(* Author: Florian Haftmann, TUM
*)
section \<open>Proof(s) of concept for algebraically founded lists of bits\<close>
theory Bit_Lists
imports
Main
"HOL-Library.Boolean_Algebra"
begin
context ab_group_add
begin
lemma minus_diff_commute:
"- b - a = - a - b"
by (simp only: diff_conv_add_uminus add.commute)
end
subsection \<open>Bit representations\<close>
subsubsection \<open>Mere syntactic bit representation\<close>
class bit_representation =
fixes bits_of :: "'a \<Rightarrow> bool list"
and of_bits :: "bool list \<Rightarrow> 'a"
assumes of_bits_of [simp]: "of_bits (bits_of a) = a"
subsubsection \<open>Algebraic bit representation\<close>
context comm_semiring_1
begin
primrec radix_value :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'a"
where "radix_value f b [] = 0"
| "radix_value f b (a # as) = f a + radix_value f b as * b"
abbreviation (input) unsigned_of_bits :: "bool list \<Rightarrow> 'a"
where "unsigned_of_bits \<equiv> radix_value of_bool 2"
lemma unsigned_of_bits_replicate_False [simp]:
"unsigned_of_bits (replicate n False) = 0"
by (induction n) simp_all
end
context unique_euclidean_semiring_with_nat
begin
lemma unsigned_of_bits_append [simp]:
"unsigned_of_bits (bs @ cs) = unsigned_of_bits bs
+ push_bit (length bs) (unsigned_of_bits cs)"
by (induction bs) (simp_all add: push_bit_double,
simp_all add: algebra_simps)
lemma unsigned_of_bits_take [simp]:
"unsigned_of_bits (take n bs) = take_bit n (unsigned_of_bits bs)"
proof (induction bs arbitrary: n)
case Nil
then show ?case
by simp
next
case (Cons b bs)
then show ?case
by (cases n) (simp_all add: ac_simps)
qed
lemma unsigned_of_bits_drop [simp]:
"unsigned_of_bits (drop n bs) = drop_bit n (unsigned_of_bits bs)"
proof (induction bs arbitrary: n)
case Nil
then show ?case
by simp
next
case (Cons b bs)
then show ?case
by (cases n) simp_all
qed
end
subsubsection \<open>Instances\<close>
text \<open>Unclear whether a \<^typ>\<open>bool\<close> instantiation is needed or not\<close>
instantiation nat :: bit_representation
begin
fun bits_of_nat :: "nat \<Rightarrow> bool list"
where "bits_of (n::nat) =
(if n = 0 then [] else odd n # bits_of (n div 2))"
lemma bits_of_nat_simps [simp]:
"bits_of (0::nat) = []"
"n > 0 \<Longrightarrow> bits_of n = odd n # bits_of (n div 2)" for n :: nat
by simp_all
declare bits_of_nat.simps [simp del]
definition of_bits_nat :: "bool list \<Rightarrow> nat"
where [simp]: "of_bits_nat = unsigned_of_bits"
\<comment> \<open>remove simp\<close>
instance proof
show "of_bits (bits_of n) = n" for n :: nat
by (induction n rule: nat_bit_induct) simp_all
qed
end
lemma bits_of_Suc_0 [simp]:
"bits_of (Suc 0) = [True]"
by simp
lemma bits_of_1_nat [simp]:
"bits_of (1 :: nat) = [True]"
by simp
lemma bits_of_nat_numeral_simps [simp]:
"bits_of (numeral Num.One :: nat) = [True]" (is ?One)
"bits_of (numeral (Num.Bit0 n) :: nat) = False # bits_of (numeral n :: nat)" (is ?Bit0)
"bits_of (numeral (Num.Bit1 n) :: nat) = True # bits_of (numeral n :: nat)" (is ?Bit1)
proof -
show ?One
by simp
define m :: nat where "m = numeral n"
then have "m > 0" and *: "numeral n = m" "numeral (Num.Bit0 n) = 2 * m" "numeral (Num.Bit1 n) = Suc (2 * m)"
by simp_all
from \<open>m > 0\<close> show ?Bit0 ?Bit1
by (simp_all add: *)
qed
lemma unsigned_of_bits_of_nat [simp]:
"unsigned_of_bits (bits_of n) = n" for n :: nat
using of_bits_of [of n] by simp
instantiation int :: bit_representation
begin
fun bits_of_int :: "int \<Rightarrow> bool list"
where "bits_of_int k = odd k #
(if k = 0 \<or> k = - 1 then [] else bits_of_int (k div 2))"
lemma bits_of_int_simps [simp]:
"bits_of (0 :: int) = [False]"
"bits_of (- 1 :: int) = [True]"
"k \<noteq> 0 \<Longrightarrow> k \<noteq> - 1 \<Longrightarrow> bits_of k = odd k # bits_of (k div 2)" for k :: int
by simp_all
lemma bits_of_not_Nil [simp]:
"bits_of k \<noteq> []" for k :: int
by simp
declare bits_of_int.simps [simp del]
