(* Author: Florian Haftmann, TUM
*)
section \<open>Proof of concept for algebraically founded bit word types\<close>
theory Word_Type
imports
Main
"HOL-ex.Bit_Lists"
"HOL-Library.Type_Length"
begin
subsection \<open>Preliminaries\<close>
lemma take_bit_uminus:
"take_bit n (- (take_bit n k)) = take_bit n (- k)" for k :: int
by (simp add: take_bit_eq_mod mod_minus_eq)
lemma take_bit_minus:
"take_bit n (take_bit n k - take_bit n l) = take_bit n (k - l)" for k l :: int
by (simp add: take_bit_eq_mod mod_diff_eq)
lemma take_bit_nonnegative [simp]:
"take_bit n k \<ge> 0" for k :: int
by (simp add: take_bit_eq_mod)
definition signed_take_bit :: "nat \<Rightarrow> int \<Rightarrow> int"
where signed_take_bit_eq_take_bit:
"signed_take_bit n k = take_bit (Suc n) (k + 2 ^ n) - 2 ^ n"
lemma signed_take_bit_eq_take_bit':
"signed_take_bit (n - Suc 0) k = take_bit n (k + 2 ^ (n - 1)) - 2 ^ (n - 1)" if "n > 0"
using that by (simp add: signed_take_bit_eq_take_bit)
lemma signed_take_bit_0 [simp]:
"signed_take_bit 0 k = - (k mod 2)"
proof (cases "even k")
case True
then have "odd (k + 1)"
by simp
then have "(k + 1) mod 2 = 1"
by (simp add: even_iff_mod_2_eq_zero)
with True show ?thesis
by (simp add: signed_take_bit_eq_take_bit)
next
case False
then show ?thesis
by (simp add: signed_take_bit_eq_take_bit odd_iff_mod_2_eq_one)
qed
lemma signed_take_bit_Suc [simp]:
"signed_take_bit (Suc n) k = signed_take_bit n (k div 2) * 2 + k mod 2"
by (simp add: odd_iff_mod_2_eq_one signed_take_bit_eq_take_bit algebra_simps)
lemma signed_take_bit_of_0 [simp]:
"signed_take_bit n 0 = 0"
by (simp add: signed_take_bit_eq_take_bit take_bit_eq_mod)
lemma signed_take_bit_of_minus_1 [simp]:
"signed_take_bit n (- 1) = - 1"
by (induct n) simp_all
lemma signed_take_bit_eq_iff_take_bit_eq:
"signed_take_bit (n - Suc 0) k = signed_take_bit (n - Suc 0) l \<longleftrightarrow> take_bit n k = take_bit n l" (is "?P \<longleftrightarrow> ?Q")
if "n > 0"
proof -
from that obtain m where m: "n = Suc m"
by (cases n) auto
show ?thesis
proof
assume ?Q
have "take_bit (Suc m) (k + 2 ^ m) =
take_bit (Suc m) (take_bit (Suc m) k + take_bit (Suc m) (2 ^ m))"
by (simp only: take_bit_add)
also have "\<dots> =
take_bit (Suc m) (take_bit (Suc m) l + take_bit (Suc m) (2 ^ m))"
by (simp only: \<open>?Q\<close> m [symmetric])
also have "\<dots> = take_bit (Suc m) (l + 2 ^ m)"
by (simp only: take_bit_add)
finally show ?P
by (simp only: signed_take_bit_eq_take_bit m) simp
next
assume ?P
with that have "(k + 2 ^ (n - Suc 0)) mod 2 ^ n = (l + 2 ^ (n - Suc 0)) mod 2 ^ n"
by (simp add: signed_take_bit_eq_take_bit' take_bit_eq_mod)
then have "(i + (k + 2 ^ (n - Suc 0))) mod 2 ^ n = (i + (l + 2 ^ (n - Suc 0))) mod 2 ^ n" for i
by (metis mod_add_eq)
then have "k mod 2 ^ n = l mod 2 ^ n"
by (metis add_diff_cancel_right' uminus_add_conv_diff)
then show ?Q
by (simp add: take_bit_eq_mod)
qed
qed
subsection \<open>Bit strings as quotient type\<close>
subsubsection \<open>Basic properties\<close>
quotient_type (overloaded) 'a word = int / "\<lambda>k l. take_bit LENGTH('a) k = take_bit LENGTH('a::len0) l"
by (auto intro!: equivpI reflpI sympI transpI)
instantiation word :: (len0) "{semiring_numeral, comm_semiring_0, comm_ring}"
begin
lift_definition zero_word :: "'a word"
is 0
.
