  (binder ‹BITS › 10)
assumes set_bits_bit_eq: ‹set_bits (bit a) = a›
begin

lemma set_bits_False_eq [simp]:
‹(BITS _. False) = 0›
using set_bits_bit_eq [of 0] by (simp add: bot_fun_def)

end

instantiation int :: bit_comprehension
begin

definition
‹set_bits f = (
if ∃n. ∀m≥n. f m = f n then
let n = LEAST n. ∀m≥n. f m = f n
in signed_take_bit n (horner_sum of_bool 2 (map f [0..<Suc n]))
else 0 :: int)›

instance proof
fix k :: int
from int_bit_bound [of k]
obtain n where *: ‹⋀m. n ≤ m ⟹ bit k m ⟷ bit k n›
and **: ‹n > 0 ⟹ bit k (n - 1) ≠ bit k n›
by blast
then have ***: ‹∃n. ∀n'≥n. bit k n' ⟷ bit k n›
by meson
have l: ‹(LEAST q. ∀m≥q. bit k m ⟷ bit k q) = n›
apply (rule Least_equality)
using * apply blast
apply (metis "**" One_nat_def Suc_pred le_cases le0 neq0_conv not_less_eq_eq)
done
show ‹set_bits (bit k) = k›
apply (simp only: *** set_bits_int_def horner_sum_bit_eq_take_bit l)
apply simp
apply (rule bit_eqI)
apply (auto simp add: not_le bit_take_bit_iff dest: *<`