definition of_bits_int :: "bool list \<Rightarrow> int"
where "of_bits_int bs = (if bs = [] \<or> \<not> last bs then unsigned_of_bits bs
else unsigned_of_bits bs - 2 ^ length bs)"
lemma of_bits_int_simps [simp]:
"of_bits [] = (0 :: int)"
"of_bits [False] = (0 :: int)"
"of_bits [True] = (- 1 :: int)"
"of_bits (bs @ [b]) = (unsigned_of_bits bs :: int) - (2 ^ length bs) * of_bool b"
"of_bits (False # bs) = 2 * (of_bits bs :: int)"
"bs \<noteq> [] \<Longrightarrow> of_bits (True # bs) = 1 + 2 * (of_bits bs :: int)"
by (simp_all add: of_bits_int_def push_bit_of_1)
instance proof
show "of_bits (bits_of k) = k" for k :: int
by (induction k rule: int_bit_induct) simp_all
qed
lemma bits_of_1_int [simp]:
"bits_of (1 :: int) = [True, False]"
by simp
lemma bits_of_int_numeral_simps [simp]:
"bits_of (numeral Num.One :: int) = [True, False]" (is ?One)
"bits_of (numeral (Num.Bit0 n) :: int) = False # bits_of (numeral n :: int)" (is ?Bit0)
"bits_of (numeral (Num.Bit1 n) :: int) = True # bits_of (numeral n :: int)" (is ?Bit1)
"bits_of (- numeral (Num.Bit0 n) :: int) = False # bits_of (- numeral n :: int)" (is ?nBit0)
"bits_of (- numeral (Num.Bit1 n) :: int) = True # bits_of (- numeral (Num.inc n) :: int)" (is ?nBit1)
proof -
show ?One
by simp
define k :: int where "k = numeral n"
then have "k > 0" and *: "numeral n = k" "numeral (Num.Bit0 n) = 2 * k" "numeral (Num.Bit1 n) = 2 * k + 1"
"numeral (Num.inc n) = k + 1"
by (simp_all add: add_One)
have "- (2 * k) div 2 = - k" "(- (2 * k) - 1) div 2 = - k - 1"
by simp_all
with \<open>k > 0\<close> show ?Bit0 ?Bit1 ?nBit0 ?nBit1
by (simp_all add: *)
qed
lemma of_bits_append [simp]:
"of_bits (bs @ cs) = of_bits bs + push_bit (length bs) (of_bits cs :: int)"
if "bs \<noteq> []" "\<not> last bs"
using that proof (induction bs rule: list_nonempty_induct)
case (single b)
then show ?case
by simp
next
case (cons b bs)
then show ?case
by (cases b) (simp_all add: push_bit_double)
qed
lemma of_bits_replicate_False [simp]:
"of_bits (replicate n False) = (0 :: int)"
by (auto simp add: of_bits_int_def)
lemma of_bits_drop [simp]:
"of_bits (drop n bs) = drop_bit n (of_bits bs :: int)"
if "n < length bs"
using that proof (induction bs arbitrary: n)
case Nil
then show ?case
by simp
next
case (Cons b bs)
show ?case
proof (cases n)
case 0
then show ?thesis
by simp
next
case (Suc n)
with Cons.prems have "bs \<noteq> []"
by auto
with Suc Cons.IH [of n] Cons.prems show ?thesis
by (cases b) simp_all
qed
qed
end
subsection \<open>Syntactic bit operations\<close>
class bit_operations = bit_representation +
fixes not :: "'a \<Rightarrow> 'a" ("NOT")
and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "AND" 64)
and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "OR" 59)
and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "XOR" 59)
and shift_left :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixl "<<" 55)
and shift_right :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixl ">>" 55)
assumes not_eq: "not = of_bits \<circ> map Not \<circ> bits_of"
and and_eq: "length bs = length cs \<Longrightarrow>
of_bits bs AND of_bits cs = of_bits (map2 (\<and>) bs cs)"
and semilattice_and: "semilattice (AND)"
and or_eq: "length bs = length cs \<Longrightarrow>
of_bits bs OR of_bits cs = of_bits (map2 (\<or>) bs cs)"
and semilattice_or: "semilattice (OR)"
and xor_eq: "length bs = length cs \<Longrightarrow>
of_bits bs XOR of_bits cs = of_bits (map2 (\<noteq>) bs cs)"
and abel_semigroup_xor: "abel_semigroup (XOR)"
and shift_right_eq: "a << n = of_bits (replicate n False @ bits_of a)"
and shift_left_eq: "n < length (bits_of a) \<Longrightarrow> a >> n = of_bits (drop n (bits_of a))"
begin
text \<open>
We want the bitwise operations to bind slightly weaker
than \<open>+\<close> and \<open>-\<close>, but \<open>~~\<close> to
bind slightly stronger than \<open>*\<close>.