lift_definition one_word :: "'a word"
is 1
.
lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
is plus
by (subst take_bit_add [symmetric]) (simp add: take_bit_add)
lift_definition uminus_word :: "'a word \<Rightarrow> 'a word"
is uminus
by (subst take_bit_uminus [symmetric]) (simp add: take_bit_uminus)
lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
is minus
by (subst take_bit_minus [symmetric]) (simp add: take_bit_minus)
lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
is times
by (auto simp add: take_bit_eq_mod intro: mod_mult_cong)
instance
by standard (transfer; simp add: algebra_simps)+
end
instance word :: (len) comm_ring_1
by standard (transfer; simp)+
quickcheck_generator word
constructors:
"zero_class.zero :: ('a::len0) word",
"numeral :: num \<Rightarrow> ('a::len0) word",
"uminus :: ('a::len0) word \<Rightarrow> ('a::len0) word"
context
includes lifting_syntax
notes power_transfer [transfer_rule]
begin
lemma power_transfer_word [transfer_rule]:
\<open>(pcr_word ===> (=) ===> pcr_word) (^) (^)\<close>
by transfer_prover
end
subsubsection \<open>Conversions\<close>
context
includes lifting_syntax
notes transfer_rule_numeral [transfer_rule]
transfer_rule_of_nat [transfer_rule]
transfer_rule_of_int [transfer_rule]
begin
lemma [transfer_rule]:
"((=) ===> (pcr_word :: int \<Rightarrow> 'a::len word \<Rightarrow> bool)) numeral numeral"
by transfer_prover
lemma [transfer_rule]:
"((=) ===> pcr_word) int of_nat"
by transfer_prover
lemma [transfer_rule]:
"((=) ===> pcr_word) (\<lambda>k. k) of_int"
proof -
have "((=) ===> pcr_word) of_int of_int"
by transfer_prover
then show ?thesis by (simp add: id_def)
qed
end
lemma abs_word_eq:
"abs_word = of_int"
by (rule ext) (transfer, rule)
context semiring_1
begin
lift_definition unsigned :: "'b::len0 word \<Rightarrow> 'a"
is "of_nat \<circ> nat \<circ> take_bit LENGTH('b)"
by simp
lemma unsigned_0 [simp]:
"unsigned 0 = 0"
by transfer simp
end
context semiring_char_0
begin
lemma word_eq_iff_unsigned:
"a = b \<longleftrightarrow> unsigned a = unsigned b"
by safe (transfer; simp add: eq_nat_nat_iff)
end
instantiation word :: (len0) equal
begin
definition equal_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
where "equal_word a b \<longleftrightarrow> (unsigned a :: int) = unsigned b"
instance proof
fix a b :: "'a word"
show "HOL.equal a b \<longleftrightarrow> a = b"
using word_eq_iff_unsigned [of a b] by (auto simp add: equal_word_def)
qed
end
context ring_1
begin
lift_definition signed :: "'b::len word \<Rightarrow> 'a"
is "of_int \<circ> signed_take_bit (LENGTH('b) - 1)"
by (simp add: signed_take_bit_eq_iff_take_bit_eq [symmetric])
lemma signed_0 [simp]:
"signed 0 = 0"
by transfer simp
end