\<close>
sublocale "and": semilattice "(AND)"
by (fact semilattice_and)
sublocale or: semilattice "(OR)"
by (fact semilattice_or)
sublocale xor: abel_semigroup "(XOR)"
by (fact abel_semigroup_xor)
end
subsubsection \<open>Instance \<^typ>\<open>nat\<close>\<close>
locale zip_nat = single: abel_semigroup f
for f :: "bool \<Rightarrow> bool \<Rightarrow> bool" (infixl "\<^bold>*" 70) +
assumes end_of_bits: "\<not> False \<^bold>* False"
begin
lemma False_P_imp:
"False \<^bold>* True \<and> P" if "False \<^bold>* P"
using that end_of_bits by (cases P) simp_all
function F :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "\<^bold>\<times>" 70)
where "m \<^bold>\<times> n = (if m = 0 \<and> n = 0 then 0
else of_bool (odd m \<^bold>* odd n) + (m div 2) \<^bold>\<times> (n div 2) * 2)"
by auto
termination
by (relation "measure (case_prod (+))") auto
lemma zero_left_eq:
"0 \<^bold>\<times> n = of_bool (False \<^bold>* True) * n"
by (induction n rule: nat_bit_induct) (simp_all add: end_of_bits)
lemma zero_right_eq:
"m \<^bold>\<times> 0 = of_bool (True \<^bold>* False) * m"
by (induction m rule: nat_bit_induct) (simp_all add: end_of_bits)
lemma simps [simp]:
"0 \<^bold>\<times> 0 = 0"
"0 \<^bold>\<times> n = of_bool (False \<^bold>* True) * n"
"m \<^bold>\<times> 0 = of_bool (True \<^bold>* False) * m"
"m > 0 \<Longrightarrow> n > 0 \<Longrightarrow> m \<^bold>\<times> n = of_bool (odd m \<^bold>* odd n) + (m div 2) \<^bold>\<times> (n div 2) * 2"
by (simp_all only: zero_left_eq zero_right_eq) simp
lemma rec:
"m \<^bold>\<times> n = of_bool (odd m \<^bold>* odd n) + (m div 2) \<^bold>\<times> (n div 2) * 2"
by (cases "m = 0 \<and> n = 0") (auto simp add: end_of_bits)
declare F.simps [simp del]
sublocale abel_semigroup F
proof
show "m \<^bold>\<times> n \<^bold>\<times> q = m \<^bold>\<times> (n \<^bold>\<times> q)" for m n q :: nat
proof (induction m arbitrary: n q rule: nat_bit_induct)
case zero
show ?case
by simp
next
case (even m)
with rec [of "2 * m"] rec [of _ q] show ?case
by (cases "even n") (auto dest: False_P_imp)
next
case (odd m)
with rec [of "Suc (2 * m)"] rec [of _ q] show ?case
by (cases "even n"; cases "even q")
(auto dest: False_P_imp simp add: ac_simps)
qed
show "m \<^bold>\<times> n = n \<^bold>\<times> m" for m n :: nat
proof (induction m arbitrary: n rule: nat_bit_induct)
case zero
show ?case
by (simp add: ac_simps)
next
case (even m)
with rec [of "2 * m" n] rec [of n "2 * m"] show ?case
by (simp add: ac_simps)
next
case (odd m)
with rec [of "Suc (2 * m)" n] rec [of n "Suc (2 * m)"] show ?case
by (simp add: ac_simps)
qed
qed
lemma self [simp]:
"n \<^bold>\<times> n = of_bool (True \<^bold>* True) * n"
by (induction n rule: nat_bit_induct) (simp_all add: end_of_bits)
lemma even_iff [simp]:
"even (m \<^bold>\<times> n) \<longleftrightarrow> \<not> (odd m \<^bold>* odd n)"
proof (induction m arbitrary: n rule: nat_bit_induct)
case zero
show ?case
by (cases "even n") (simp_all add: end_of_bits)
next
case (even m)
then show ?case
by (simp add: rec [of "2 * m"])
next
case (odd m)
then show ?case
by (simp add: rec [of "Suc (2 * m)"])
qed
lemma of_bits:
"of_bits bs \<^bold>\<times> of_bits cs = (of_bits (map2 (\<^bold>*) bs cs) :: nat)"
if "length bs = length cs" for bs cs
using that proof (induction bs cs rule: list_induct2)
case Nil
then show ?case
by simp
next
case (Cons b bs c cs)
then show ?case
by (cases "of_bits bs = (0::nat) \<or> of_bits cs = (0::nat)")
(auto simp add: ac_simps end_of_bits rec [of "Suc 0"])
qed
end
instantiation nat :: bit_operations
begin
definition not_nat :: "nat \<Rightarrow> nat"
where "not_nat = of_bits \<circ> map Not \<circ> bits_of"
global_interpretation and_nat: zip_nat "(\<and>)"
defines and_nat = and_nat.F
by standard auto
global_interpretation and_nat: semilattice "(AND) :: nat \<Rightarrow> nat \<Rightarrow> nat"
proof (rule semilattice.intro, fact and_nat.abel_semigroup_axioms, standard)
show "n AND n = n" for n :: nat
by (simp add: and_nat.self)
qed
declare and_nat.simps [simp] \<comment> \<open>inconsistent declaration handling by
\<open>global_interpretation\<close> in \<open>instantiation\<close>\<close>
lemma zero_nat_and_eq [simp]:
"0 AND n = 0" for n :: nat
by simp
lemma and_zero_nat_eq [simp]:
"n AND 0 = 0" for n :: nat
by simp
global_interpretation or_nat: zip_nat "(\<or>)"
defines or_nat = or_nat.F
by standard auto
global_interpretation or_nat: semilattice "(OR) :: nat \<Rightarrow> nat \<Rightarrow> nat"
proof (rule semilattice.intro, fact or_nat.abel_semigroup_axioms, standard)
show "n OR n = n" for n :: nat
by (simp add: or_nat.self)
qed
declare or_nat.simps [simp] \<comment> \<open>inconsistent declaration handling by
\<open>global_interpretation\<close> in \<open>instantiation\<close>\<close>
lemma zero_nat_or_eq [simp]:
"0 OR n = n" for n :: nat
by simp
lemma or_zero_nat_eq [simp]:
"n OR 0 = n" for n :: nat
by simp
global_interpretation xor_nat: zip_nat "(\<noteq>)"
defines xor_nat = xor_nat.F
by standard auto
declare xor_nat.simps [simp] \<comment> \<open>inconsistent declaration handling by
\<open>global_interpretation\<close> in \<open>instantiation\<close>\<close>
lemma zero_nat_xor_eq [simp]:
"0 XOR n = n" for n :: nat
by simp
lemma xor_zero_nat_eq [simp]:
"n XOR 0 = n" for n :: nat
by simp
definition shift_left_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
where [simp]: "m << n = push_bit n m" for m :: nat
definition shift_right_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
where [simp]: "m >> n = drop_bit n m" for m :: nat
instance proof
show "semilattice ((AND) :: nat \<Rightarrow> _)"
by (fact and_nat.semilattice_axioms)
show "semilattice ((OR):: nat \<Rightarrow> _)"
by (fact or_nat.semilattice_axioms)
show "abel_semigroup ((XOR):: nat \<Rightarrow> _)"
by (fact xor_nat.abel_semigroup_axioms)
show "(not :: nat \<Rightarrow> _) = of_bits \<circ> map Not \<circ> bits_of"
by (fact not_nat_def)
show "of_bits bs AND of_bits cs = (of_bits (map2 (\<and>) bs cs) :: nat)"
if "length bs = length cs" for bs cs
using that by (fact and_nat.of_bits)
show "of_bits bs OR of_bits cs = (of_bits (map2 (\<or>) bs cs) :: nat)"
if "length bs = length cs" for bs cs
using that by (fact or_nat.of_bits)
show "of_bits bs XOR of_bits cs = (of_bits (map2 (\<noteq>) bs cs) :: nat)"
if "length bs = length cs" for bs cs
using that by (fact xor_nat.of_bits)
show "m << n = of_bits (replicate n False @ bits_of m)"
for m n :: nat
by simp
show "m >> n = of_bits (drop n (bits_of m))"
for m n :: nat
by simp
qed
end
global_interpretation or_nat: semilattice_neutr "(OR)" "0 :: nat"
by standard simp
global_interpretation xor_nat: comm_monoid "(XOR)" "0 :: nat"
by standard simp
lemma not_nat_simps [simp]:
"NOT 0 = (0::nat)"
"n > 0 \<Longrightarrow> NOT n = of_bool (even n) + 2 * NOT (n div 2)" for n :: nat
by (simp_all add: not_eq)
lemma not_1_nat [simp]:
"NOT 1 = (0::nat)"
by simp
lemma not_Suc_0 [simp]:
"NOT (Suc 0) = (0::nat)"
by simp
lemma Suc_0_and_eq [simp]:
"Suc 0 AND n = n mod 2"
by (cases n) auto
lemma and_Suc_0_eq [simp]:
"n AND Suc 0 = n mod 2"
using Suc_0_and_eq [of n] by (simp add: ac_simps)
lemma Suc_0_or_eq [simp]:
"Suc 0 OR n = n + of_bool (even n)"
by (cases n) (simp_all add: ac_simps)
lemma or_Suc_0_eq [simp]:
"n OR Suc 0 = n + of_bool (even n)"
using Suc_0_or_eq [of n] by (simp add: ac_simps)
lemma Suc_0_xor_eq [simp]:
"Suc 0 XOR n = n + of_bool (even n) - of_bool (odd n)"
by (cases n) (simp_all add: ac_simps)
lemma xor_Suc_0_eq [simp]:
"n XOR Suc 0 = n + of_bool (even n) - of_bool (odd n)"
using Suc_0_xor_eq [of n] by (simp add: ac_simps)
subsubsection \<open>Instance \<^typ>\<open>int\<close>\<close>
abbreviation (input) complement :: "int \<Rightarrow> int"
where "complement k \<equiv> - k - 1"
lemma complement_half:
"complement (k * 2) div 2 = complement k"
by simp
lemma complement_div_2:
"complement (k div 2) = complement k div 2"
by linarith
locale zip_int = single: abel_semigroup f
for f :: "bool \<Rightarrow> bool \<Rightarrow> bool" (infixl "\<^bold>*" 70)
begin
lemma False_False_imp_True_True:
"True \<^bold>* True" if "False \<^bold>* False"
proof (rule ccontr)
assume "\<not> True \<^bold>* True"
with that show False
using single.assoc [of False True True]
by (cases "False \<^bold>* True") simp_all
qed
function F :: "int \<Rightarrow> int \<Rightarrow> int" (infixl "\<^bold>\<times>" 70)
where "k \<^bold>\<times> l = (if k \<in> {0, - 1} \<and> l \<in> {0, - 1}
then - of_bool (odd k \<^bold>* odd l)
else of_bool (odd k \<^bold>* odd l) + (k div 2) \<^bold>\<times> (l div 2) * 2)"
by auto
termination
by (relation "measure (\<lambda>(k, l). nat (\<bar>k\<bar> + \<bar>l\<bar>))") auto
lemma zero_left_eq:
"0 \<^bold>\<times> l = (case (False \<^bold>* False, False \<^bold>* True)
of (False, False) \<Rightarrow> 0
| (False, True) \<Rightarrow> l
| (True, False) \<Rightarrow> complement l
| (True, True) \<Rightarrow> - 1)"
by (induction l rule: int_bit_induct)
(simp_all split: bool.split)
lemma minus_left_eq:
"- 1 \<^bold>\<times> l = (case (True \<^bold>* False, True \<^bold>* True)
of (False, False) \<Rightarrow> 0
| (False, True) \<Rightarrow> l
| (True, False) \<Rightarrow> complement l
| (True, True) \<Rightarrow> - 1)"
by (induction l rule: int_bit_induct)
(simp_all split: bool.split)
lemma zero_right_eq:
"k \<^bold>\<times> 0 = (case (False \<^bold>* False, False \<^bold>* True)
of (False, False) \<Rightarrow> 0
| (False, True) \<Rightarrow> k
| (True, False) \<Rightarrow> complement k
| (True, True) \<Rightarrow> - 1)"
by (induction k rule: int_bit_induct)
(simp_all add: ac_simps split: bool.split)
lemma minus_right_eq:
"k \<^bold>\<times> - 1 = (case (True \<^bold>* False, True \<^bold>* True)
of (False, False) \<Rightarrow> 0
| (False, True) \<Rightarrow> k
| (True, False) \<Rightarrow> complement k
| (True, True) \<Rightarrow> - 1)"
by (induction k rule: int_bit_induct)
(simp_all add: ac_simps split: bool.split)
lemma simps [simp]:
"0 \<^bold>\<times> 0 = - of_bool (False \<^bold>* False)"
"- 1 \<^bold>\<times> 0 = - of_bool (True \<^bold>* False)"
"0 \<^bold>\<times> - 1 = - of_bool (False \<^bold>* True)"
"- 1 \<^bold>\<times> - 1 = - of_bool (True \<^bold>* True)"
"0 \<^bold>\<times> l = (case (False \<^bold>* False, False \<^bold>* True)
of (False, False) \<Rightarrow> 0
| (False, True) \<Rightarrow> l
| (True, False) \<Rightarrow> complement l
| (True, True) \<Rightarrow> - 1)"
"- 1 \<^bold>\<times> l = (case (True \<^bold>* False, True \<^bold>* True)
of (False, False) \<Rightarrow> 0
| (False, True) \<Rightarrow> l
| (True, False) \<Rightarrow> complement l
| (True, True) \<Rightarrow> - 1)"
"k \<^bold>\<times> 0 = (case (False \<^bold>* False, False \<^bold>* True)
of (False, False) \<Rightarrow> 0
| (False, True) \<Rightarrow> k
| (True, False) \<Rightarrow> complement k
| (True, True) \<Rightarrow> - 1)"
"k \<^bold>\<times> - 1 = (case (True \<^bold>* False, True \<^bold>* True)
of (False, False) \<Rightarrow> 0
| (False, True) \<Rightarrow> k
| (True, False) \<Rightarrow> complement k
| (True, True) \<Rightarrow> - 1)"
"k \<noteq> 0 \<Longrightarrow> k \<noteq> - 1 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> l \<noteq> - 1
\<Longrightarrow> k \<^bold>\<times> l = of_bool (odd k \<^bold>* odd l) + (k div 2) \<^bold>\<times> (l div 2) * 2"
by simp_all[4] (simp_all only: zero_left_eq minus_left_eq zero_right_eq minus_right_eq, simp)
declare F.simps [simp del]
lemma rec:
"k \<^bold>\<times> l = of_bool (odd k \<^bold>* odd l) + (k div 2) \<^bold>\<times> (l div 2) * 2"
by (cases "k \<in> {0, - 1} \<and> l \<in> {0, - 1}") (auto simp add: ac_simps F.simps [of k l] split: bool.split)
sublocale abel_semigroup F
proof
show "k \<^bold>\<times> l \<^bold>\<times> r = k \<^bold>\<times> (l \<^bold>\<times> r)" for k l r :: int