lemma unsigned_of_nat [simp]:
"unsigned (of_nat n :: 'a word) = take_bit LENGTH('a::len) n"
by transfer (simp add: nat_eq_iff take_bit_eq_mod zmod_int)
lemma of_nat_unsigned [simp]:
"of_nat (unsigned a) = a"
by transfer simp
lemma of_int_unsigned [simp]:
"of_int (unsigned a) = a"
by transfer simp
lemma unsigned_nat_less:
\<open>unsigned a < (2 ^ LENGTH('a) :: nat)\<close> for a :: \<open>'a::len0 word\<close>
by transfer (simp add: take_bit_eq_mod)
lemma unsigned_int_less:
\<open>unsigned a < (2 ^ LENGTH('a) :: int)\<close> for a :: \<open>'a::len0 word\<close>
by transfer (simp add: take_bit_eq_mod)
context ring_char_0
begin
lemma word_eq_iff_signed:
"a = b \<longleftrightarrow> signed a = signed b"
by safe (transfer; auto simp add: signed_take_bit_eq_iff_take_bit_eq)
end
lemma signed_of_int [simp]:
"signed (of_int k :: 'a word) = signed_take_bit (LENGTH('a::len) - 1) k"
by transfer simp
lemma of_int_signed [simp]:
"of_int (signed a) = a"
by transfer (simp add: signed_take_bit_eq_take_bit take_bit_eq_mod mod_simps)
subsubsection \<open>Properties\<close>
lemma length_cases:
obtains (triv) "LENGTH('a::len) = 1" "take_bit LENGTH('a) 2 = (0 :: int)"
| (take_bit_2) "take_bit LENGTH('a) 2 = (2 :: int)"
proof (cases "LENGTH('a) \<ge> 2")
case False
then have "LENGTH('a) = 1"
by (auto simp add: not_le dest: less_2_cases)
then have "take_bit LENGTH('a) 2 = (0 :: int)"
by simp
with \<open>LENGTH('a) = 1\<close> triv show ?thesis
by simp
next
case True
then obtain n where "LENGTH('a) = Suc (Suc n)"
by (auto dest: le_Suc_ex)
then have "take_bit LENGTH('a) 2 = (2 :: int)"
by simp
with take_bit_2 show ?thesis
by simp
qed
subsubsection \<open>Division\<close>
instantiation word :: (len0) modulo
begin
lift_definition divide_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
is "\<lambda>a b. take_bit LENGTH('a) a div take_bit LENGTH('a) b"
by simp
lift_definition modulo_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
is "\<lambda>a b. take_bit LENGTH('a) a mod take_bit LENGTH('a) b"
by simp
instance ..
end
lemma zero_word_div_eq [simp]:
\<open>0 div a = 0\<close> for a :: \<open>'a::len0 word\<close>
by transfer simp
lemma div_zero_word_eq [simp]:
\<open>a div 0 = 0\<close> for a :: \<open>'a::len0 word\<close>
by transfer simp
context
includes lifting_syntax
begin
lemma [transfer_rule]:
"(pcr_word ===> (\<longleftrightarrow>)) even ((dvd) 2 :: 'a::len word \<Rightarrow> bool)"
proof -
have even_word_unfold: "even k \<longleftrightarrow> (\<exists>l. take_bit LENGTH('a) k = take_bit LENGTH('a) (2 * l))" (is "?P \<longleftrightarrow> ?Q")
for k :: int
proof
assume ?P
then show ?Q
by auto
next
assume ?Q
then obtain l where "take_bit LENGTH('a) k = take_bit LENGTH('a) (2 * l)" ..