proof (induction k arbitrary: l r rule: int_bit_induct)
case zero
have "complement l \<^bold>\<times> r = complement (l \<^bold>\<times> r)" if "False \<^bold>* False" "\<not> False \<^bold>* True"
proof (induction l arbitrary: r rule: int_bit_induct)
case zero
from that show ?case
by (auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
next
case minus
from that show ?case
by (auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
next
case (even l)
with that rec [of _ r] show ?case
by (cases "even r")
(auto simp add: complement_half ac_simps False_False_imp_True_True split: bool.splits)
next
case (odd l)
moreover have "- l - 1 = - 1 - l"
by simp
ultimately show ?case
using that rec [of _ r]
by (cases "even r")
(auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
qed
then show ?case
by (auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
next
case minus
have "complement l \<^bold>\<times> r = complement (l \<^bold>\<times> r)" if "\<not> True \<^bold>* True" "False \<^bold>* True"
proof (induction l arbitrary: r rule: int_bit_induct)
case zero
from that show ?case
by (auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
next
case minus
from that show ?case
by (auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
next
case (even l)
with that rec [of _ r] show ?case
by (cases "even r")
(auto simp add: complement_half ac_simps False_False_imp_True_True split: bool.splits)
next
case (odd l)
moreover have "- l - 1 = - 1 - l"
by simp
ultimately show ?case
using that rec [of _ r]
by (cases "even r")
(auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
qed
then show ?case
by (auto simp add: ac_simps False_False_imp_True_True split: bool.splits)
next
case (even k)
with rec [of "k * 2"] rec [of _ r] show ?case
by (cases "even r"; cases "even l") (auto simp add: ac_simps False_False_imp_True_True)
next
case (odd k)
with rec [of "1 + k * 2"] rec [of _ r] show ?case
by (cases "even r"; cases "even l") (auto simp add: ac_simps False_False_imp_True_True)
qed
show "k \<^bold>\<times> l = l \<^bold>\<times> k" for k l :: int
proof (induction k arbitrary: l rule: int_bit_induct)
case zero
show ?case
by simp
next
case minus
show ?case
by simp
next
case (even k)
with rec [of "k * 2" l] rec [of l "k * 2"] show ?case
by (simp add: ac_simps)
next
case (odd k)
with rec [of "k * 2 + 1" l] rec [of l "k * 2 + 1"] show ?case
by (simp add: ac_simps)
qed
qed
lemma self [simp]:
"k \<^bold>\<times> k = (case (False \<^bold>* False, True \<^bold>* True)
of (False, False) \<Rightarrow> 0
| (False, True) \<Rightarrow> k
| (True, True) \<Rightarrow> - 1)"
by (induction k rule: int_bit_induct) (auto simp add: False_False_imp_True_True split: bool.split)
lemma even_iff [simp]:
"even (k \<^bold>\<times> l) \<longleftrightarrow> \<not> (odd k \<^bold>* odd l)"
proof (induction k arbitrary: l rule: int_bit_induct)
case zero
show ?case
by (cases "even l") (simp_all split: bool.splits)
next
case minus
show ?case
by (cases "even l") (simp_all split: bool.splits)
next
case (even k)
then show ?case
by (simp add: rec [of "k * 2"])
next
case (odd k)
then show ?case
by (simp add: rec [of "1 + k * 2"])
qed
lemma of_bits:
"of_bits bs \<^bold>\<times> of_bits cs = (of_bits (map2 (\<^bold>*) bs cs) :: int)"
if "length bs = length cs" and "\<not> False \<^bold>* False" for bs cs
using \<open>length bs = length cs\<close> proof (induction bs cs rule: list_induct2)
case Nil
then show ?case
using \<open>\<not> False \<^bold>* False\<close> by (auto simp add: False_False_imp_True_True split: bool.splits)
next
case (Cons b bs c cs)
show ?case
proof (cases "bs = []")
case True
with Cons show ?thesis
using \<open>\<not> False \<^bold>* False\<close> by (cases b; cases c)
(auto simp add: ac_simps split: bool.splits)
next
case False
with Cons.hyps have "cs \<noteq> []"
by auto
with \<open>bs \<noteq> []\<close> have "map2 (\<^bold>*) bs cs \<noteq> []"
by (simp add: zip_eq_Nil_iff)
from \<open>bs \<noteq> []\<close> \<open>cs \<noteq> []\<close> \<open>map2 (\<^bold>*) bs cs \<noteq> []\<close> Cons show ?thesis
by (cases b; cases c)
(auto simp add: \<open>\<not> False \<^bold>* False\<close> ac_simps
rec [of "of_bits bs * 2"] rec [of "of_bits cs * 2"]
rec [of "1 + of_bits bs * 2"])
qed
qed
end
instantiation int :: bit_operations
begin
definition not_int :: "int \<Rightarrow> int"
where "not_int = complement"
global_interpretation and_int: zip_int "(\<and>)"
defines and_int = and_int.F
by standard
declare and_int.simps [simp] \<comment> \<open>inconsistent declaration handling by
\<open>global_interpretation\<close> in \<open>instantiation\<close>\<close>
global_interpretation and_int: semilattice "(AND) :: int \<Rightarrow> int \<Rightarrow> int"
proof (rule semilattice.intro, fact and_int.abel_semigroup_axioms, standard)
show "k AND k = k" for k :: int
by (simp add: and_int.self)
qed
lemma zero_int_and_eq [simp]:
"0 AND k = 0" for k :: int
by simp
lemma and_zero_int_eq [simp]:
"k AND 0 = 0" for k :: int
by simp
lemma minus_int_and_eq [simp]:
"- 1 AND k = k" for k :: int
by simp
lemma and_minus_int_eq [simp]:
"k AND - 1 = k" for k :: int
by simp
global_interpretation or_int: zip_int "(\<or>)"
defines or_int = or_int.F
by standard
declare or_int.simps [simp] \<comment> \<open>inconsistent declaration handling by
\<open>global_interpretation\<close> in \<open>instantiation\<close>\<close>
global_interpretation or_int: semilattice "(OR) :: int \<Rightarrow> int \<Rightarrow> int"
proof (rule semilattice.intro, fact or_int.abel_semigroup_axioms, standard)
show "k OR k = k" for k :: int
by (simp add: or_int.self)
qed
lemma zero_int_or_eq [simp]:
"0 OR k = k" for k :: int
by simp
lemma and_zero_or_eq [simp]:
"k OR 0 = k" for k :: int
by simp
lemma minus_int_or_eq [simp]:
"- 1 OR k = - 1" for k :: int
by simp
lemma or_minus_int_eq [simp]:
"k OR - 1 = - 1" for k :: int
by simp
global_interpretation xor_int: zip_int "(\<noteq>)"
defines xor_int = xor_int.F
by standard
declare xor_int.simps [simp] \<comment> \<open>inconsistent declaration handling by
\<open>global_interpretation\<close> in \<open>instantiation\<close>\<close>
lemma zero_int_xor_eq [simp]:
"0 XOR k = k" for k :: int
by simp
lemma and_zero_xor_eq [simp]:
"k XOR 0 = k" for k :: int
by simp
lemma minus_int_xor_eq [simp]:
"- 1 XOR k = complement k" for k :: int
by simp
lemma xor_minus_int_eq [simp]:
"k XOR - 1 = complement k" for k :: int
by simp
definition shift_left_int :: "int \<Rightarrow> nat \<Rightarrow> int"
where [simp]: "k << n = push_bit n k" for k :: int
definition shift_right_int :: "int \<Rightarrow> nat \<Rightarrow> int"
where [simp]: "k >> n = drop_bit n k" for k :: int
instance proof
show "semilattice ((AND) :: int \<Rightarrow> _)"
by (fact and_int.semilattice_axioms)
show "semilattice ((OR) :: int \<Rightarrow> _)"
by (fact or_int.semilattice_axioms)
show "abel_semigroup ((XOR) :: int \<Rightarrow> _)"
by (fact xor_int.abel_semigroup_axioms)
show "(not :: int \<Rightarrow> _) = of_bits \<circ> map Not \<circ> bits_of"
proof (rule sym, rule ext)
fix k :: int
show "(of_bits \<circ> map Not \<circ> bits_of) k = NOT k"
by (induction k rule: int_bit_induct) (simp_all add: not_int_def)
qed
show "of_bits bs AND of_bits cs = (of_bits (map2 (\<and>) bs cs) :: int)"
if "length bs = length cs" for bs cs
using that by (rule and_int.of_bits) simp
show "of_bits bs OR of_bits cs = (of_bits (map2 (\<or>) bs cs) :: int)"
if "length bs = length cs" for bs cs
using that by (rule or_int.of_bits) simp
show "of_bits bs XOR of_bits cs = (of_bits (map2 (\<noteq>) bs cs) :: int)"
if "length bs = length cs" for bs cs
using that by (rule xor_int.of_bits) simp
show "k << n = of_bits (replicate n False @ bits_of k)"
for k :: int and n :: nat
by (cases "n = 0") simp_all
show "k >> n = of_bits (drop n (bits_of k))"
if "n < length (bits_of k)"