then have "even (take_bit LENGTH('a) k)"
by simp
then show ?P
by simp
qed
show ?thesis by (simp only: even_word_unfold [abs_def] dvd_def [where ?'a = "'a word", abs_def])
transfer_prover
qed
end
instance word :: (len) semiring_modulo
proof
show "a div b * b + a mod b = a" for a b :: "'a word"
proof transfer
fix k l :: int
define r :: int where "r = 2 ^ LENGTH('a)"
then have r: "take_bit LENGTH('a) k = k mod r" for k
by (simp add: take_bit_eq_mod)
have "k mod r = ((k mod r) div (l mod r) * (l mod r)
+ (k mod r) mod (l mod r)) mod r"
by (simp add: div_mult_mod_eq)
also have "... = (((k mod r) div (l mod r) * (l mod r)) mod r
+ (k mod r) mod (l mod r)) mod r"
by (simp add: mod_add_left_eq)
also have "... = (((k mod r) div (l mod r) * l) mod r
+ (k mod r) mod (l mod r)) mod r"
by (simp add: mod_mult_right_eq)
finally have "k mod r = ((k mod r) div (l mod r) * l
+ (k mod r) mod (l mod r)) mod r"
by (simp add: mod_simps)
with r show "take_bit LENGTH('a) (take_bit LENGTH('a) k div take_bit LENGTH('a) l * l
+ take_bit LENGTH('a) k mod take_bit LENGTH('a) l) = take_bit LENGTH('a) k"
by simp
qed
qed
instance word :: (len) semiring_parity
proof
show "\<not> 2 dvd (1::'a word)"
by transfer simp
show even_iff_mod_2_eq_0: "2 dvd a \<longleftrightarrow> a mod 2 = 0"
for a :: "'a word"
by (transfer; cases rule: length_cases [where ?'a = 'a]) (simp_all add: mod_2_eq_odd)
show "\<not> 2 dvd a \<longleftrightarrow> a mod 2 = 1"
for a :: "'a word"
by (transfer; cases rule: length_cases [where ?'a = 'a]) (simp_all add: mod_2_eq_odd)
qed
subsubsection \<open>Orderings\<close>
instantiation word :: (len0) linorder
begin
lift_definition less_eq_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
is "\<lambda>a b. take_bit LENGTH('a) a \<le> take_bit LENGTH('a) b"
by simp
lift_definition less_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
is "\<lambda>a b. take_bit LENGTH('a) a < take_bit LENGTH('a) b"
by simp
instance
by standard (transfer; auto)+
end
context linordered_semidom
begin
lemma word_less_eq_iff_unsigned:
"a \<le> b \<longleftrightarrow> unsigned a \<le> unsigned b"
by (transfer fixing: less_eq) (simp add: nat_le_eq_zle)
lemma word_less_iff_unsigned:
"a < b \<longleftrightarrow> unsigned a < unsigned b"
by (transfer fixing: less) (auto dest: preorder_class.le_less_trans [OF take_bit_nonnegative])
end
lemma word_greater_zero_iff:
\<open>a > 0 \<longleftrightarrow> a \<noteq> 0\<close> for a :: \<open>'a::len0 word\<close>
by transfer (simp add: less_le)
lemma of_nat_word_eq_iff:
\<open>of_nat m = (of_nat n :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) m = take_bit LENGTH('a) n\<close>
by transfer (simp add: take_bit_of_nat)
lemma of_nat_word_less_eq_iff:
\<open>of_nat m \<le> (of_nat n :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) m \<le> take_bit LENGTH('a) n\<close>
by transfer (simp add: take_bit_of_nat)
lemma of_nat_word_less_iff:
\<open>of_nat m < (of_nat n :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) m < take_bit LENGTH('a) n\<close>
by transfer (simp add: take_bit_of_nat)
lemma of_nat_word_eq_0_iff:
\<open>of_nat n = (0 :: 'a::len word) \<longleftrightarrow> 2 ^ LENGTH('a) dvd n\<close>
using of_nat_word_eq_iff [where ?'a = 'a, of n 0] by (simp add: take_bit_eq_0_iff)
lemma of_int_word_eq_iff:
\<open>of_int k = (of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close>
by transfer rule
lemma of_int_word_less_eq_iff:
\<open>of_int k \<le> (of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k \<le> take_bit LENGTH('a) l\<close>
by transfer rule
lemma of_int_word_less_iff:
\<open>of_int k < (of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k < take_bit LENGTH('a) l\<close>
by transfer rule
lemma of_int_word_eq_0_iff:
\<open>of_int k = (0 :: 'a::len word) \<longleftrightarrow> 2 ^ LENGTH('a) dvd k\<close>
using of_int_word_eq_iff [where ?'a = 'a, of k 0] by (simp add: take_bit_eq_0_iff)
subsection \<open>Bit operation on \<^typ>\<open>'a word\<close>\<close>
context unique_euclidean_semiring_with_nat
begin
primrec n_bits_of :: "nat \<Rightarrow> 'a \<Rightarrow> bool list"
where
"n_bits_of 0 a = []"
| "n_bits_of (Suc n) a = odd a # n_bits_of n (a div 2)"
lemma n_bits_of_eq_iff:
"n_bits_of n a = n_bits_of n b \<longleftrightarrow> take_bit n a = take_bit n b"
apply (induction n arbitrary: a b)
apply (auto elim!: evenE oddE)
apply (metis dvd_triv_right even_plus_one_iff)
apply (metis dvd_triv_right even_plus_one_iff)
done
lemma take_n_bits_of [simp]:
"take m (n_bits_of n a) = n_bits_of (min m n) a"
proof -
define q and v and w where "q = min m n" and "v = m - q" and "w = n - q"
then have "v = 0 \<or> w = 0"
by auto
then have "take (q + v) (n_bits_of (q + w) a) = n_bits_of q a"
by (induction q arbitrary: a) auto
with q_def v_def w_def show ?thesis
by simp
qed
lemma unsigned_of_bits_n_bits_of [simp]:
"unsigned_of_bits (n_bits_of n a) = take_bit n a"
by (induction n arbitrary: a) (simp_all add: ac_simps)
end
lemma unsigned_of_bits_eq_of_bits:
"unsigned_of_bits bs = (of_bits (bs @ [False]) :: int)"
by (simp add: of_bits_int_def)
instantiation word :: (len) bit_representation
begin
lift_definition bits_of_word :: "'a word \<Rightarrow> bool list"
is "n_bits_of LENGTH('a)"
by (simp add: n_bits_of_eq_iff)
lift_definition of_bits_word :: "bool list \<Rightarrow> 'a word"
is unsigned_of_bits .
instance proof
fix a :: "'a word"
show "of_bits (bits_of a) = a"
by transfer simp
qed
end
lemma take_bit_complement_iff:
"take_bit n (complement k) = take_bit n (complement l) \<longleftrightarrow> take_bit n k = take_bit n l"
for k l :: int
by (simp add: take_bit_eq_mod mod_eq_dvd_iff dvd_diff_commute)
lemma take_bit_not_iff:
"take_bit n (NOT k) = take_bit n (NOT l) \<longleftrightarrow> take_bit n k = take_bit n l"
for k l :: int
by (simp add: not_int_def take_bit_complement_iff)
lemma n_bits_of_not:
"n_bits_of n (NOT k) = map Not (n_bits_of n k)"
for k :: int
by (induction n arbitrary: k) (simp_all add: not_div_2)
lemma take_bit_and [simp]:
"take_bit n (k AND l) = take_bit n k AND take_bit n l"
for k l :: int
apply (induction n arbitrary: k l)
apply simp
apply (subst and_int.rec)
apply (subst (2) and_int.rec)
apply simp
done
lemma take_bit_or [simp]:
"take_bit n (k OR l) = take_bit n k OR take_bit n l"
for k l :: int
apply (induction n arbitrary: k l)
apply simp
apply (subst or_int.rec)
apply (subst (2) or_int.rec)
apply simp
done
lemma take_bit_xor [simp]:
"take_bit n (k XOR l) = take_bit n k XOR take_bit n l"
for k l :: int
apply (induction n arbitrary: k l)
apply simp
apply (subst xor_int.rec)
apply (subst (2) xor_int.rec)
apply simp
done
instantiation word :: (len) bit_operations
begin
lift_definition not_word :: "'a word \<Rightarrow> 'a word"
is not
by (simp add: take_bit_not_iff)
lift_definition and_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
is "and"
by simp
lift_definition or_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
is or
by simp
lift_definition xor_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
is xor
by simp
lift_definition shift_left_word :: "'a word \<Rightarrow> nat \<Rightarrow> 'a word"
is shift_left
proof -
show "take_bit LENGTH('a) (k << n) = take_bit LENGTH('a) (l << n)"
if "take_bit LENGTH('a) k = take_bit LENGTH('a) l" for k l :: int and n :: nat
proof -
from that
have "take_bit (LENGTH('a) - n) (take_bit LENGTH('a) k)
= take_bit (LENGTH('a) - n) (take_bit LENGTH('a) l)"
by simp
moreover have "min (LENGTH('a) - n) LENGTH('a) = LENGTH('a) - n"
by simp
ultimately show ?thesis by (simp add: take_bit_push_bit)
qed
qed
lift_definition shift_right_word :: "'a word \<Rightarrow> nat \<Rightarrow> 'a word"
is "\<lambda>k n. drop_bit n (take_bit LENGTH('a) k)"
by simp
instance proof
show "semilattice ((AND) :: 'a word \<Rightarrow> _)"
by standard (transfer; simp add: ac_simps)+
show "semilattice ((OR) :: 'a word \<Rightarrow> _)"
by standard (transfer; simp add: ac_simps)+
show "abel_semigroup ((XOR) :: 'a word \<Rightarrow> _)"
by standard (transfer; simp add: ac_simps)+
show "not = (of_bits \<circ> map Not \<circ> bits_of :: 'a word \<Rightarrow> 'a word)"
proof
fix a :: "'a word"
have "NOT a = of_bits (map Not (bits_of a))"
by transfer (simp flip: unsigned_of_bits_take n_bits_of_not add: take_map)
then show "NOT a = (of_bits \<circ> map Not \<circ> bits_of) a"
by simp
qed
show "of_bits bs AND of_bits cs = (of_bits (map2 (\<and>) bs cs) :: 'a word)"
if "length bs = length cs" for bs cs
using that apply transfer
apply (simp only: unsigned_of_bits_eq_of_bits)
apply (subst and_eq)
apply simp_all
done
show "of_bits bs OR of_bits cs = (of_bits (map2 (\<or>) bs cs) :: 'a word)"
if "length bs = length cs" for bs cs
using that apply transfer
apply (simp only: unsigned_of_bits_eq_of_bits)
apply (subst or_eq)
apply simp_all
done
show "of_bits bs XOR of_bits cs = (of_bits (map2 (\<noteq>) bs cs) :: 'a word)"
if "length bs = length cs" for bs cs
using that apply transfer
apply (simp only: unsigned_of_bits_eq_of_bits)
apply (subst xor_eq)
apply simp_all
done
show "a << n = of_bits (replicate n False @ bits_of a)"
for a :: "'a word" and n :: nat
by transfer (simp add: push_bit_take_bit)
show "a >> n = of_bits (drop n (bits_of a))"
if "n < length (bits_of a)"
for a :: "'a word" and n :: nat
using that by transfer simp