for k :: int and n :: nat
using that by simp
qed
end
global_interpretation and_int: semilattice_neutr "(AND)" "- 1 :: int"
by standard simp
global_interpretation or_int: semilattice_neutr "(OR)" "0 :: int"
by standard simp
global_interpretation xor_int: comm_monoid "(XOR)" "0 :: int"
by standard simp
lemma not_int_simps [simp]:
"NOT 0 = (- 1 :: int)"
"NOT (- 1) = (0 :: int)"
"k \<noteq> 0 \<Longrightarrow> k \<noteq> - 1 \<Longrightarrow> NOT k = of_bool (even k) + 2 * NOT (k div 2)" for k :: int
by (auto simp add: not_int_def elim: oddE)
lemma not_one_int [simp]:
"NOT 1 = (- 2 :: int)"
by simp
lemma even_not_iff [simp]:
"even (NOT k) \<longleftrightarrow> odd k"
for k :: int
by (simp add: not_int_def)
lemma not_div_2:
"NOT k div 2 = NOT (k div 2)"
for k :: int
by (simp add: complement_div_2 not_int_def)
lemma one_and_int_eq [simp]:
"1 AND k = k mod 2" for k :: int
using and_int.rec [of 1] by (simp add: mod2_eq_if)
lemma and_one_int_eq [simp]:
"k AND 1 = k mod 2" for k :: int
using one_and_int_eq [of 1] by (simp add: ac_simps)
lemma one_or_int_eq [simp]:
"1 OR k = k + of_bool (even k)" for k :: int
using or_int.rec [of 1] by (auto elim: oddE)
lemma or_one_int_eq [simp]:
"k OR 1 = k + of_bool (even k)" for k :: int
using one_or_int_eq [of k] by (simp add: ac_simps)
lemma one_xor_int_eq [simp]:
"1 XOR k = k + of_bool (even k) - of_bool (odd k)" for k :: int
using xor_int.rec [of 1] by (auto elim: oddE)
lemma xor_one_int_eq [simp]:
"k XOR 1 = k + of_bool (even k) - of_bool (odd k)" for k :: int
using one_xor_int_eq [of k] by (simp add: ac_simps)
global_interpretation bit_int: boolean_algebra "(AND)" "(OR)" NOT 0 "- 1 :: int"
rewrites "bit_int.xor = ((XOR) :: int \<Rightarrow> _)"
proof -
interpret bit_int: boolean_algebra "(AND)" "(OR)" NOT 0 "- 1 :: int"
proof
show "k AND (l OR r) = k AND l OR k AND r"
for k l r :: int
proof (induction k arbitrary: l r rule: int_bit_induct)
case zero
show ?case
by simp
next
case minus
show ?case
by simp
next
case (even k)
then show ?case by (simp add: and_int.rec [of "k * 2"]
or_int.rec [of "(k AND l div 2) * 2"] or_int.rec [of l])
next
case (odd k)
then show ?case by (simp add: and_int.rec [of "1 + k * 2"]
or_int.rec [of "(k AND l div 2) * 2"] or_int.rec [of "1 + (k AND l div 2) * 2"] or_int.rec [of l])
qed
show "k OR l AND r = (k OR l) AND (k OR r)"
for k l r :: int
proof (induction k arbitrary: l r rule: int_bit_induct)
case zero
then show ?case
by simp
next
case minus
then show ?case
by simp
next
case (even k)
then show ?case by (simp add: or_int.rec [of "k * 2"]
and_int.rec [of "(k OR l div 2) * 2"] and_int.rec [of "1 + (k OR l div 2) * 2"] and_int.rec [of l])
next
case (odd k)
then show ?case by (simp add: or_int.rec [of "1 + k * 2"]
and_int.rec [of "1 + (k OR l div 2) * 2"] and_int.rec [of l])
qed
show "k AND NOT k = 0" for k :: int
by (induction k rule: int_bit_induct)
(simp_all add: not_int_def complement_half minus_diff_commute [of 1])
show "k OR NOT k = - 1" for k :: int
by (induction k rule: int_bit_induct)
(simp_all add: not_int_def complement_half minus_diff_commute [of 1])
qed simp_all
show "boolean_algebra (AND) (OR) NOT 0 (- 1 :: int)"
by (fact bit_int.boolean_algebra_axioms)
show "bit_int.xor = ((XOR) :: int \<Rightarrow> _)"
proof (rule ext)+
fix k l :: int
have "k XOR l = k AND NOT l OR NOT k AND l"
proof (induction k arbitrary: l rule: int_bit_induct)
case zero
show ?case
by simp
next
case minus
show ?case
by (simp add: not_int_def)
next
case (even k)
then show ?case
by (simp add: xor_int.rec [of "k * 2"] and_int.rec [of "k * 2"]
or_int.rec [of _ "1 + 2 * NOT k AND l"] not_div_2)
(simp add: and_int.rec [of _ l])
next
case (odd k)
then show ?case
by (simp add: xor_int.rec [of "1 + k * 2"] and_int.rec [of "1 + k * 2"]
or_int.rec [of _ "2 * NOT k AND l"] not_div_2)
(simp add: and_int.rec [of _ l])
qed
then show "bit_int.xor k l = k XOR l"
by (simp add: bit_int.xor_def)
qed
qed
end