qed
subsection \<open>Bit structure on \<^typ>\<open>'a word\<close>\<close>
lemma word_bit_induct [case_names zero even odd]:
\<open>P a\<close> if word_zero: \<open>P 0\<close>
and word_even: \<open>\<And>a. P a \<Longrightarrow> 0 < a \<Longrightarrow> a < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> P (2 * a)\<close>
and word_odd: \<open>\<And>a. P a \<Longrightarrow> a < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> P (1 + 2 * a)\<close>
for P and a :: \<open>'a::len word\<close>
proof -
define m :: nat where \<open>m = LENGTH('a) - 1\<close>
then have l: \<open>LENGTH('a) = Suc m\<close>
by simp
define n :: nat where \<open>n = unsigned a\<close>
then have \<open>n < 2 ^ LENGTH('a)\<close>
by (simp add: unsigned_nat_less)
then have \<open>n < 2 * 2 ^ m\<close>
by (simp add: l)
then have \<open>P (of_nat n)\<close>
proof (induction n rule: nat_bit_induct)
case zero
show ?case
by simp (rule word_zero)
next
case (even n)
then have \<open>n < 2 ^ m\<close>
by simp
with even.IH have \<open>P (of_nat n)\<close>
by simp
moreover from \<open>n < 2 ^ m\<close> even.hyps have \<open>0 < (of_nat n :: 'a word)\<close>
by (auto simp add: word_greater_zero_iff of_nat_word_eq_0_iff l)
moreover from \<open>n < 2 ^ m\<close> have \<open>(of_nat n :: 'a word) < 2 ^ (LENGTH('a) - 1)\<close>
using of_nat_word_less_iff [where ?'a = 'a, of n \<open>2 ^ m\<close>]
by (cases \<open>m = 0\<close>) (simp_all add: not_less take_bit_eq_self ac_simps l)
ultimately have \<open>P (2 * of_nat n)\<close>
by (rule word_even)
then show ?case
by simp
next
case (odd n)
then have \<open>Suc n \<le> 2 ^ m\<close>
by simp
with odd.IH have \<open>P (of_nat n)\<close>
by simp
moreover from \<open>Suc n \<le> 2 ^ m\<close> have \<open>(of_nat n :: 'a word) < 2 ^ (LENGTH('a) - 1)\<close>
using of_nat_word_less_iff [where ?'a = 'a, of n \<open>2 ^ m\<close>]
by (cases \<open>m = 0\<close>) (simp_all add: not_less take_bit_eq_self ac_simps l)
ultimately have \<open>P (1 + 2 * of_nat n)\<close>
by (rule word_odd)
then show ?case
by simp
qed
then show ?thesis
by (simp add: n_def)
qed
end
global_interpretation bit_word: boolean_algebra "(AND)" "(OR)" NOT 0 "- 1 :: 'a::len word"
rewrites "bit_word.xor = ((XOR) :: 'a word \<Rightarrow> _)"
proof -
interpret bit_word: boolean_algebra "(AND)" "(OR)" NOT 0 "- 1 :: 'a word"
proof
show "a AND (b OR c) = a AND b OR a AND c"
for a b c :: "'a word"
by transfer (simp add: bit_int.conj_disj_distrib)
show "a OR b AND c = (a OR b) AND (a OR c)"
for a b c :: "'a word"
by transfer (simp add: bit_int.disj_conj_distrib)
show "a AND NOT a = 0" for a :: "'a word"
by transfer simp
show "a OR NOT a = - 1" for a :: "'a word"
by transfer simp
qed (transfer; simp)+
show "boolean_algebra (AND) (OR) NOT 0 (- 1 :: 'a word)"
by (fact bit_word.boolean_algebra_axioms)
show "bit_word.xor = ((XOR) :: 'a word \<Rightarrow> _)"
proof (rule ext)+
fix a b :: "'a word"
have "a XOR b = a AND NOT b OR NOT a AND b"
by transfer (simp add: bit_int.xor_def)
then show "bit_word.xor a b = a XOR b"
by (simp add: bit_word.xor_def)
qed
qed
end